Measures Of Central Tendency And Dispersion

1. Quartiles are the values dividing a given set of observations into

Two equal parts

Four equal parts

Five equal parts

None of these

Quartiles divide a given set of observations into four equal parts. This means that the data is divided into four groups, each containing an equal number of observations. The first quartile (Q1) represents the lower 25% of the data, the second quartile (Q2) represents the median or middle value, and the third quartile (Q3) represents the upper 25% of the data. The fourth quartile would represent the highest 25% of the data. Therefore, the correct answer is "Four equal parts."

Explanation

Quartiles divide a given set of observations into four equal parts. This means that the data is divided into four groups, each containing an equal number of observations. The first quartile (Q1) represents the lower 25% of the data, the second quartile (Q2) represents the median or middle value, and the third quartile (Q3) represents the upper 25% of the data. The fourth quartile would represent the highest 25% of the data. Therefore, the correct answer is "Four equal parts."

2. If there are two groups with 75 and 65 as harmonic means and containing 15 and 13 observation then the combined HM is given by

65

70.36

70

71

The harmonic mean is calculated by dividing the number of observations by the sum of the reciprocals of the observations. In this case, we have two groups with 15 and 13 observations and harmonic means of 75 and 65 respectively. To find the combined harmonic mean, we need to calculate the sum of the reciprocals of the observations in each group and divide the total number of observations by this sum. The combined harmonic mean is therefore 70.36.

Explanation

The harmonic mean is calculated by dividing the number of observations by the sum of the reciprocals of the observations. In this case, we have two groups with 15 and 13 observations and harmonic means of 75 and 65 respectively. To find the combined harmonic mean, we need to calculate the sum of the reciprocals of the observations in each group and divide the total number of observations by this sum. The combined harmonic mean is therefore 70.36.

3. The range of 15, 12, 10, 9, 17, 20 is

5

12

13

11

The range of a set of numbers is calculated by subtracting the smallest number from the largest number. In this case, the smallest number is 9 and the largest number is 20. Therefore, the range is 20 - 9 = 11.

Explanation

The range of a set of numbers is calculated by subtracting the smallest number from the largest number. In this case, the smallest number is 9 and the largest number is 20. Therefore, the range is 20 - 9 = 11.

4. In case of an even number of observations which of the following is median ?

Any of the two middle-most value

The simple average of these two middle values

The weighted average of these two middle values

Any of these

In case of an even number of observations, the median is the simple average of the two middle values. This is because the median represents the middle value of a dataset, and when there is an even number of observations, there are two middle values. Taking the simple average of these two values ensures that the median falls exactly between them, providing a balanced representation of the central tendency of the data.

Explanation

In case of an even number of observations, the median is the simple average of the two middle values. This is because the median represents the middle value of a dataset, and when there is an even number of observations, there are two middle values. Taking the simple average of these two values ensures that the median falls exactly between them, providing a balanced representation of the central tendency of the data.

5. If x and y are related by x-y-10 = 0 and mode of x is known to be 23, then the mode of y is

20

13

3

23

The equation x-y-10=0 can be rearranged to y=x-10. Since the mode of x is known to be 23, it means that 23 occurs most frequently in the dataset. Therefore, the mode of y can be found by substituting 23 into the equation for x, giving y=23-10=13. Hence, the mode of y is 13.

Explanation

The equation x-y-10=0 can be rearranged to y=x-10. Since the mode of x is known to be 23, it means that 23 occurs most frequently in the dataset. Therefore, the mode of y can be found by substituting 23 into the equation for x, giving y=23-10=13. Hence, the mode of y is 13.

6. Measures of central tendency for a given set of observations measures

The scatterness of the observations

The central location of the observations

Both (i) and (ii)

None of these.

The measures of central tendency, such as mean, median, and mode, provide information about the central location of the observations in a given set. They help to identify the typical or average value of the data. Therefore, the correct answer is "The central location of the observations."

Explanation

The measures of central tendency, such as mean, median, and mode, provide information about the central location of the observations in a given set. They help to identify the typical or average value of the data. Therefore, the correct answer is "The central location of the observations."

7. What is the median for the following observations? 5, 8, 6, 9, 11, 4.

6

7

8

None of these

The median is the middle value of a set of numbers when they are arranged in order. In this case, the numbers are already in order: 4, 5, 6, 8, 9, 11. The middle value is 6 because it is the third number in the set. Therefore, the median for the given observations is 6.

Explanation

The median is the middle value of a set of numbers when they are arranged in order. In this case, the numbers are already in order: 4, 5, 6, 8, 9, 11. The middle value is 6 because it is the third number in the set. Therefore, the median for the given observations is 6.

8. What is the modal value for the numbers 5, 8, 6, 4, 10, 15, 18, 10?

18

10

14

None of these

The modal value is the value that appears most frequently in a set of numbers. In this case, the number 10 appears twice, which is more than any other number in the set. Therefore, the modal value for the given numbers is 10.

Explanation

The modal value is the value that appears most frequently in a set of numbers. In this case, the number 10 appears twice, which is more than any other number in the set. Therefore, the modal value for the given numbers is 10.

9. Which of he following measures of central tendency is based on only fifty percent of the central values?

Mean

Median

Mode

Both (i) and(ii)

The median is the measure of central tendency that is based on only fifty percent of the central values. It is the middle value in a set of data when the data is arranged in ascending or descending order. Unlike the mean, which takes into account all the values in the data set, the median only considers the middle value(s). Therefore, it is not influenced by extreme values or outliers, making it a useful measure in skewed distributions. The mode, on the other hand, represents the most frequently occurring value(s) in the data set and is not based on fifty percent of the central values.

Explanation

The median is the measure of central tendency that is based on only fifty percent of the central values. It is the middle value in a set of data when the data is arranged in ascending or descending order. Unlike the mean, which takes into account all the values in the data set, the median only considers the middle value(s). Therefore, it is not influenced by extreme values or outliers, making it a useful measure in skewed distributions. The mode, on the other hand, represents the most frequently occurring value(s) in the data set and is not based on fifty percent of the central values.

10. Which measure of dispersion is the quickest to compute?

Standard deviation

Quartile deviation

Mean deviation

Range

The range is the simplest measure of dispersion to compute because it only requires finding the difference between the maximum and minimum values in a dataset. It does not involve any complex calculations or require the use of all the data points. Therefore, it can be quickly calculated even with large datasets, making it the quickest measure of dispersion to compute.

Explanation

The range is the simplest measure of dispersion to compute because it only requires finding the difference between the maximum and minimum values in a dataset. It does not involve any complex calculations or require the use of all the data points. Therefore, it can be quickly calculated even with large datasets, making it the quickest measure of dispersion to compute.

11. If GM of x is 10 and GM of y is 15, then the GM of xy is

150

Log 10 x Log 15

Log 150

None of these.

The geometric mean (GM) of two numbers is the square root of their product. In this question, the GM of x is 10 and the GM of y is 15. To find the GM of xy, we need to find the square root of the product of x and y. Since the GM of x is 10 and the GM of y is 15, we can multiply these two values to get the product of xy, which is 150. Therefore, the GM of xy is 150.

Explanation

The geometric mean (GM) of two numbers is the square root of their product. In this question, the GM of x is 10 and the GM of y is 15. To find the GM of xy, we need to find the square root of the product of x and y. Since the GM of x is 10 and the GM of y is 15, we can multiply these two values to get the product of xy, which is 150. Therefore, the GM of xy is 150.

12. If the AM and GM for 10 observations are both 15, then the value of HM is

Less than 15

More than 15

15

Can not be determined

If the arithmetic mean (AM) and geometric mean (GM) of 10 observations are both 15, it means that the sum of the 10 observations is 150 and their product is also 150. The harmonic mean (HM) is the reciprocal of the arithmetic mean of the reciprocals of the observations. Since the sum of the reciprocals of the observations is equal to the reciprocal of their product, which is 1/150, the arithmetic mean of these reciprocals is 1/150 divided by 10, which is 1/1500. Taking the reciprocal of this value gives us the harmonic mean, which is 1500. Therefore, the value of HM is 15.

Explanation

If the arithmetic mean (AM) and geometric mean (GM) of 10 observations are both 15, it means that the sum of the 10 observations is 150 and their product is also 150. The harmonic mean (HM) is the reciprocal of the arithmetic mean of the reciprocals of the observations. Since the sum of the reciprocals of the observations is equal to the reciprocal of their product, which is 1/150, the arithmetic mean of these reciprocals is 1/150 divided by 10, which is 1/1500. Taking the reciprocal of this value gives us the harmonic mean, which is 1500. Therefore, the value of HM is 15.

13. Which of the following measure(s) possesses (possess) mathematical properties?

AM

GM

HM

All of these

All of the measures mentioned (AM, GM, HM) possess mathematical properties. AM (Arithmetic Mean) is the sum of a set of numbers divided by the count of those numbers. GM (Geometric Mean) is the nth root of the product of n numbers. HM (Harmonic Mean) is the reciprocal of the arithmetic mean of the reciprocals of a set of numbers. These measures are all mathematical calculations that can be used to analyze and describe data.

Explanation

All of the measures mentioned (AM, GM, HM) possess mathematical properties. AM (Arithmetic Mean) is the sum of a set of numbers divided by the count of those numbers. GM (Geometric Mean) is the nth root of the product of n numbers. HM (Harmonic Mean) is the reciprocal of the arithmetic mean of the reciprocals of a set of numbers. These measures are all mathematical calculations that can be used to analyze and describe data.

14. For any two numbers SD is always

Twice the range

Half of the range

Square of the range

None of these

The correct answer is "Half of the range." This means that for any two numbers, the standard deviation (SD) will always be equal to half of the range between those numbers. The range is the difference between the maximum and minimum values in a set of numbers. Therefore, the standard deviation will be half of this difference.

Explanation

The correct answer is "Half of the range." This means that for any two numbers, the standard deviation (SD) will always be equal to half of the range between those numbers. The range is the difference between the maximum and minimum values in a set of numbers. Therefore, the standard deviation will be half of this difference.

15. Which of the following measure of the central tendency is difficult to compute?

Mean

Median

Mode

Gm

The geometric mean (Gm) is difficult to compute compared to the other measures of central tendency (mean, median, and mode). This is because it involves taking the nth root of the product of n numbers, which can be complex and time-consuming. Additionally, the geometric mean is sensitive to extreme values in the data set, making it less robust and potentially less representative of the central tendency.

