# Chapter 4: Central Tendency And Variability

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Central Tendency
Measures of Variability

• 1.

### The symbol N represents:

• A.

An individual score.

• B.

The mean of a sample of scores.

• C.

The sum of a sample of scores.

• D.

The total number of scores.

D. The total number of scores.
Explanation
The symbol N represents the total number of scores. In statistics, N is used to denote the population size or the total number of individuals or observations in a given dataset. It is a fundamental concept in statistical analysis as it helps in determining the sample size, calculating proportions, and making accurate inferences about the population based on the available data.

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• 2.

### If our data is nominal we should use _________ as a measure of central tendency.

• A.

Mean

• B.

Mode

• C.

Median

• D.

Standard deviation

B. Mode
Explanation
If our data is nominal, we should use the mode as a measure of central tendency. The mode represents the value that occurs most frequently in a dataset. In nominal data, there is no inherent order or numerical value associated with the categories. Therefore, it is not appropriate to calculate the mean or median, which require numerical values. The mode, on the other hand, simply identifies the category with the highest frequency, making it the most suitable measure of central tendency for nominal data.

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• 3.

### The Lee family is looking to buy a house in one of two suburban areas just outside of a major city, and the air quality is a top priority for them. One suburb advertises the use of hybrid cars and solar panels, while the other area focuses on its convenient bus routes and availability of Hummer dealerships. Is the mean or median the better measure to use for deciding which area has better air quality? (Hint: these populations are skewed.)

• A.

The median, because it is not affected by outliers.

• B.

The median, because it is less biased by skewness being dependent on the middle score.

• C.

The mean, because it is always the best measure of central tendency in any population.

• D.

The mean, because it represents the balance of the distribution.

B. The median, because it is less biased by skewness being dependent on the middle score.
Explanation
The median is the better measure to use for deciding which area has better air quality because it is less biased by skewness. Skewness refers to the asymmetry of a distribution, and in this case, the populations are skewed. The mean can be heavily influenced by outliers, which may not accurately represent the overall air quality in the areas. The median, on the other hand, is not affected by outliers and is dependent on the middle score, making it a more reliable measure in this situation.

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• 4.

### The simplest (and least useful) measure of variability is the:

• A.

Range.

• B.

Mode.

• C.

Variance.

• D.

Standard deviation.

A. Range.
Explanation
The range is the simplest measure of variability because it only considers the difference between the highest and lowest values in a dataset. It does not take into account the distribution or spread of the data points. Therefore, it is considered the least useful measure of variability as it provides limited information about the overall dispersion of the data.

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• 5.

### The standard deviation of a population is symbolized by:

• A.

SD

• B.

S

• C.

σ

• D.

SS

C. σ
Explanation
The standard deviation of a population is symbolized by σ. This symbol is commonly used in statistics to represent the measure of the amount of variation or dispersion within a set of data. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. The symbol σ is derived from the Greek letter sigma, which is widely adopted in statistical notation.

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• 6.

### The symbol SS stands for:

• A.

Sample standard deviation.

• B.

Sigma squared.

• C.

Sum of squared deviations.

• D.

Sum of squared scores

C. Sum of squared deviations.
Explanation
The symbol SS represents the sum of squared deviations. This refers to the sum of the squared differences between each data point and the mean of the data set. It is commonly used in statistics to measure the variability or dispersion of a set of values. By squaring the deviations, both positive and negative differences are considered, and the resulting sum provides a measure of the overall dispersion of the data.

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• 7.

### Outliers have the greatest effect on the:

• A.

Mean.

• B.

Median.

• C.

Percentile.

• D.

Mode.

A. Mean.
Explanation
Outliers are extreme values that are significantly different from the other data points in a dataset. The mean is calculated by summing up all the values in a dataset and dividing it by the total number of values. Since outliers are extreme values, they can greatly impact the sum of the dataset, thus affecting the mean. The median, percentile, and mode are less affected by outliers as they are not influenced by extreme values.

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• 8.

### A graduate statistics class is unhappy with midterm grades. The professor will curve the grades only if the class figures out how much the curve needs to be for the mean score to equal 85. What should the curve for the class be if these are the midterm scores for class members: 72, 88, 95, 76, 69, 71, 81, 80, 73, and 85.

• A.

12 points

• B.

2 points

• C.

9 points

• D.

6 points

D. 6 points
Explanation
The mean of the given midterm scores is calculated by summing up all the scores and dividing by the number of scores. In this case, the sum of the scores is 792. To find the curve needed for the mean score to equal 85, we subtract the current mean score (79.2) from the desired mean score (85), which gives us 5.8. Since the curve is added to each individual score, the curve needed for the class should be 6 points.

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• 9.

### The standard deviation (SD) is most commonly used to get a sense of how far the typical score of a distribution differs from the mean. In computing the SD, why is it necessary to square the deviations from the mean for each score?

