Measures of Central Location

This section elaborates on mean, median, and mode at the population level and sample level. This section also contains many interesting examples of range, variance, and standard deviation. Complete the exercises and check your answers.

Measures of Variability

EXERCISES

BASIC

1. Find the range, the variance, and the standard deviation for the following sample.

1 2 3 4

Find the range, the variance, and the standard deviation for the following sample.

$\begin{array}{llll}2 & 1 & 2 & 7\end{array}$

5. Find the range, the variance, and the standard deviation for the sample represented by the data frequency table.

$\begin{array}{l|lll} x & 1 & 2 & 7 \\ \hline f & 1 & 2 & 1 \end{array}$

APPLICATIONS

7. Find the range, the variance, and the standard deviation for the sample of ten IQ scores randomly selected from a school for academically gifted students.

$\begin{array}{lllll}132 & 162 & 133 & 145 & 148 \\ 139 & 147 & 160 & 150 & 153\end{array}$

ADDITIONAL EXERCISES

9. Consider the data set represented by the table

$\begin{array}{c|ccccccc} x & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\ \hline f & 3 & 4 & 16 & 12 & 6 & 2 & 1 \end{array}$

Use the frequency table to find that $\Sigma x=1256$ and $\Sigma x^{2}=35,926$ .
Use the information in part (a) to compute the sample mean and the sample standard deviation.

11. A random sample of 49 invoices for repairs at an automotive body shop is taken. The data are arrayed in the stem and leaf diagram shown. (Stems are thousands of dollars, leaves are hundreds, so that for example the largest observation is 3,800).

$\begin{array}{l|lllllllllllll}\ 3 & 5 & 6 & 8 & & & & & & \\ 3 & 0 & 0 & 1 & 1 & 2 & 4 & & & & \\ 2 & 5 & 6 & 6 & 7 & 7 & 8 & 8 & 9 & 9 & \\ 2 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 & & & \\ 1 & 5 & 5 & 5 & 6 & 6 & 7 & 7 & 7 & 8 & 8 & 9 \\ 1 & 0 & 0 & 1 & 3 & 4 & 4 & 4 & & & \\\ 0 & 5 & 6 & 8 & 8 & & & & & \\ 0 & 4 & & & & & & & \end{array}$

Compute the mean, median, and mode.
Compute the range.
Compute the sample standard deviation.

13. A data set consisting of 25 measurements has standard deviation $\mathrm{0}$ . One of the measurements has value 17. What are the other 24 measurements?

15. Create a sample data set of size $\mathrm{n = 3}$ for which the sample variance is $\mathrm{0}$ and the sample mean is $\mathrm{1}$ .

17. The sample $\{-1,0,1\}$ has mean $\bar{x}=0$ and standard deviation $s=1$ . Create a sample data set of size $n=3$ for which $\bar{x}=0$ and the standard deviation $s$ is less than $\mathrm{1}$ .

LARGE DATA SET EXERCISES

19. Large Data Set 1 lists the SAT scores and GPAs of 1,000 students.

http://www.gone.2012books.lardbucket.org/sites/all/files/data1.xls

Compute the range and sample standard deviation of the 1,000 SAT scores.
Compute the range and sample standard deviation of the 1,000 GPAs.

21. a. Regard the data as arising from a census of all freshman at a small college at the end of their first academic year of college study, in which the GPA of every such person was measured. Compute the population range and population standard deviation $\sigma$ .

b. Regard the first 25 observations as a random sample drawn from this population. Compute the sample range and sample standard deviation $s$ and compare them to the population range and $\sigma$ .

c. Regard the next 25 observations as a random sample drawn from this population. Compute the sample range and sample standard deviation $s$ and compare them to the population range and $\sigma$ .