Explanation

The geometric mean (Gm) is difficult to compute compared to the other measures of central tendency (mean, median, and mode). This is because it involves taking the nth root of the product of n numbers, which can be complex and time-consuming. Additionally, the geometric mean is sensitive to extreme values in the data set, making it less robust and potentially less representative of the central tendency.

16. Two variables x and y are given by y= 2x - 3. If the median of x is 20, what is the median of y?

20

40

37

35

The equation given, y = 2x - 3, shows a linear relationship between x and y. To find the median of y, we need to determine the value of y when x is at its median, which is 20. Substituting x = 20 into the equation, we get y = 2(20) - 3 = 37. Therefore, the median of y is 37.

Explanation

The equation given, y = 2x - 3, shows a linear relationship between x and y. To find the median of y, we need to determine the value of y when x is at its median, which is 20. Substituting x = 20 into the equation, we get y = 2(20) - 3 = 37. Therefore, the median of y is 37.

17. If the relationship between two variables u and v are given by 2u + v + 7 = 0 and if the AM of u is 10, then the AM of v is

17

-17

-27

27

The given equation is 2u + v + 7 = 0. To find the arithmetic mean (AM) of v, we need to solve for v. Rearranging the equation, we get v = -2u - 7. Since the AM of u is given as 10, we substitute u = 10 into the equation for v. Therefore, v = -2(10) - 7 = -27. Hence, the AM of v is -27.

Explanation

The given equation is 2u + v + 7 = 0. To find the arithmetic mean (AM) of v, we need to solve for v. Rearranging the equation, we get v = -2u - 7. Since the AM of u is given as 10, we substitute u = 10 into the equation for v. Therefore, v = -2(10) - 7 = -27. Hence, the AM of v is -27.

18. If the profitsof a company remains the same for the last ten months,then the standard deviation of profits for these ten months would be ?

Positive

Negative

Zero

(a) or (c)

If the profits of a company remain the same for the last ten months, it means that there is no variation or change in the profits. In other words, all the profit values are identical. When calculating the standard deviation, it measures the dispersion or variability of a set of values from the mean. However, if all the values are the same, the deviation from the mean is zero for each value, resulting in a standard deviation of zero. Therefore, the correct answer is zero.

Explanation

If the profits of a company remain the same for the last ten months, it means that there is no variation or change in the profits. In other words, all the profit values are identical. When calculating the standard deviation, it measures the dispersion or variability of a set of values from the mean. However, if all the values are the same, the deviation from the mean is zero for each value, resulting in a standard deviation of zero. Therefore, the correct answer is zero.

19. The mean and SD of a sample of 100 observations were calculated as 40 and 5.1 respectively by a CA student who took one observation as 50 instead of 40 by mistake. The current value of SD would be

4.90

5.00

4.88

4.85

When calculating the standard deviation, each observation is subtracted from the mean, squared, and then the squared differences are averaged. Since one observation was mistakenly recorded as 50 instead of 40, this will affect the calculation of the standard deviation. The correct value for this observation is 40, so the squared difference between the correct value and the mean will be smaller than if it were 50. As a result, the overall sum of squared differences will be smaller, leading to a slightly smaller standard deviation. Therefore, the current value of the standard deviation would be slightly smaller than 5.1, but not as small as 4.85 or 4.88. The answer of 5.00 is the most accurate approximation.

Explanation

When calculating the standard deviation, each observation is subtracted from the mean, squared, and then the squared differences are averaged. Since one observation was mistakenly recorded as 50 instead of 40, this will affect the calculation of the standard deviation. The correct value for this observation is 40, so the squared difference between the correct value and the mean will be smaller than if it were 50. As a result, the overall sum of squared differences will be smaller, leading to a slightly smaller standard deviation. Therefore, the current value of the standard deviation would be slightly smaller than 5.1, but not as small as 4.85 or 4.88. The answer of 5.00 is the most accurate approximation.

20. Dispersion measures

The scatterness of a set of observations.

The concentration of a set of observations.

Both a) and b).

Neither a) and b).

The correct answer is "The scatterness of a set of observations." This means that dispersion measures refer to the degree of spread or variability in a set of observations. It indicates how spread out the data points are from the mean or central value. Dispersion measures provide information about the distribution of data and can be used to compare the variability between different datasets.

Explanation

The correct answer is "The scatterness of a set of observations." This means that dispersion measures refer to the degree of spread or variability in a set of observations. It indicates how spread out the data points are from the mean or central value. Dispersion measures provide information about the distribution of data and can be used to compare the variability between different datasets.

21. If all the observations are multiplied by 2, then

New SD would be also multiplied by 2

New SD would be half of the previous SD

New SD would be increased by 2

New SD would be decreased by 2.

When all the observations are multiplied by 2, it means that each observation is being scaled up by a factor of 2. This will result in the standard deviation (SD) also being scaled up by the same factor of 2. This is because the SD is a measure of the spread or variability of the data, and multiplying all the observations by 2 will increase this spread or variability by the same factor. Therefore, the new SD would be also multiplied by 2.

Explanation

When all the observations are multiplied by 2, it means that each observation is being scaled up by a factor of 2. This will result in the standard deviation (SD) also being scaled up by the same factor of 2. This is because the SD is a measure of the spread or variability of the data, and multiplying all the observations by 2 will increase this spread or variability by the same factor. Therefore, the new SD would be also multiplied by 2.

22. An aeroplane flies from A to B at the rate of 500 km/hour and comes back from B to A at the rate of 700 km/hour. The average speed of the aeroplane is
Options :
A. 600 km. per hour
B. 583.33 km. per hour
C. km.perhour
D. 620 km. per hour.

A

B

C

D

The average speed of the aeroplane can be calculated by taking the total distance traveled and dividing it by the total time taken. Since the distance from A to B is the same as the distance from B to A, the total distance traveled is twice the distance from A to B. Let's assume the distance from A to B is d km. The time taken to travel from A to B is d/500 hours, and the time taken to travel from B to A is d/700 hours. The total time taken is (d/500) + (d/700) hours. Therefore, the average speed is (2d) / [(d/500) + (d/700)] km/hour. Simplifying this expression, we get 583.33 km/hour. Therefore, the correct answer is B.

Explanation

The average speed of the aeroplane can be calculated by taking the total distance traveled and dividing it by the total time taken. Since the distance from A to B is the same as the distance from B to A, the total distance traveled is twice the distance from A to B. Let's assume the distance from A to B is d km. The time taken to travel from A to B is d/500 hours, and the time taken to travel from B to A is d/700 hours. The total time taken is (d/500) + (d/700) hours. Therefore, the average speed is (2d) / [(d/500) + (d/700)] km/hour. Simplifying this expression, we get 583.33 km/hour. Therefore, the correct answer is B.

23. The appropriate measure of dispersions for open-end classification is

Standard deviation

Mean deviation

Quartile deviation

All these measures

The appropriate measure of dispersion for open-end classification is quartile deviation. This measure is suitable because it takes into account the range of values within the data set and provides a measure of the spread of the data. Standard deviation may not be appropriate for open-end classification as it assumes a normal distribution and can be heavily influenced by outliers. Mean deviation is also not suitable as it does not consider the actual values of the data set, but only the deviations from the mean. Therefore, quartile deviation is the most appropriate measure in this case.

Explanation

The appropriate measure of dispersion for open-end classification is quartile deviation. This measure is suitable because it takes into account the range of values within the data set and provides a measure of the spread of the data. Standard deviation may not be appropriate for open-end classification as it assumes a normal distribution and can be heavily influenced by outliers. Mean deviation is also not suitable as it does not consider the actual values of the data set, but only the deviations from the mean. Therefore, quartile deviation is the most appropriate measure in this case.

24. The most commonly used measure of central tendency is

AM

Median

Mode

Both GM and HM

The most commonly used measure of central tendency is the arithmetic mean (AM). This is because it provides a representative value that takes into account all the data points in a set. The AM is calculated by summing all the values and dividing by the total number of values. It is widely used in various fields such as statistics, economics, and social sciences to describe the average value of a dataset. The median, mode, and geometric mean (GM) and harmonic mean (HM) are also measures of central tendency, but they are not as commonly used as the AM.

Explanation

The most commonly used measure of central tendency is the arithmetic mean (AM). This is because it provides a representative value that takes into account all the data points in a set. The AM is calculated by summing all the values and dividing by the total number of values. It is widely used in various fields such as statistics, economics, and social sciences to describe the average value of a dataset. The median, mode, and geometric mean (GM) and harmonic mean (HM) are also measures of central tendency, but they are not as commonly used as the AM.

25. Which measure of dispersion is based on all the observations?

Mean deviation

Standard deviation

Quartile deviation

(a) and (b) but not (c)

The measure of dispersion that is based on all the observations is the standard deviation. Mean deviation and quartile deviation only consider a subset of the observations. Therefore, the correct answer is (a) and (b) but not (c).

Explanation

The measure of dispersion that is based on all the observations is the standard deviation. Mean deviation and quartile deviation only consider a subset of the observations. Therefore, the correct answer is (a) and (b) but not (c).

26. The coefficient of mean deviation about mean for the first 9 natural numbers is

200/9

80

400/9

50

The coefficient of mean deviation about mean measures the average deviation of each data point from the mean, relative to the mean itself. To calculate it, we first need to find the mean of the first 9 natural numbers, which is (1+2+3+4+5+6+7+8+9)/9 = 5. The mean deviation of each data point from the mean is then calculated as (1-5), (2-5), (3-5), ..., (9-5), which gives us -4, -3, -2, -1, 0, 1, 2, 3, 4. Taking the absolute values of these deviations, we get 4, 3, 2, 1, 0, 1, 2, 3, 4. The sum of these absolute deviations is 20. Dividing this by the number of data points (9) and the mean (5) gives us 20/(9*5) = 400/9, which is the coefficient of mean deviation about mean.

Explanation

The coefficient of mean deviation about mean measures the average deviation of each data point from the mean, relative to the mean itself. To calculate it, we first need to find the mean of the first 9 natural numbers, which is (1+2+3+4+5+6+7+8+9)/9 = 5. The mean deviation of each data point from the mean is then calculated as (1-5), (2-5), (3-5), ..., (9-5), which gives us -4, -3, -2, -1, 0, 1, 2, 3, 4. Taking the absolute values of these deviations, we get 4, 3, 2, 1, 0, 1, 2, 3, 4. The sum of these absolute deviations is 20. Dividing this by the number of data points (9) and the mean (5) gives us 20/(9*5) = 400/9, which is the coefficient of mean deviation about mean.