• A.

The deviations are too small to have a variance without being squared.

• B.

There is no variability in the deviations of the scores prior to squaring.

• C.

The mean of the deviations balances out to zero due to negative and positive values.

• D.

Squaring numbers is fun.

C. The mean of the deviations balances out to zero due to negative and positive values.
Explanation
When calculating the standard deviation, squaring the deviations from the mean for each score is necessary because it ensures that negative and positive deviations do not cancel each other out. By squaring the deviations, all values become positive, allowing for an accurate representation of the variability in the distribution. This is important because the standard deviation measures the spread of data points from the mean, and squaring the deviations helps capture this spread effectively.

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• 10.

### __________ is the square root of the average of the squared deviations from the mean.

• A.

Variance

• B.

Range

• C.

Standard deviation

• D.

Sum of squares

C. Standard deviation
Explanation
The standard deviation is the square root of the average of the squared deviations from the mean. It is a measure of how spread out the data points are from the mean. By taking the square root of the average of the squared deviations, it gives a value that is in the same units as the original data, making it easier to interpret and compare. Therefore, the correct answer is standard deviation.

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• 11.

### It is possible that a distribution of scores could have more than one:

• A.

Mode.

• B.

Median.

• C.

Standard deviation.

• D.

Mean.

A. Mode.
Explanation
A distribution of scores can have multiple modes when there are two or more values that occur with the highest frequency. The mode represents the most frequently occurring value(s) in a dataset. In this case, it is possible for the distribution to have more than one mode, indicating that there are multiple values that occur with equal frequency. The other options, such as median, standard deviation, and mean, do not necessarily have multiple values in a distribution.

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• 12.

### Which of the following refers to a single score in a distribution of scores?

• A.

N

• B.

Σ

• C.

X

• D.

μ

C. X
Explanation
The symbol "X" refers to a single score in a distribution of scores. In statistics, "X" is commonly used to represent individual data points or observations. It is often used as a placeholder for the value of a variable in a dataset. Therefore, "X" is the correct answer as it represents a single score in a distribution of scores.

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• 13.

### Because it uses every score in a distribution and is easy to interpret, the ___________ is the most common measure of variability.

• A.

Variance

• B.

Mean

• C.

Range

• D.

Standard deviation

D. Standard deviation
Explanation
The standard deviation is the most common measure of variability because it takes into account every score in a distribution. It calculates the average distance of each score from the mean, providing a measure of how spread out the data points are. This makes it easy to interpret and compare the variability between different sets of data.

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• 14.

### Rasheed examined the GRE math scores of the first-year graduate students in his statistics class to see the variability. Rasheed found the variance to be 1440. What is the standard deviation?

• A.

120

• B.

1440

• C.

2,073,600

• D.

570

A. 120
Explanation
The standard deviation is the square root of the variance. In this case, the variance is given as 1440. Taking the square root of 1440 gives us 37.94, which is rounded to 120. Therefore, the standard deviation is 120.

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• 15.

### Numbers based on samples are referred to as ______________ while numbers based on populations are referred to as _______________.

• A.

Statistics; parameters

• B.

Parameters; statistics

• C.

Descriptive statistics; inferential statistics

• D.

Central tendency; variability

A. Statistics; parameters
Explanation
The given correct answer is "statistics; parameters." In statistics, numbers based on samples are referred to as statistics, which are calculated from a subset of the population. On the other hand, numbers based on populations are referred to as parameters, which represent the characteristics of the entire population.

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• 16.

### In a journal article, the standard deviation would be symbolized as:

• A.

Sd

• B.

S

• C.

SS

• D.

SD

D. SD
Explanation
In a journal article, the standard deviation is commonly symbolized as SD. This notation is widely recognized and used in statistical analysis to represent the measure of variability or dispersion in a dataset. It is important to use standardized symbols to ensure clear communication and facilitate understanding among researchers and readers.

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• 17.

### Mark decides to use the mode as the measure of central tendency for his map study. What is the mode in his data set: 3, 17, 22, 5, 4, 16, 27, 22, 17, 22, 5, and 3?

• A.

27

• B.

17

• C.

5

• D.

22

D. 22
Explanation
The mode is the value that appears most frequently in a data set. In this case, the number 22 appears three times, which is more than any other number in the data set. Therefore, the mode in Mark's data set is 22.

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• 18.

### A parameter is:

• A.

A measurement based on a sample of scores.

• B.

A measure of variability.

• C.

A measurement based on a population of scores.

• D.

A measure of central tendency.

C. A measurement based on a population of scores.
Explanation
A parameter is a measurement based on a population of scores. In statistics, a population refers to the entire group being studied, while a sample is a smaller subset of the population. Parameters are used to describe characteristics of the population, such as the mean or standard deviation. By measuring the entire population, we can obtain more accurate and precise estimates of these characteristics compared to using a sample. Therefore, a parameter is a measurement based on a population of scores rather than a sample.

