# 3.10: Measures of Central Tendency and Variability Exercises

• • Michelle Oja
• Taft College
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## Exercise $$\PageIndex{1}$$

If the mean time to respond to a stimulus is much higher than the median time to respond, what can you say about the shape of the distribution of response times?

If the mean is higher, that means it is farther out into the right-hand tail of the distribution. Therefore, we know this distribution is positively skewed.

## Exercise $$\PageIndex{2}$$

Your younger sibling comes home one day after taking a science test. He says that some- one at school told him that “60% of the students in the class scored above the median test grade.” What is wrong with this statement?

What if they had said “60% of the students scored above the mean?”

The median is defined as the value with 50% of scores above it and 50% of scores below it; therefore, 60% of score cannot fall above the median. If 60% of scores fall above the mean, that would indicate that the mean has been pulled down below the value of the median, which means that the distribution is negatively skewed.

## Exercise $$\PageIndex{4}$$

Compute the population mean and population standard deviation for the following scores: 5, 7, 8, 3, 4, 4, 2, 7, 1, 6

Remember,you can use the Sum of Squares table.

$$\mu=4.80, \sigma^{2}=2.36$$

## Exercise $$\PageIndex{5}$$

If two normal distributions have exactly the same mean, but one has a standard deviation of 20 and the other has a standard deviation of 10, how would the shapes of the two distributions compare?

If both distributions are normal, then they are both symmetrical, and having the same mean causes them to overlap with one another. The distribution with the standard deviation of 10 will be narrower than the other distribution.

## Exercise $$\PageIndex{6}$$

Use the following dataset for the computations below:

Table $$\PageIndex{1}$$- Sample Data for Two Variables
$$\mathrm{X}$$ $$\mathrm{Y}$$
2 8
3 8
7 4
5 1
9 4
1. $$\sum \mathrm{X}$$
2. $$\sum \mathrm{Y}^2$$
3. $$\sum \mathrm{XY}$$
4. $$(\sum \mathrm{Y})^2$$