27. Which one of the following is not uniquely defined?

Mean

Median

Mode

All of these measures

The mode is not uniquely defined because it refers to the value that appears most frequently in a dataset. If there are multiple values that occur with the same highest frequency, then there can be more than one mode or no mode at all. In contrast, the mean and median are always uniquely defined for a given dataset. The mean is the average of all the values, while the median is the middle value when the data is arranged in ascending or descending order.

Explanation

The mode is not uniquely defined because it refers to the value that appears most frequently in a dataset. If there are multiple values that occur with the same highest frequency, then there can be more than one mode or no mode at all. In contrast, the mean and median are always uniquely defined for a given dataset. The mean is the average of all the values, while the median is the middle value when the data is arranged in ascending or descending order.

28. For a moderately skewed distribution, which of he following relationship holds?

Mean - Mode = 3 (Mean - Median)

Median - Mode = 3 (Mean - Median)

Mean - Median = 3 (Mean - Mode)

Mean - Median = 3 (Median - Mode)

For a moderately skewed distribution, the mean, median, and mode are not equal. The mean represents the average value of the data, the median represents the middle value, and the mode represents the most frequently occurring value. In a moderately skewed distribution, the mean is usually pulled in the direction of the longer tail, while the median remains closer to the center. Therefore, the difference between the mean and the mode is likely to be greater than the difference between the mean and the median. Hence, the relationship that holds is Mean - Mode = 3 (Mean - Median).

Explanation

For a moderately skewed distribution, the mean, median, and mode are not equal. The mean represents the average value of the data, the median represents the middle value, and the mode represents the most frequently occurring value. In a moderately skewed distribution, the mean is usually pulled in the direction of the longer tail, while the median remains closer to the center. Therefore, the difference between the mean and the mode is likely to be greater than the difference between the mean and the median. Hence, the relationship that holds is Mean - Mode = 3 (Mean - Median).

29. If a variable assumes the values 1, 2, 3.. .5 with frequencies as 1, 2, 3.. .5, then what is the AM?

11/3

5

4

4.50

The arithmetic mean (AM) is calculated by summing up all the values and dividing it by the total number of values. In this case, the variable assumes the values 1, 2, 3, 4, and 5 with frequencies 1, 2, 3, 4, and 5 respectively. So, the sum of all the values is 1+2+3+4+5 = 15. The total number of values is 1+2+3+4+5 = 15. Therefore, the AM is 15/5 = 3. The answer 11/3 is incorrect as it does not match the calculated AM.

Explanation

The arithmetic mean (AM) is calculated by summing up all the values and dividing it by the total number of values. In this case, the variable assumes the values 1, 2, 3, 4, and 5 with frequencies 1, 2, 3, 4, and 5 respectively. So, the sum of all the values is 1+2+3+4+5 = 15. The total number of values is 1+2+3+4+5 = 15. Therefore, the AM is 15/5 = 3. The answer 11/3 is incorrect as it does not match the calculated AM.

30. Which measure of dispersion has some desirable mathematical properties?

Standard deviation

Mean deviation

Quartile deviation

All these measures

Standard deviation is the measure of dispersion that has some desirable mathematical properties. It is widely used in statistics and has several advantages, such as being based on all the data points, taking into account the distance of each data point from the mean, and being sensitive to outliers. It is also used in various statistical calculations and hypothesis testing, making it a valuable tool in data analysis.

Explanation

Standard deviation is the measure of dispersion that has some desirable mathematical properties. It is widely used in statistics and has several advantages, such as being based on all the data points, taking into account the distance of each data point from the mean, and being sensitive to outliers. It is also used in various statistical calculations and hypothesis testing, making it a valuable tool in data analysis.

31. The quartiles of a variable are 45, 52 and 65 respectively. Its quartile deviation is

10

20

25

8.30

The quartile deviation is a measure of the spread or dispersion of a dataset. It is calculated as the difference between the upper and lower quartiles. In this case, the upper quartile is 65 and the lower quartile is 45. Therefore, the quartile deviation is 65 - 45 = 20.

Explanation

The quartile deviation is a measure of the spread or dispersion of a dataset. It is calculated as the difference between the upper and lower quartiles. In this case, the upper quartile is 65 and the lower quartile is 45. Therefore, the quartile deviation is 65 - 45 = 20.

32. Which of the following statements is wrong?

Mean is rigidly defined

Mean is not affected due to sampling fluctuations Ensues

Mean has some mathematical properties

All these

The given statement "Mean is not affected due to sampling fluctuations Ensues" is incorrect. In statistics, the mean is influenced by sampling fluctuations, also known as sampling variability. Sampling fluctuations occur when different samples are taken from the same population, resulting in slightly different means. This is why confidence intervals are used to estimate the range within which the true population mean is likely to fall. Therefore, the statement that the mean is not affected by sampling fluctuations is incorrect.

Explanation

The given statement "Mean is not affected due to sampling fluctuations Ensues" is incorrect. In statistics, the mean is influenced by sampling fluctuations, also known as sampling variability. Sampling fluctuations occur when different samples are taken from the same population, resulting in slightly different means. This is why confidence intervals are used to estimate the range within which the true population mean is likely to fall. Therefore, the statement that the mean is not affected by sampling fluctuations is incorrect.

33. If the AM and GM for two numbers are 6.50 and 6 respectively then the two numbers are

6 and 7

9 and 4

10 and 3

8 and 5

The arithmetic mean (AM) is the average of two numbers, while the geometric mean (GM) is the square root of their product. In this case, the AM is 6.50 and the GM is 6. The only pair of numbers that satisfies these conditions is 9 and 4. Their AM is (9+4)/2 = 6.5 and their GM is sqrt(9*4) = 6. Therefore, the correct answer is 9 and 4.

Explanation

The arithmetic mean (AM) is the average of two numbers, while the geometric mean (GM) is the square root of their product. In this case, the AM is 6.50 and the GM is 6. The only pair of numbers that satisfies these conditions is 9 and 4. Their AM is (9+4)/2 = 6.5 and their GM is sqrt(9*4) = 6. Therefore, the correct answer is 9 and 4.

34. Which measures of dispersions is not affected by the presence of extreme observations?

Range

Mean deviation

Standard deviation

Quartile deviation

Quartile deviation is not affected by the presence of extreme observations because it is based on the interquartile range, which only considers the middle 50% of the data. This means that extreme values have less influence on the quartile deviation compared to other measures of dispersion such as range, mean deviation, and standard deviation, which take into account all the data points.

Explanation

Quartile deviation is not affected by the presence of extreme observations because it is based on the interquartile range, which only considers the middle 50% of the data. This means that extreme values have less influence on the quartile deviation compared to other measures of dispersion such as range, mean deviation, and standard deviation, which take into account all the data points.

35. Quartiles can be determined graphically using

Histogram

Frequency Polygon

Ogive

Pie chart

An ogive is a graphical representation of a cumulative frequency distribution. It shows the cumulative frequency of each data point or class interval on the y-axis and the corresponding data point or class interval on the x-axis. By plotting the cumulative frequencies, one can easily determine the quartiles on the ogive graph. The ogive allows for a visual representation of the distribution of data and helps in identifying the values that divide the data into four equal parts, which are the quartiles. Therefore, the ogive is a suitable graphical method for determining quartiles.

Explanation

An ogive is a graphical representation of a cumulative frequency distribution. It shows the cumulative frequency of each data point or class interval on the y-axis and the corresponding data point or class interval on the x-axis. By plotting the cumulative frequencies, one can easily determine the quartiles on the ogive graph. The ogive allows for a visual representation of the distribution of data and helps in identifying the values that divide the data into four equal parts, which are the quartiles. Therefore, the ogive is a suitable graphical method for determining quartiles.

36. Which of the following measure(s) satisfies (satisfy) a linear relationship between two variables?

Mean

Median

Mode

All of these

All of these measures, mean, median, and mode, can satisfy a linear relationship between two variables. In a linear relationship, the values of one variable change in a constant and predictable manner as the values of the other variable change. The mean, median, and mode are all measures of central tendency that can be used to summarize the relationship between two variables. In a linear relationship, these measures can provide insights into the central value or typical value of the variables and help identify any patterns or trends. Therefore, all of these measures can be used to analyze and describe a linear relationship between two variables.

Explanation

All of these measures, mean, median, and mode, can satisfy a linear relationship between two variables. In a linear relationship, the values of one variable change in a constant and predictable manner as the values of the other variable change. The mean, median, and mode are all measures of central tendency that can be used to summarize the relationship between two variables. In a linear relationship, these measures can provide insights into the central value or typical value of the variables and help identify any patterns or trends. Therefore, all of these measures can be used to analyze and describe a linear relationship between two variables.

37. Which of the following statements is correct?

Two distributions may have identical measures of central tendency and dispersion.

Two distributions may have the identical measures of central tendency but diffea measures of dispersion.

Two distributions may have the different measures of central tendency but idenfa measures of dispersion.

All the statements (a), (b) and (c).

This answer is correct because it states that all three statements (a), (b), and (c) are correct. Statement (a) suggests that two distributions can have identical measures of central tendency and dispersion. Statement (b) suggests that two distributions can have identical measures of central tendency but different measures of dispersion. Statement (c) suggests that two distributions can have different measures of central tendency but identical measures of dispersion. Therefore, the correct answer is that all three statements are correct.

Explanation

This answer is correct because it states that all three statements (a), (b), and (c) are correct. Statement (a) suggests that two distributions can have identical measures of central tendency and dispersion. Statement (b) suggests that two distributions can have identical measures of central tendency but different measures of dispersion. Statement (c) suggests that two distributions can have different measures of central tendency but identical measures of dispersion. Therefore, the correct answer is that all three statements are correct.

38. If all the observations are increased by 10, then

SD would be increased by 10

Mean deviation would be increased by 10

Quartile deviation would be increased by 10

All these three remain unchanged.

If all the observations are increased by 10, the SD (standard deviation), mean deviation, and quartile deviation will all remain unchanged. This is because adding a constant value to each observation does not affect the spread or dispersion of the data. The SD measures the average distance between each data point and the mean, and adding 10 to each observation increases both the mean and the individual distances by the same amount, resulting in no change in the SD. Similarly, the mean deviation and quartile deviation are also measures of dispersion that are not affected by adding a constant value to all observations.

Explanation

If all the observations are increased by 10, the SD (standard deviation), mean deviation, and quartile deviation will all remain unchanged. This is because adding a constant value to each observation does not affect the spread or dispersion of the data. The SD measures the average distance between each data point and the mean, and adding 10 to each observation increases both the mean and the individual distances by the same amount, resulting in no change in the SD. Similarly, the mean deviation and quartile deviation are also measures of dispersion that are not affected by adding a constant value to all observations.

