3.2 Measures Of Central Tendency

1. What is the mode of the following set of data? 5, 7, 7, 8, 12, 16

The mode of a set of data is the value that appears most frequently. In this set of data, the number 7 appears twice, which is more than any other number. Therefore, the mode of the set is 7.

Explanation

The mode of a set of data is the value that appears most frequently. In this set of data, the number 7 appears twice, which is more than any other number. Therefore, the mode of the set is 7.

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2. What is the median of the following set of data? 5, 7, 7, 8, 12, 16

The median of a set of data is the middle value when the data is arranged in ascending or descending order. In this case, the data set is already in ascending order: 5, 7, 7, 8, 12, 16. Since there is an even number of values, the median is the average of the two middle values, which are 7 and 8. Therefore, the median is 7.5.

Explanation

The median of a set of data is the middle value when the data is arranged in ascending or descending order. In this case, the data set is already in ascending order: 5, 7, 7, 8, 12, 16. Since there is an even number of values, the median is the average of the two middle values, which are 7 and 8. Therefore, the median is 7.5.

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3. What is the mean of the following set of data? (round to 1 decimal place) 5, 7, 7, 8, 12, 16

The mean of a set of data is the average value, calculated by summing all the values and dividing by the total number of values. In this case, the sum of the data set is 55, and there are 6 values. Dividing 55 by 6 gives us 9.2, which is the mean of the given set of data.

Explanation

The mean of a set of data is the average value, calculated by summing all the values and dividing by the total number of values. In this case, the sum of the data set is 55, and there are 6 values. Dividing 55 by 6 gives us 9.2, which is the mean of the given set of data.

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4. In a distribution that is skewed left, the mean is...

Less than the median

Greater than the median

Equal to the median

It depends

In a distribution that is skewed left, the mean is less than the median. This is because the skewness is caused by a few extremely low values that pull the mean towards the left side of the distribution. The median, on the other hand, is less affected by extreme values and represents the middle value of the data set. Therefore, in a left-skewed distribution, the mean is lower than the median.

Explanation

In a distribution that is skewed left, the mean is less than the median. This is because the skewness is caused by a few extremely low values that pull the mean towards the left side of the distribution. The median, on the other hand, is less affected by extreme values and represents the middle value of the data set. Therefore, in a left-skewed distribution, the mean is lower than the median.

5. For qualitative data, which measure of central tendency is most appropriate to use?

Mean

Median

Mode

The mode is the most appropriate measure of central tendency for qualitative data because it represents the value that occurs most frequently in the data set. Since qualitative data does not have numerical values, the mean and median cannot be calculated. Therefore, the mode provides the best representation of the central value in qualitative data.

Explanation

The mode is the most appropriate measure of central tendency for qualitative data because it represents the value that occurs most frequently in the data set. Since qualitative data does not have numerical values, the mean and median cannot be calculated. Therefore, the mode provides the best representation of the central value in qualitative data.

6. If there are outliers in your data, which measure of central tendency is the most appropriate to use?

Mean

Median

Mode

When there are outliers in the data, the most appropriate measure of central tendency to use is the median. The median is not affected by extreme values or outliers because it only considers the middle value in the data set. This makes it a more robust measure compared to the mean, which can be heavily influenced by outliers. The mode, on the other hand, is not affected by outliers either, but it only represents the most frequently occurring value and may not provide a representative measure of central tendency for the entire data set.

Explanation

When there are outliers in the data, the most appropriate measure of central tendency to use is the median. The median is not affected by extreme values or outliers because it only considers the middle value in the data set. This makes it a more robust measure compared to the mean, which can be heavily influenced by outliers. The mode, on the other hand, is not affected by outliers either, but it only represents the most frequently occurring value and may not provide a representative measure of central tendency for the entire data set.

7. The MDM4U mark breakdown is as follows: 40% tests, 20% assignments, 10% ISU, 30% exam. If you have the following averages for each category, what mark would you receive at the end of the course? Round the PERCENT to 1 decimal place. Tests - 87% Assignments - 90% ISU - 80% Exam - 71%

The mark breakdown for the course is given as 40% tests, 20% assignments, 10% ISU, and 30% exam. To calculate the final mark, we need to find the weighted average of each category.

The weighted average for tests is 87% * 40% = 34.8%
The weighted average for assignments is 90% * 20% = 18%
The weighted average for ISU is 80% * 10% = 8%
The weighted average for the exam is 71% * 30% = 21.3%

Adding up these weighted averages, we get 34.8% + 18% + 8% + 21.3% = 82.1%. Therefore, the final mark for the course would be 82.1%.

Explanation

The mark breakdown for the course is given as 40% tests, 20% assignments, 10% ISU, and 30% exam. To calculate the final mark, we need to find the weighted average of each category.

The weighted average for tests is 87% * 40% = 34.8%
The weighted average for assignments is 90% * 20% = 18%
The weighted average for ISU is 80% * 10% = 8%
The weighted average for the exam is 71% * 30% = 21.3%

Adding up these weighted averages, we get 34.8% + 18% + 8% + 21.3% = 82.1%. Therefore, the final mark for the course would be 82.1%.

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8. A sample of hockey players were asked how old they were when they learned to skate. The results were then reported in a frequency distribution. Calculate the mean age to one decimal place.

Age	0-4	5-9	10-14	15-19	20-24	25-29
Frequency	6	8	3	5	2	1

The mean age is calculated by taking the sum of the products of each age category and its corresponding frequency, and then dividing that sum by the total number of players. In this case, the calculation would be (4.5 * 6 + 7 * 8 + 12.5 * 3 + 17 * 5 + 22 * 2 + 27 * 1) / (6 + 8 + 3 + 5 + 2 + 1) = 10.4.

Explanation

The mean age is calculated by taking the sum of the products of each age category and its corresponding frequency, and then dividing that sum by the total number of players. In this case, the calculation would be (4.5 * 6 + 7 * 8 + 12.5 * 3 + 17 * 5 + 22 * 2 + 27 * 1) / (6 + 8 + 3 + 5 + 2 + 1) = 10.4.

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