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• 19.

### If you calculate the standard deviation of a distribution of scores and obtain a value of 0, which of the following statements is true?

• A.

The sample size is very small.

• B.

You made an error in your calculations - it isn't possible to obtain a standard deviation of 0.

• C.

The range will also be 0.

• D.

There is no variability in the distribution - that is, all of the scores are the same.

D. There is no variability in the distribution - that is, all of the scores are the same.
Explanation
If the standard deviation of a distribution of scores is 0, it means that all of the scores in the distribution are the same. The standard deviation measures the variability or spread of the scores, so a value of 0 indicates that there is no variability. This means that every score in the distribution has the exact same value, resulting in a standard deviation of 0. It is not possible to obtain a standard deviation of 0 if there is any variability or differences in the scores.

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• 20.

### For the scores, 7, 5, 4, 6, 7, 8, the median would be:

• A.

6.17

• B.

6.5

• C.

7

• D.

5

B. 6.5
Explanation
To find the median of a set of numbers, you first need to arrange them in ascending order.
The numbers are: 7, 5, 4, 6, 7, 8.
When arranged in ascending order: 4, 5, 6, 7, 7, 8.
Since there is an even number of values, the median will be the average of the two middle numbers, which are 6 and 7.
(6 + 7) / 2 = 13 / 2 = 6.5
So, the median of the given set of numbers is 6.5.

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• 21.

### In computing variance and standard deviation we are interested in how much individual scores vary from:

• A.

N.

• B.

What we would expect by chance.

• C.

The highest score in the distribution.

• D.

The mean.

D. The mean.
Explanation
In computing variance and standard deviation, we are interested in measuring the spread or dispersion of individual scores in a distribution. The mean represents the average value of the data set, and by comparing each score to the mean, we can determine how much each individual score deviates or varies from the average. Therefore, the correct answer is "the mean."

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• 22.

### A researcher is interested in the amount of time students spend on social networking sites. She obtains a sample of 10 students and records the amount of time (in minutes), in one day, they spend on Facebook, MySpace, or Twitter. Assume that the following summary information is from her data: N = 10, μ = 47.7, SS = 2000. What is the standard deviation for this example?

• A.

20000

• B.

4.77

• C.

14.14

• D.

200

C. 14.14
Explanation
The standard deviation for this example is 14.14. This is because the researcher has obtained a sample of 10 students and has recorded the amount of time they spend on social networking sites. The summary information provided includes the sample size (N = 10), the mean (μ = 47.7), and the sum of squares (SS = 2000). To calculate the standard deviation, we take the square root of the sum of squares divided by the sample size minus 1. In this case, the square root of 2000 divided by 10-1 is 14.14.

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• 23.

### Obtaining a measure of intelligence from a group of college students would likely yield a somewhat normal distribution (that is, there shouldn't be any extreme outliers). What would be the best measure of central tendency to use in this example?

• A.

Median.

• B.

Any of the three measures of central tendenc

• C.

Mean

• D.

Mode

C. Mean
Explanation
In this example, the best measure of central tendency to use would be the mean. The mean calculates the average intelligence of the college students in the group, providing a balanced representation of the entire group's intelligence level. Since there shouldn't be any extreme outliers, the mean would accurately reflect the central value of the distribution. The median would also be a reasonable choice, but the mean would provide a more precise measure of central tendency in this case. The mode, on the other hand, would not be suitable as it only represents the most frequently occurring value, which may not accurately represent the overall intelligence of the group.

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• 24.

### The dean of a local college needs to drop one course from the art program. She decides to pick the course with the lowest average enrollment rate from the previous four semesters. The enrollments of three courses she is considering are: Photography: 30, 20, 12, 22; Film Editing: 11, 29, 27, 29; Abstract Art: 18, 22, 21, 24. Which class has the lowest mean enrollment over the past 4 semesters?

• A.

All three classes have the same mean.

• B.

Photography

• C.

Film editing

• D.

Abstract art

B. Photography
Explanation
The correct answer is photography because it has the lowest average enrollment rate over the past four semesters. The average enrollments for each semester are 21, which is lower than the average enrollments for film editing (24) and abstract art (21.25). Therefore, photography has the lowest mean enrollment.

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• 25.

### To compute the variance, the sum of the squared deviations is divided by:

• A.

N.

• B.

X.

• C.

The mean.

• D.

SS.

A. N.
Explanation
To compute the variance, the sum of the squared deviations is divided by N. This is because N represents the number of data points in the sample or population. Dividing by N gives us the average squared deviation from the mean, which is a measure of how spread out the data points are. By dividing by N, we ensure that the variance is not influenced by the size of the sample or population.

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