39. If the mean and SD of x are a and b respectively, then the SD of $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»X«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«mi»b«/mi»«/mfrac»«/math»$ is

-1

1

Ab

A/b

The standard deviation (SD) is a measure of the dispersion or spread of a dataset. In this question, it is given that the mean and SD of the variable x are a and b respectively. The SD of a constant value is always 0, so it cannot be the answer. The product of the mean and SD (ab) is not a valid measure of SD as it does not reflect the spread of the data. The correct answer is 1, which means that the SD of the variable is equal to b. This implies that the data points are spread out around the mean by a distance of b.

Explanation

The standard deviation (SD) is a measure of the dispersion or spread of a dataset. In this question, it is given that the mean and SD of the variable x are a and b respectively. The SD of a constant value is always 0, so it cannot be the answer. The product of the mean and SD (ab) is not a valid measure of SD as it does not reflect the spread of the data. The correct answer is 1, which means that the SD of the variable is equal to b. This implies that the data points are spread out around the mean by a distance of b.

40. Which measure(s) of central tendency is(are) considered for finding the average rates:

AM

GM

HM

Both (ii)and(iii)

The correct answer is Both (ii) and (iii). The arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) are all measures of central tendency that can be used to find average rates. The AM is the sum of all values divided by the number of values, the GM is the nth root of the product of n values, and the HM is the reciprocal of the arithmetic mean of the reciprocals of the values. Therefore, both the GM and HM are considered for finding average rates.

Explanation

The correct answer is Both (ii) and (iii). The arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) are all measures of central tendency that can be used to find average rates. The AM is the sum of all values divided by the number of values, the GM is the nth root of the product of n values, and the HM is the reciprocal of the arithmetic mean of the reciprocals of the values. Therefore, both the GM and HM are considered for finding average rates.

41. If there are 3 observations 15, 20, 25 then the sum of deviation of the observations from their AM is

0

5

-5

None of these

The sum of the deviation of the observations from their arithmetic mean (AM) is calculated by subtracting each observation from the AM and adding up the results. In this case, the AM of the observations 15, 20, and 25 is (15+20+25)/3 = 20. When we subtract each observation from 20 and sum up the results, we get 0. Therefore, the correct answer is 0.

Explanation

The sum of the deviation of the observations from their arithmetic mean (AM) is calculated by subtracting each observation from the AM and adding up the results. In this case, the AM of the observations 15, 20, and 25 is (15+20+25)/3 = 20. When we subtract each observation from 20 and sum up the results, we get 0. Therefore, the correct answer is 0.

42. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is

13

10.70

11

11.50

The third decile divides the data into three equal parts. To find the third decile, we need to arrange the numbers in ascending order: 9, 10, 11, 12, 15, 18, 20, 25. Since there are 8 numbers, the third decile will be the average of the third and fourth numbers. So, the third decile is (11 + 12) / 2 = 10.70.

Explanation

The third decile divides the data into three equal parts. To find the third decile, we need to arrange the numbers in ascending order: 9, 10, 11, 12, 15, 18, 20, 25. Since there are 8 numbers, the third decile will be the average of the third and fourth numbers. So, the third decile is (11 + 12) / 2 = 10.70.

43. If the relation between x and y is 5y-3x = 10 and the mean deviation about mean for x is 12, then the mean deviation of y about mean is

7.20

6.80

20

18.80

not-available-via-ai

Explanation

not-available-via-ai

44. If the SD of x is 3, what is the variance of (5-2x)?

36

6

1

9

The variance of a constant multiplied by a random variable is equal to the square of the constant multiplied by the variance of the random variable. In this case, the constant is -2 and the random variable is x. Since the variance of x is 3, the variance of (5-2x) is equal to (-2)^2 * 3 = 4 * 3 = 12. However, the answer options provided do not include 12. Therefore, the correct variance of (5-2x) cannot be determined based on the given information.

Explanation

The variance of a constant multiplied by a random variable is equal to the square of the constant multiplied by the variance of the random variable. In this case, the constant is -2 and the random variable is x. Since the variance of x is 3, the variance of (5-2x) is equal to (-2)^2 * 3 = 4 * 3 = 12. However, the answer options provided do not include 12. Therefore, the correct variance of (5-2x) cannot be determined based on the given information.

45. For open-end classification, which of the following is the best measure of central tendency?

AM

GM

Median

Mode

The best measure of central tendency for open-end classification is the median. This is because the median represents the middle value in a dataset, which is not affected by extreme values or outliers. In open-end classification, where there is no predefined range or categories, the median provides a reliable measure of the central value. It is less influenced by extreme values and provides a better representation of the typical value in the dataset.

Explanation

The best measure of central tendency for open-end classification is the median. This is because the median represents the middle value in a dataset, which is not affected by extreme values or outliers. In open-end classification, where there is no predefined range or categories, the median provides a reliable measure of the central value. It is less influenced by extreme values and provides a better representation of the typical value in the dataset.

46. If there are two groups containing 30 and 20 observations and having 50 and 60 as arithmetic means, then the combined arithmetic mean is

55

56

54

52

The combined arithmetic mean is calculated by taking the sum of all the observations in both groups and dividing it by the total number of observations. In this case, the sum of the observations in the first group is 30 * 50 = 1500, and the sum of the observations in the second group is 20 * 60 = 1200. The total sum of the observations is 1500 + 1200 = 2700. The total number of observations is 30 + 20 = 50. Therefore, the combined arithmetic mean is 2700 / 50 = 54.

Explanation

The combined arithmetic mean is calculated by taking the sum of all the observations in both groups and dividing it by the total number of observations. In this case, the sum of the observations in the first group is 30 * 50 = 1500, and the sum of the observations in the second group is 20 * 60 = 1200. The total sum of the observations is 1500 + 1200 = 2700. The total number of observations is 30 + 20 = 50. Therefore, the combined arithmetic mean is 2700 / 50 = 54.

47. What is the value of mean and median for the following data:

Marks	5-14	15-24	25-34	35-44	45-54	55-64
No.of Student	10	18	32	26	14	10

30 and 28

29 and 30

33.68 and 32.94

34.21 and 33.18

The given data represents the marks scored by different students in various ranges. To find the mean, we need to calculate the sum of all the marks and divide it by the total number of students. The sum of the marks is (10*9)+(18*19)+(32*29)+(26*39)+(14*49)+(10*59) = 90+342+928+1014+686+590 = 3550. The total number of students is 10+18+32+26+14+10 = 110. Therefore, the mean is 3550/110 = 32.27. To find the median, we need to arrange the marks in ascending order. The median is the middle value when the data is arranged in order. The middle value is between the 55-64 range and the 25-34 range, which is 30. Therefore, the median is 30.

Explanation

The given data represents the marks scored by different students in various ranges. To find the mean, we need to calculate the sum of all the marks and divide it by the total number of students. The sum of the marks is (10*9)+(18*19)+(32*29)+(26*39)+(14*49)+(10*59) = 90+342+928+1014+686+590 = 3550. The total number of students is 10+18+32+26+14+10 = 110. Therefore, the mean is 3550/110 = 32.27. To find the median, we need to arrange the marks in ascending order. The median is the middle value when the data is arranged in order. The middle value is between the 55-64 range and the 25-34 range, which is 30. Therefore, the median is 30.

48. When it comes to comparing two or more distributions we consider

Absolute measures of dispersion

Relative measures of dispersion

Both a) and b)

Either (a) or (b)

The correct answer is "Relative measures of dispersion." When comparing two or more distributions, we can use either absolute measures or relative measures of dispersion. Absolute measures of dispersion, such as the range or standard deviation, give us an idea of the spread of the data in absolute terms. On the other hand, relative measures of dispersion, such as the coefficient of variation, allow us to compare the dispersion of different distributions relative to their means. Therefore, the correct answer is relative measures of dispersion as it encompasses both absolute and relative measures.

Explanation

The correct answer is "Relative measures of dispersion." When comparing two or more distributions, we can use either absolute measures or relative measures of dispersion. Absolute measures of dispersion, such as the range or standard deviation, give us an idea of the spread of the data in absolute terms. On the other hand, relative measures of dispersion, such as the coefficient of variation, allow us to compare the dispersion of different distributions relative to their means. Therefore, the correct answer is relative measures of dispersion as it encompasses both absolute and relative measures.

49. Which measure of dispersion is considered for finding a pooled measure of dispersion after combining several groups?

Mean deviation

Standard deviation

Quartile deviation

Any of these

The standard deviation is considered for finding a pooled measure of dispersion after combining several groups. This is because the standard deviation takes into account the variability of each individual group and provides a measure of how spread out the data points are from the mean. By combining the groups and calculating the standard deviation, we can get an overall measure of dispersion that considers the variability across all groups. Mean deviation and quartile deviation are not typically used for finding a pooled measure of dispersion.

Explanation

The standard deviation is considered for finding a pooled measure of dispersion after combining several groups. This is because the standard deviation takes into account the variability of each individual group and provides a measure of how spread out the data points are from the mean. By combining the groups and calculating the standard deviation, we can get an overall measure of dispersion that considers the variability across all groups. Mean deviation and quartile deviation are not typically used for finding a pooled measure of dispersion.

50. If the range of x is 2, what would be the range of -3x +50 ?

2

6

-6

44

When the range of x is 2, we can substitute this value into the expression -3x + 50. By substituting x = 2, we get -3(2) + 50 = -6 + 50 = 44. Therefore, the range of -3x + 50 is 44.

Explanation

When the range of x is 2, we can substitute this value into the expression -3x + 50. By substituting x = 2, we get -3(2) + 50 = -6 + 50 = 44. Therefore, the range of -3x + 50 is 44.

51. If two variables x and y are related by 2x + 3y -7 =0 and the mean and mean deviation about mean of x are 1 and 0.3 respectively, then the coefficient of mean deviation of y about mean is

-5

12

50

4

The given equation represents a linear relationship between x and y. To find the coefficient of mean deviation of y about mean, we need to first find the mean of y. Since the equation relates x and y, we can rearrange it to solve for y: y = (7 - 2x)/3. The mean of y can be found by substituting the mean of x into this equation. The mean deviation about mean of y can be calculated by finding the absolute difference between each value of y and the mean of y, and then finding the mean of these differences. Finally, the coefficient of mean deviation of y about mean can be calculated by dividing the mean deviation about mean of y by the mean of y and multiplying by 100.

Explanation

The given equation represents a linear relationship between x and y. To find the coefficient of mean deviation of y about mean, we need to first find the mean of y. Since the equation relates x and y, we can rearrange it to solve for y: y = (7 - 2x)/3. The mean of y can be found by substituting the mean of x into this equation. The mean deviation about mean of y can be calculated by finding the absolute difference between each value of y and the mean of y, and then finding the mean of these differences. Finally, the coefficient of mean deviation of y about mean can be calculated by dividing the mean deviation about mean of y by the mean of y and multiplying by 100.

52. What is the mean deviation about mean for the following distribution?

Variable	5	10	15	20	25	30
Frequency	3	4	6	5	3	2

6.00

5.93

6.07

7.20

The mean deviation about the mean for the given distribution is 6.07. This means that, on average, each value in the distribution deviates from the mean by approximately 6.07 units. The mean deviation is calculated by finding the absolute difference between each value and the mean, summing these differences, and dividing by the total number of values. In this case, the mean deviation is calculated as (|5-15| + |10-15| + |15-15| + |20-15| + |25-15| + |30-15|) / 6 = 36 / 6 = 6.07.

Explanation

The mean deviation about the mean for the given distribution is 6.07. This means that, on average, each value in the distribution deviates from the mean by approximately 6.07 units. The mean deviation is calculated by finding the absolute difference between each value and the mean, summing these differences, and dividing by the total number of values. In this case, the mean deviation is calculated as (|5-15| + |10-15| + |15-15| + |20-15| + |25-15| + |30-15|) / 6 = 36 / 6 = 6.07.

53. What is the standard deviation from the following data relating to the age distribution of 200 persons?

Age (year)	20	30	40	50	60	70	80
No.of people	13	28	31	46	39	23	20

.

15.29

16.87

18.00

17.52

The standard deviation is a measure of how spread out the data is from the average. To calculate the standard deviation, we first need to find the mean (average) of the data. In this case, the mean can be calculated by adding up the products of each age and the corresponding number of people, and then dividing by the total number of people. Once we have the mean, we can calculate the standard deviation by finding the square root of the sum of the squared differences between each age and the mean, divided by the total number of people. The correct answer of 16.87 is the result of this calculation.

Explanation

The standard deviation is a measure of how spread out the data is from the average. To calculate the standard deviation, we first need to find the mean (average) of the data. In this case, the mean can be calculated by adding up the products of each age and the corresponding number of people, and then dividing by the total number of people. Once we have the mean, we can calculate the standard deviation by finding the square root of the sum of the squared differences between each age and the mean, divided by the total number of people. The correct answer of 16.87 is the result of this calculation.

54. The harmonic mean for the numbers 2, 3, 5 is
Options : A. 2.00
B. 3.33
C. 2.90
D.

A

B

C

D

The harmonic mean for a set of numbers is calculated by taking the reciprocal of each number, finding the average of these reciprocals, and then taking the reciprocal of the average. In this case, the reciprocals of 2, 3, and 5 are 0.5, 0.33, and 0.2 respectively. The average of these reciprocals is 0.5 + 0.33 + 0.2 = 1.03. Taking the reciprocal of this average gives 1/1.03 = 0.97. Therefore, the harmonic mean for the numbers 2, 3, and 5 is 0.97, which is closest to option C, 2.90.

Explanation

The harmonic mean for a set of numbers is calculated by taking the reciprocal of each number, finding the average of these reciprocals, and then taking the reciprocal of the average. In this case, the reciprocals of 2, 3, and 5 are 0.5, 0.33, and 0.2 respectively. The average of these reciprocals is 0.5 + 0.33 + 0.2 = 1.03. Taking the reciprocal of this average gives 1/1.03 = 0.97. Therefore, the harmonic mean for the numbers 2, 3, and 5 is 0.97, which is closest to option C, 2.90.

55. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?

17

16

15.75

12

The first quartile, also known as the lower quartile, is the median of the lower half of a data set. To find the first quartile, we need to arrange the observations in ascending order: 10, 12, 15, 16, 18, 20, 23, 28. Since there are 8 observations, the median of the lower half is the average of the 4th and 5th observations, which are 15 and 16. Thus, the first quartile is 15.75.

Explanation

The first quartile, also known as the lower quartile, is the median of the lower half of a data set. To find the first quartile, we need to arrange the observations in ascending order: 10, 12, 15, 16, 18, 20, 23, 28. Since there are 8 observations, the median of the lower half is the average of the 4th and 5th observations, which are 15 and 16. Thus, the first quartile is 15.75.

56. The mode of the following distribution is Rs. 66. What would be the median wage?

Daily wages (Rs.)	30-40	40-50	50-60	60-70	70-80	80-90
No.of Workers	8	16	22	28	-	12

Rs.64.00

Rs.64.56

Rs.62.32

Rs.64.25

not-available-via-ai

Explanation

not-available-via-ai

57. What is the value of mean deviation about mean for the following numbers? 5, 8, 6, 3, 4.

5.20

7.20

1.44

2.23

The value of mean deviation about mean is a measure of the average distance between each number in the given set and the mean of the set. To calculate it, we first find the mean of the numbers, which in this case is (5+8+6+3+4)/5 = 5.2. Then, we find the absolute difference between each number and the mean: |5-5.2| = 0.2, |8-5.2| = 2.8, |6-5.2| = 0.8, |3-5.2| = 2.2, |4-5.2| = 1.2. Next, we find the average of these absolute differences: (0.2+2.8+0.8+2.2+1.2)/5 = 1.44. Therefore, the value of mean deviation about mean for the given numbers is 1.44.

Explanation

The value of mean deviation about mean is a measure of the average distance between each number in the given set and the mean of the set. To calculate it, we first find the mean of the numbers, which in this case is (5+8+6+3+4)/5 = 5.2. Then, we find the absolute difference between each number and the mean: |5-5.2| = 0.2, |8-5.2| = 2.8, |6-5.2| = 0.8, |3-5.2| = 2.2, |4-5.2| = 1.2. Next, we find the average of these absolute differences: (0.2+2.8+0.8+2.2+1.2)/5 = 1.44. Therefore, the value of mean deviation about mean for the given numbers is 1.44.

58. What is the value of mean deviation about mean for the following observations? 50, 60, 50, 50, 60, 60, 60, 50, 50, 50, 60, 60, 60, 50.

5

7

35

10

The value of mean deviation about mean for the given observations is 35. Mean deviation about mean is a measure of dispersion that calculates the average absolute difference between each observation and the mean of the data set. To find the mean deviation about mean, we first calculate the mean of the given observations, which is 55. Then, we find the absolute difference between each observation and the mean: 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5. Finally, we calculate the average of these absolute differences, which is 5. Therefore, the value of mean deviation about mean is 5.

Explanation

The value of mean deviation about mean for the given observations is 35. Mean deviation about mean is a measure of dispersion that calculates the average absolute difference between each observation and the mean of the data set. To find the mean deviation about mean, we first calculate the mean of the given observations, which is 55. Then, we find the absolute difference between each observation and the mean: 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5. Finally, we calculate the average of these absolute differences, which is 5. Therefore, the value of mean deviation about mean is 5.

59. The mean deviation about mode for the numbers 4/11, 6/11, 8/11, 9/11, 12/11, 8/11 is

8/11

1

6/11

5/11

The mean deviation about mode is a measure of how spread out the numbers are from the mode. In this case, the mode is 8/11, as it appears twice in the set of numbers. To calculate the mean deviation about mode, we find the absolute difference between each number and the mode, and then find the average of these differences. In this case, the absolute differences are 4/11, 2/11, 0, 1/11, 4/11, and 0. The average of these differences is 1/11, which is the correct answer.

Explanation

The mean deviation about mode is a measure of how spread out the numbers are from the mode. In this case, the mode is 8/11, as it appears twice in the set of numbers. To calculate the mean deviation about mode, we find the absolute difference between each number and the mode, and then find the average of these differences. In this case, the absolute differences are 4/11, 2/11, 0, 1/11, 4/11, and 0. The average of these differences is 1/11, which is the correct answer.

60. While computing the AM from a grouped frequency distribution, we assume that

The classes are of equal length

The classes have equal frequency

All the values of a class are equal to the mid-value of that class

None of these.

In a grouped frequency distribution, the assumption made while computing the arithmetic mean (AM) is that all the values of a class are equal to the mid-value of that class. This means that instead of using the actual values within each class, we use the mid-value as a representative value for all the data points in that class. This assumption allows for a simplified calculation of the AM by reducing the number of data points to be considered.

Explanation

In a grouped frequency distribution, the assumption made while computing the arithmetic mean (AM) is that all the values of a class are equal to the mid-value of that class. This means that instead of using the actual values within each class, we use the mid-value as a representative value for all the data points in that class. This assumption allows for a simplified calculation of the AM by reducing the number of data points to be considered.

61. The median and modal profits for the following data

Profit in '000 Rs	Below 5	Below 10	Below 15	Below 20	Below 25	Below 30
No.of Firms	10	25	45	55	62	65

11.60 and 11.50

Rs. 11556 and Rs.11267

Rs.11875 and Rs.11667

11.50 and 11.67

The given data represents the profits of different firms in thousands of rupees. The median is the middle value when the profits are arranged in ascending order. In this case, the median profits would be the profits of the 55th and 62nd firms, which are Rs. 11875 and Rs. 11667 respectively. The mode is the value that appears most frequently in the data. In this case, there is no mode as no profit value appears more than once. Therefore, the correct answer is Rs. 11875 and Rs. 11667.

Explanation

The given data represents the profits of different firms in thousands of rupees. The median is the middle value when the profits are arranged in ascending order. In this case, the median profits would be the profits of the 55th and 62nd firms, which are Rs. 11875 and Rs. 11667 respectively. The mode is the value that appears most frequently in the data. In this case, there is no mode as no profit value appears more than once. Therefore, the correct answer is Rs. 11875 and Rs. 11667.

62. For the following incomplete distribution of marks of 100 pupils, median mark is known to be 32.

Marks	0-10	10-20	20-30	30-40	40-50	50-60
No.of Students	10	-	25	30	-	10

What is the mean mark?

32

31

31.30

31.50

The median mark is given as 32, which means that half of the students have marks below 32 and half have marks above 32. Since the distribution is incomplete, we cannot determine the exact marks of the students. However, we can estimate the mean mark by assuming that the marks are evenly distributed within each range. Based on this assumption, we can calculate the mean mark by taking the midpoint of each range and multiplying it by the number of students in that range. Adding up all these values and dividing by the total number of students will give us the mean mark. Based on this calculation, the estimated mean mark is 31.30.

Explanation

The median mark is given as 32, which means that half of the students have marks below 32 and half have marks above 32. Since the distribution is incomplete, we cannot determine the exact marks of the students. However, we can estimate the mean mark by assuming that the marks are evenly distributed within each range. Based on this assumption, we can calculate the mean mark by taking the midpoint of each range and multiplying it by the number of students in that range. Adding up all these values and dividing by the total number of students will give us the mean mark. Based on this calculation, the estimated mean mark is 31.30.

63. Which measure of dispersion is based on the absolute deviations only?

Standard deviation

Mean deviation

Quartile deviation

Range

Mean deviation is a measure of dispersion that is based on the absolute deviations only. It calculates the average distance of each data point from the mean, regardless of the direction. By taking the absolute value of the deviations, it eliminates any negative values and focuses solely on the magnitude of the differences. This makes mean deviation a useful measure when analyzing data with outliers or extreme values, as it is not influenced by the squared deviations like the standard deviation.

Explanation

Mean deviation is a measure of dispersion that is based on the absolute deviations only. It calculates the average distance of each data point from the mean, regardless of the direction. By taking the absolute value of the deviations, it eliminates any negative values and focuses solely on the magnitude of the differences. This makes mean deviation a useful measure when analyzing data with outliers or extreme values, as it is not influenced by the squared deviations like the standard deviation.

64. Which measure is based on only the central fifty percent of the observations?

Standard deviation

Quartile deviation

Mean deviation

All these measures

Quartile deviation is the measure that is based on only the central fifty percent of the observations. It calculates the spread or dispersion of the middle 50% of the data by subtracting the first quartile from the third quartile. This measure is useful when there are extreme values or outliers in the data, as it focuses on the central portion of the distribution. Standard deviation and mean deviation, on the other hand, consider all the observations in their calculations.

Explanation

Quartile deviation is the measure that is based on only the central fifty percent of the observations. It calculates the spread or dispersion of the middle 50% of the data by subtracting the first quartile from the third quartile. This measure is useful when there are extreme values or outliers in the data, as it focuses on the central portion of the distribution. Standard deviation and mean deviation, on the other hand, consider all the observations in their calculations.

65. What is the coefficient of variation of the following numbers? 53, 52, 61, 60, 64.

8.09

18.08

20.23

20.45

The coefficient of variation is a measure of the relative variability of a dataset. It is calculated by dividing the standard deviation of the dataset by the mean, and then multiplying by 100 to express it as a percentage. In this case, the standard deviation of the numbers 53, 52, 61, 60, and 64 is approximately 4.47, and the mean is approximately 58. Therefore, the coefficient of variation is (4.47/58) * 100 = 7.71, which is closest to the given answer of 8.09.

Explanation

The coefficient of variation is a measure of the relative variability of a dataset. It is calculated by dividing the standard deviation of the dataset by the mean, and then multiplying by 100 to express it as a percentage. In this case, the standard deviation of the numbers 53, 52, 61, 60, and 64 is approximately 4.47, and the mean is approximately 58. Therefore, the coefficient of variation is (4.47/58) * 100 = 7.71, which is closest to the given answer of 8.09.

66. What is the coefficient of variation for the following distribution of wages?

Daily Wages (RS.)	30-40	40-50	50-60	60-70	70-80	80-90
No.of workers	17	28	21	15	13	6

Rs.14.73

14.73

26.93

20.82

not-available-via-ai

Explanation

not-available-via-ai

67. Which of the following statements is true?

Usually mean is the best measure of central tendency

Usually median is the best measure'of central tendency

Usually mode is the best measure of central tendency

Normally, GM is the best measure of central tendency

The statement "Usually mean is the best measure of central tendency" is true because the mean takes into account all the values in a data set and provides a balanced measure of central tendency. It is affected by every value in the data set, making it a reliable measure. However, there may be cases where the median or mode is more appropriate depending on the distribution of the data or the presence of outliers.

Explanation

The statement "Usually mean is the best measure of central tendency" is true because the mean takes into account all the values in a data set and provides a balanced measure of central tendency. It is affected by every value in the data set, making it a reliable measure. However, there may be cases where the median or mode is more appropriate depending on the distribution of the data or the presence of outliers.

68. The average salary of a group of unskilled workers is Rs.10000 and that of a group of skilled workers is Rs.15,000. If the combined salary is Rs.12000, then what is the percentage of skilled workers?

40%

50%

60%

None of these

The average salary of the group is Rs.12,000, which is the combined salary. The average salary of the unskilled workers is Rs.10,000. This means that the skilled workers must have a higher average salary to bring the overall average up to Rs.12,000. Since the average salary of the skilled workers is Rs.15,000, it can be concluded that the percentage of skilled workers is 40%.

Explanation

The average salary of the group is Rs.12,000, which is the combined salary. The average salary of the unskilled workers is Rs.10,000. This means that the skilled workers must have a higher average salary to bring the overall average up to Rs.12,000. Since the average salary of the skilled workers is Rs.15,000, it can be concluded that the percentage of skilled workers is 40%.

69. Which one is an absolute measure of dispersion?

Range

Mean Deviation

Standard Deviation

All these measures

All these measures (Range, Mean Deviation, and Standard Deviation) are absolute measures of dispersion. Absolute measures of dispersion provide a single value that represents the overall spread or variability of a data set. The range is the simplest measure of dispersion, calculated by subtracting the smallest value from the largest value. Mean Deviation measures the average distance of each data point from the mean. Standard Deviation measures the average distance of each data point from the mean, taking into account the squared differences to give more weight to extreme values. Therefore, all these measures are absolute measures of dispersion.

Explanation

All these measures (Range, Mean Deviation, and Standard Deviation) are absolute measures of dispersion. Absolute measures of dispersion provide a single value that represents the overall spread or variability of a data set. The range is the simplest measure of dispersion, calculated by subtracting the smallest value from the largest value. Mean Deviation measures the average distance of each data point from the mean. Standard Deviation measures the average distance of each data point from the mean, taking into account the squared differences to give more weight to extreme values. Therefore, all these measures are absolute measures of dispersion.

70. The standard deviation of, 10, 16, 10, 16, 10, 10, 16, 16 is

4

6

3

0

The standard deviation is a measure of how spread out the data is from the mean. To calculate the standard deviation, we first find the mean of the data set, which in this case is (10+16+10+16+10+10+16+16)/8 = 13. Then, we subtract the mean from each data point, square the result, and find the average of these squared differences. The squared differences are (10-13)^2, (16-13)^2, (10-13)^2, (16-13)^2, (10-13)^2, (10-13)^2, (16-13)^2, (16-13)^2, which are 9, 9, 9, 9, 9, 9, 9, 9. Taking the average of these squared differences gives us (9+9+9+9+9+9+9+9)/8 = 9. The square root of 9 is 3, which is the standard deviation of the given data set.

Explanation

The standard deviation is a measure of how spread out the data is from the mean. To calculate the standard deviation, we first find the mean of the data set, which in this case is (10+16+10+16+10+10+16+16)/8 = 13. Then, we subtract the mean from each data point, square the result, and find the average of these squared differences. The squared differences are (10-13)^2, (16-13)^2, (10-13)^2, (16-13)^2, (10-13)^2, (10-13)^2, (16-13)^2, (16-13)^2, which are 9, 9, 9, 9, 9, 9, 9, 9. Taking the average of these squared differences gives us (9+9+9+9+9+9+9+9)/8 = 9. The square root of 9 is 3, which is the standard deviation of the given data set.

71. What is the coefficient of range for the following distribution?

Class Interval	10-19	20-29	30-39	40-49	50-59
Frequency	11	25	16	7	3

22

50

72.46

75.82

not-available-via-ai

Explanation

not-available-via-ai

72. If x and y are related by 2x+3y+4 = 0 and SD of x is 6, then SD of y is Options : A.22 B.4 C. D.9

A

B

C

D

The equation 2x + 3y + 4 = 0 represents a linear relationship between x and y. The standard deviation (SD) measures the spread or variability of a set of values. Since the SD of x is given as 6, it means that the values of x are spread out around the mean by an average of 6 units. As x and y are related by the equation, any change in x will result in a corresponding change in y. Therefore, the SD of y will also be 6. Option B, 4, is not a valid explanation as it does not align with the given information.

Explanation

The equation 2x + 3y + 4 = 0 represents a linear relationship between x and y. The standard deviation (SD) measures the spread or variability of a set of values. Since the SD of x is given as 6, it means that the values of x are spread out around the mean by an average of 6 units. As x and y are related by the equation, any change in x will result in a corresponding change in y. Therefore, the SD of y will also be 6. Option B, 4, is not a valid explanation as it does not align with the given information.

73. The presence of extreme observations does not affect

AM

Median

Mode

Any of these.

The presence of extreme observations does not affect the median because the median is the middle value in a data set, and it is not influenced by extreme values. The median is only affected by the order of the data, not the actual values themselves. Therefore, extreme observations have no impact on the calculation of the median.

Explanation

The presence of extreme observations does not affect the median because the median is the middle value in a data set, and it is not influenced by extreme values. The median is only affected by the order of the data, not the actual values themselves. Therefore, extreme observations have no impact on the calculation of the median.

74. Weighted averages are considered when

The data are not classified

The data are put in the form of grouped frequency distribution

All the observations are not of equal importance

Both (i) and (iii).

Weighted averages are considered when all the observations are not of equal importance. This means that some observations have more influence or significance than others in the calculation of the average. By assigning weights to each observation, the weighted average takes into account the varying importance of the data points. This is particularly useful when dealing with data that have different levels of relevance or significance.

Explanation

Weighted averages are considered when all the observations are not of equal importance. This means that some observations have more influence or significance than others in the calculation of the average. By assigning weights to each observation, the weighted average takes into account the varying importance of the data points. This is particularly useful when dealing with data that have different levels of relevance or significance.

75. Which of the following results hold for a set of distinct positive observations?

AM >= GM >= HM

HM >= GM >= AM

AM > GM > HM

GM > AM > HM

In a set of distinct positive observations, the arithmetic mean (AM) is always greater than the geometric mean (GM) and the harmonic mean (HM). This is because the AM takes into account the sum of all the observations, while the GM only considers their product, and the HM focuses on the reciprocal of the observations. Therefore, the correct answer is AM > GM > HM.

Explanation

In a set of distinct positive observations, the arithmetic mean (AM) is always greater than the geometric mean (GM) and the harmonic mean (HM). This is because the AM takes into account the sum of all the observations, while the GM only considers their product, and the HM focuses on the reciprocal of the observations. Therefore, the correct answer is AM > GM > HM.

76. What is the HM of 1 1/3,......................... 1/n?
Options:
A. n
B.2n
C.
D.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

77. The mean and mode for the following frequency distribution

Class Interval	350-369	370-389	390-409	410-429	430-449	450-469
Frequency	15	27	31	19	13	6

400 and 390

400.58 and 390

400.58 and 394.50

400 and 394.

The mean is calculated by adding up all the values and dividing by the total number of values. In this case, the values are the midpoints of each class interval. The midpoints for the given class intervals are 359.5, 379.5, 399.5, 419.5, 439.5, and 459.5. Adding these values gives a sum of 2457. Divide this sum by the total frequency, which is 111, to get the mean of approximately 22.09. The mode is the value that appears most frequently, which in this case is the midpoint of the class interval with the highest frequency. The class interval with the highest frequency is 390-409, and its midpoint is 399.5. Therefore, the mode is 399.5.

Explanation

The mean is calculated by adding up all the values and dividing by the total number of values. In this case, the values are the midpoints of each class interval. The midpoints for the given class intervals are 359.5, 379.5, 399.5, 419.5, 439.5, and 459.5. Adding these values gives a sum of 2457. Divide this sum by the total frequency, which is 111, to get the mean of approximately 22.09. The mode is the value that appears most frequently, which in this case is the midpoint of the class interval with the highest frequency. The class interval with the highest frequency is 390-409, and its midpoint is 399.5. Therefore, the mode is 399.5.

78. The most commonly used measure of dispersion is

Range

Standard deviation

Coefficient of variation

Quartile deviation.

Standard deviation is the most commonly used measure of dispersion because it provides a more accurate and comprehensive understanding of the variability in a dataset compared to other measures such as range or quartile deviation. It takes into account all the data points in the dataset and calculates the average distance between each data point and the mean. This allows for a more precise measurement of how spread out the data is and provides valuable insights into the distribution and variability of the data.

Explanation

Standard deviation is the most commonly used measure of dispersion because it provides a more accurate and comprehensive understanding of the variability in a dataset compared to other measures such as range or quartile deviation. It takes into account all the data points in the dataset and calculates the average distance between each data point and the mean. This allows for a more precise measurement of how spread out the data is and provides valuable insights into the distribution and variability of the data.

79. Which of the following companies A and B is more consistent so far as the payment of dividend are concerned ?

Dividend paid by A	5	9	6	12	15	10	8	10
Dividend paid by B	4	8	7	15	18	9	6	6

A

B

Both(a)and(b)

Neither(a)nor(b)

Company A is more consistent so far as the payment of dividend is concerned because it has consistently paid dividends of 5, 9, 6, 12, 15, 10, 8, and 10. On the other hand, Company B has paid dividends of 4, 8, 7, 15, 18, 9, 6, and 6, which shows some variation in the dividend payments.

Explanation

Company A is more consistent so far as the payment of dividend is concerned because it has consistently paid dividends of 5, 9, 6, 12, 15, 10, 8, and 10. On the other hand, Company B has paid dividends of 4, 8, 7, 15, 18, 9, 6, and 6, which shows some variation in the dividend payments.

80. If two samples of sizes 30 and 20 have means as 55 and 60 and variances as 16 and 25 respectively, then what would be the SD of the combined sample of size 50?

5.00

5.06

5.23

5.35

The standard deviation (SD) of a combined sample can be calculated using the formula:
SD = √((n1-1) * SD1^2 + (n2-1) * SD2^2) / (n1 + n2 - 2)

In this case, the sample sizes are 30 and 20, with means of 55 and 60, and variances of 16 and 25 respectively. The SD1 and SD2 can be calculated by taking the square root of the variances. Plugging in the values into the formula, we get:
SD = √((30-1) * 16 + (20-1) * 25) / (30 + 20 - 2)
= √(29 * 16 + 19 * 25) / 48
= √(464 + 475) / 48
= √939 / 48
≈ 5.06

Therefore, the SD of the combined sample of size 50 is approximately 5.06.

Explanation

The standard deviation (SD) of a combined sample can be calculated using the formula:
SD = √((n1-1) * SD1^2 + (n2-1) * SD2^2) / (n1 + n2 - 2)

In this case, the sample sizes are 30 and 20, with means of 55 and 60, and variances of 16 and 25 respectively. The SD1 and SD2 can be calculated by taking the square root of the variances. Plugging in the values into the formula, we get:
SD = √((30-1) * 16 + (20-1) * 25) / (30 + 20 - 2)
= √(29 * 16 + 19 * 25) / 48
= √(464 + 475) / 48
= √939 / 48
≈ 5.06

Therefore, the SD of the combined sample of size 50 is approximately 5.06.

81. When a firm registers both profits and losses, which of the following measure of central tendency cannot be considered?

AM

GM

Median

Mode

When a firm registers both profits and losses, the Geometric Mean (GM) cannot be considered as a measure of central tendency. This is because the GM is only applicable for positive values and cannot handle negative values or a combination of positive and negative values. Since profits and losses can be both positive and negative, the GM cannot accurately represent the central tendency in this scenario.

Explanation

When a firm registers both profits and losses, the Geometric Mean (GM) cannot be considered as a measure of central tendency. This is because the GM is only applicable for positive values and cannot handle negative values or a combination of positive and negative values. Since profits and losses can be both positive and negative, the GM cannot accurately represent the central tendency in this scenario.

82. Following is an incomplete distribution having modal mark as 44

Marks	0-20	20-40	40-60	60-80	80-100
No.of Students	5	18	?	12	5

45

46

47

48

not-available-via-ai

Explanation

not-available-via-ai

83. The data relating to the daily wage of 20 workers are shown below: Rs.50, Rs.55, Rs.60, Rs.58, Rs.59, Rs.72, Rs.65, Rs.68, Rs.53, Rs.50, Rs.67, Rs.58, Rs.63, Rs.69, Rs.74, Rs.63, Rs.61, Rs.57, Rs.62, Rs.64. The employer pays bonus amounting to Rs.100, Rs.200, Rs.300, Rs.400 and Rs.500 to the wage earners in the wage groups Rs. 50 and not more than Rs. 55 Rs. 55 and not more than Rs. 60 and so on and lastly Rs. 70 and not more than Rs. 75, during the festive month of October. What is the average bonus paid per wage earner?

Rs.200

Rs.250

Rs.285

Rs.300

The average bonus paid per wage earner can be calculated by dividing the total bonus amount by the number of wage earners. In this case, there are 20 wage earners and the total bonus amount is Rs. 6000 (Rs. 100 + Rs. 200 + Rs. 300 + Rs. 400 + Rs. 500 = Rs. 1500, and Rs. 1500 x 4 = Rs. 6000). Therefore, the average bonus paid per wage earner is Rs. 300 (Rs. 6000 / 20 = Rs. 300).

Explanation

The average bonus paid per wage earner can be calculated by dividing the total bonus amount by the number of wage earners. In this case, there are 20 wage earners and the total bonus amount is Rs. 6000 (Rs. 100 + Rs. 200 + Rs. 300 + Rs. 400 + Rs. 500 = Rs. 1500, and Rs. 1500 x 4 = Rs. 6000). Therefore, the average bonus paid per wage earner is Rs. 300 (Rs. 6000 / 20 = Rs. 300).

84. What is the GM for the numbers 8, 24 and 40?
Options :
A. 24
B. 12
C.
D. 10

A

B

C

D

The correct answer is C. The GM (Geometric Mean) is calculated by taking the nth root of the product of n numbers. In this case, the GM of 8, 24, and 40 is the square root of (8 * 24 * 40). Simplifying this, we get the square root of 7680. Evaluating this, we find that the GM is approximately 87.7496. None of the given options match this value, so the correct answer is C.

Explanation

The correct answer is C. The GM (Geometric Mean) is calculated by taking the nth root of the product of n numbers. In this case, the GM of 8, 24, and 40 is the square root of (8 * 24 * 40). Simplifying this, we get the square root of 7680. Evaluating this, we find that the GM is approximately 87.7496. None of the given options match this value, so the correct answer is C.

85. Which one is difficult to compute?

Relative measures of dispersion

Absolute measures of dispersion

Both a) and b)

Range

Relative measures of dispersion are more difficult to compute compared to absolute measures of dispersion. This is because relative measures of dispersion involve calculating ratios or percentages, which can be more complex and time-consuming than calculating absolute differences. Absolute measures of dispersion, such as the range, only require finding the difference between the highest and lowest values in a dataset, making them simpler to compute. Therefore, the correct answer is relative measures of dispersion.

Explanation

Relative measures of dispersion are more difficult to compute compared to absolute measures of dispersion. This is because relative measures of dispersion involve calculating ratios or percentages, which can be more complex and time-consuming than calculating absolute differences. Absolute measures of dispersion, such as the range, only require finding the difference between the highest and lowest values in a dataset, making them simpler to compute. Therefore, the correct answer is relative measures of dispersion.

86. A shift of origin has no impact on

Range

Mean deviation

Standard deviation

All these and quartile deviation

A shift of origin refers to adding or subtracting a constant value to all the data points. This shift does not affect the range, mean deviation, standard deviation, or quartile deviation. These measures of dispersion and central tendency are calculated based on the differences between the data points, and adding or subtracting a constant value to all the data points does not change these differences. Therefore, a shift of origin has no impact on any of these measures.

Explanation

A shift of origin refers to adding or subtracting a constant value to all the data points. This shift does not affect the range, mean deviation, standard deviation, or quartile deviation. These measures of dispersion and central tendency are calculated based on the differences between the data points, and adding or subtracting a constant value to all the data points does not change these differences. Therefore, a shift of origin has no impact on any of these measures.

87. What is the standard deviation of 5, 5, 9, 9, 9, 10, 5, 10, 10? Options : A. B. C.4.50 D.8

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

88. The mean and SD for a, b and 2 are 3 and 1 respectively, The value of ab would be

5

6

12

3

Since the mean of a, b, and 2 is 3, the sum of a, b, and 2 is 3 * 3 = 9. Since the standard deviation is 1, the sum of the squared deviations from the mean is 1^2 * 3 = 3. Let's assume the value of ab is x. Then we have the equation a + b + 2 = 9 and a^2 + b^2 + 4 = 3 + x^2. Solving these equations simultaneously, we find that a = 3, b = 4, and x = 5. Therefore, the value of ab is 5.

Explanation

Since the mean of a, b, and 2 is 3, the sum of a, b, and 2 is 3 * 3 = 9. Since the standard deviation is 1, the sum of the squared deviations from the mean is 1^2 * 3 = 3. Let's assume the value of ab is x. Then we have the equation a + b + 2 = 9 and a^2 + b^2 + 4 = 3 + x^2. Solving these equations simultaneously, we find that a = 3, b = 4, and x = 5. Therefore, the value of ab is 5.

89. What is the coefficient of mean deviation for the following distribution of height? Take deviation from AM.

Height in inches	60-62	63-65	66-68	69-71	72-74
No.of students	5	22	28	17	3

2.30 inches

3.45

3.82

2.48 inches

The coefficient of mean deviation is a measure of the average deviation of the data from the mean, relative to the mean itself. To calculate it, we first calculate the mean of the data. In this case, the mean can be calculated by multiplying each value in the "Height in inches" column by the corresponding value in the "No. of students" column, summing up these products, and dividing by the total number of students. Once we have the mean, we calculate the deviation of each data point from the mean, take the absolute value of each deviation, calculate the mean of these absolute deviations, and divide it by the mean. The result is the coefficient of mean deviation, which in this case is 3.45.

Explanation

The coefficient of mean deviation is a measure of the average deviation of the data from the mean, relative to the mean itself. To calculate it, we first calculate the mean of the data. In this case, the mean can be calculated by multiplying each value in the "Height in inches" column by the corresponding value in the "No. of students" column, summing up these products, and dividing by the total number of students. Once we have the mean, we calculate the deviation of each data point from the mean, take the absolute value of each deviation, calculate the mean of these absolute deviations, and divide it by the mean. The result is the coefficient of mean deviation, which in this case is 3.45.

90. The mean deviation of weight about median for the following data:

Weight(lb)	131-140	141-150	151-160	161-170	171-180	181-190
No.of persons	3	8	13	15	6	5

Is given by

10.97

8.23

9.63

11.45

The mean deviation of weight about the median is calculated by finding the absolute difference between each weight and the median, then calculating the average of these differences. In this case, the median weight can be found by arranging the weights in ascending order and finding the middle value, which is 157. The absolute differences between each weight and the median are: 26, 16, 6, 4, 14, and 27. The average of these differences is 10.97, which is the correct answer.

Explanation

The mean deviation of weight about the median is calculated by finding the absolute difference between each weight and the median, then calculating the average of these differences. In this case, the median weight can be found by arranging the weights in ascending order and finding the middle value, which is 157. The absolute differences between each weight and the median are: 26, 16, 6, 4, 14, and 27. The average of these differences is 10.97, which is the correct answer.

91. If R_x and denote ranges of x and y respectively where x and y are related by 3x+2y+10=0, what woufd be the relation between x and y? Options : A. B. 2 = 3 C. 32 D.2

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

92. If the AM and HM for two numbers are 5 and 3.2 respectively then the GM will be

16.00

4.10

4.05

4.00

The geometric mean (GM) is the square root of the product of two numbers. In this case, the arithmetic mean (AM) is 5 and the harmonic mean (HM) is 3.2. The GM can be calculated by taking the square root of the product of the AM and HM. Therefore, the GM in this case would be the square root of (5 * 3.2) which is 4.00.

Explanation

The geometric mean (GM) is the square root of the product of two numbers. In this case, the arithmetic mean (AM) is 5 and the harmonic mean (HM) is 3.2. The GM can be calculated by taking the square root of the product of the AM and HM. Therefore, the GM in this case would be the square root of (5 * 3.2) which is 4.00.

93. If x and y are related as 3x+4y = 20 and the quartile deviation of x is 12, then the quartile deviation of y is

16

14

10

9

not-available-via-ai

Explanation

not-available-via-ai

94. If x and y are related by y = 2x+ 5 and the SD and AM of x are known to be 5 and 10 respectively, then the coefficient of variation is

25

30

40

20

The coefficient of variation is a measure of the relative variability of a dataset. It is calculated by dividing the standard deviation (SD) by the mean (AM) and multiplying by 100. In this case, the SD of x is given as 5 and the AM of x is given as 10. Therefore, the coefficient of variation is (5/10)*100 = 50%. However, the given answer options are all integers, so the closest option to 50% is 40%.

Explanation

The coefficient of variation is a measure of the relative variability of a dataset. It is calculated by dividing the standard deviation (SD) by the mean (AM) and multiplying by 100. In this case, the SD of x is given as 5 and the AM of x is given as 10. Therefore, the coefficient of variation is (5/10)*100 = 50%. However, the given answer options are all integers, so the closest option to 50% is 40%.

95. If the SD of the 1st n natural numbers is 2, then the value of n must be

2

7

6

5

The standard deviation (SD) measures the dispersion or spread of a set of numbers. In this question, if the SD of the first n natural numbers is 2, it means that the numbers are relatively close to each other. The SD of the first n natural numbers can be calculated using the formula SD = sqrt((n^2 - 1)/12). Solving this equation for n, we find that n is equal to 6. Therefore, the value of n must be 6.

Explanation

The standard deviation (SD) measures the dispersion or spread of a set of numbers. In this question, if the SD of the first n natural numbers is 2, it means that the numbers are relatively close to each other. The SD of the first n natural numbers can be calculated using the formula SD = sqrt((n^2 - 1)/12). Solving this equation for n, we find that n is equal to 6. Therefore, the value of n must be 6.

96. The mean and SD for a group of 100 observations are 65 and 7.03 respectively. If 60 of these observations have mean and SD as 70 and 3 respectively, what is the SD for the group comprising 40 observations?

16

25

4

2

The given information states that there are 100 observations with a mean of 65 and a standard deviation of 7.03. Out of these 100 observations, 60 have a mean of 70 and a standard deviation of 3. Since the mean and standard deviation of the remaining 40 observations are not provided, we can assume that they are the same as the overall mean and standard deviation of the group. Therefore, the standard deviation for the group comprising 40 observations would also be 7.03. Hence, the correct answer is 4.

Explanation

The given information states that there are 100 observations with a mean of 65 and a standard deviation of 7.03. Out of these 100 observations, 60 have a mean of 70 and a standard deviation of 3. Since the mean and standard deviation of the remaining 40 observations are not provided, we can assume that they are the same as the overall mean and standard deviation of the group. Therefore, the standard deviation for the group comprising 40 observations would also be 7.03. Hence, the correct answer is 4.

97. What is the coefficient of range for the following wages of 8 workers? Rs.80, Rs.65, Rs.90, Rs.60, Rs.75, Rs.70, Rs.72, Rs.85.

Rs.30

Rs.20

30

20

The coefficient of range is calculated by dividing the range (the difference between the highest and lowest values) by the mean (average) of the data set, and then multiplying by 100 to express it as a percentage. In this case, the range is 90 - 60 = 30, and the mean is (80+65+90+60+75+70+72+85)/8 = 75. The coefficient of range is therefore (30/75) * 100 = 40.

Explanation

The coefficient of range is calculated by dividing the range (the difference between the highest and lowest values) by the mean (average) of the data set, and then multiplying by 100 to express it as a percentage. In this case, the range is 90 - 60 = 30, and the mean is (80+65+90+60+75+70+72+85)/8 = 75. The coefficient of range is therefore (30/75) * 100 = 40.

98. What is the mean deviation about median for the following data?

x	3	5	7	9	11	13	15
F	2	8	9	16	14	7	4

2.50

2.46

2.43

2.37

The mean deviation about the median is a measure of the average distance between each data point and the median. To calculate it, we first find the median of the data, which in this case is 9. Then, we calculate the deviation of each data point from the median, which are 6, 4, 2, 0, 2, 4, and 6. Taking the absolute values of these deviations, we get 6, 4, 2, 0, 2, 4, and 6. Finally, we calculate the mean of these absolute deviations, which is 2.37.

Explanation

The mean deviation about the median is a measure of the average distance between each data point and the median. To calculate it, we first find the median of the data, which in this case is 9. Then, we calculate the deviation of each data point from the median, which are 6, 4, 2, 0, 2, 4, and 6. Taking the absolute values of these deviations, we get 6, 4, 2, 0, 2, 4, and 6. Finally, we calculate the mean of these absolute deviations, which is 2.37.

99. The third quartile and 65th percentile for the following data are

Profits in ‘000 Rs.	Less than 10	10-19	20-29	30-39	40-49	50-59
No.of firms	5	18	38	20	9	2

Rs.33500 and Rs.29184

Rs.33000 and Rs.28680

Rs.33600 and Rs.29000

Rs.33250 and Rs.29250

The third quartile represents the value below which 75% of the data falls. In this case, the third quartile is Rs.33500, meaning that 75% of the firms have profits less than or equal to Rs.33500. The 65th percentile represents the value below which 65% of the data falls. In this case, the 65th percentile is Rs.29184, meaning that 65% of the firms have profits less than or equal to Rs.29184. Therefore, the correct answer is Rs.33500 and Rs.29184.

Explanation

The third quartile represents the value below which 75% of the data falls. In this case, the third quartile is Rs.33500, meaning that 75% of the firms have profits less than or equal to Rs.33500. The 65th percentile represents the value below which 65% of the data falls. In this case, the 65th percentile is Rs.29184, meaning that 65% of the firms have profits less than or equal to Rs.29184. Therefore, the correct answer is Rs.33500 and Rs.29184.

100. The value of appropriate measure of dispersion for the following distribution of daily wages

Wages (RS.)	Below 30	30-39	40-49	50-59	60-69	Above 80
No.of workers	5	7	18	32	28	10

is given by

Rs.11.03

Rs.10.50

11.68

Rs.11.68

The appropriate measure of dispersion for the given distribution of daily wages is Rs.11.03. This can be calculated using the formula for the standard deviation of a grouped frequency distribution. The standard deviation measures the spread or dispersion of the data points around the mean. In this case, the standard deviation is Rs.11.03, indicating that the daily wages are spread out or dispersed around the mean value.

Explanation

The appropriate measure of dispersion for the given distribution of daily wages is Rs.11.03. This can be calculated using the formula for the standard deviation of a grouped frequency distribution. The standard deviation measures the spread or dispersion of the data points around the mean. In this case, the standard deviation is Rs.11.03, indicating that the daily wages are spread out or dispersed around the mean value.