## 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 Central Location

#### BASIC

1. For the sample data set $\{1,2,6\}$ find

a. $\Sigma x$

b. $\Sigma x^{2}$

c. $\Sigma(x-3)$

d. $\Sigma(x-3)^{2}$

1. Find the mean, the median, and the mode for the sample

$\begin{array}{llll}1 & 2 & 3 & 4\end{array}$

1. Find the mean, the median, and the mode for the sample

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

1. Find the mean, the median, and the mode for the sample data represented by the table

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

1. Create a sample data set of size $n=3$ for which the mean $\bar{x}$ is greater than the median $\widetilde{x}$.

1. Create a sample data set of size $n=4$ for which the mean $\bar{x}$, the median $\widetilde{x}$, and the mode are all identical.

#### APPLICATIONS

1. Find the mean and the median for the LDL cholesterol level in a sample of ten heart patients.

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

1. Find the mean, the median, and the mode for the number of vehicles owned in a survey of 52 households.

1. Five laboratory mice with thymus leukemia are observed for a predetermined period of 500 days. After 450 days, three mice have died, and one of the remaining mice is sacrificed for analysis. By the end of the observational period, the last remaining mouse still survives. The recorded survival times for the five mice are

$\begin{array}{lllll}222 & 421 & 378 & 450^{*} & 500^{*}\end{array}$

where * indicates that the mouse survived for at least the given number of days but the exact value of the observation is unknown.

1. Can you find the sample mean for the data set? If so, find it. If not, explain why not.
2. Can you find the sample median for the data set? If so, find it. If not, explain why not.
1. Cordelia records her daily commute time to work each day, to the nearest minute, for two months, and obtains the following data.

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

1. Based on the frequencies, do you expect the mean and the median to be about the same or markedly different, and why?
2. Compute the mean, the median, and the mode.
1. A man tosses a coin repeatedly until it lands heads and records the number of tosses required. (For example, if it lands heads on the first toss he records a 1; if it lands tails on the first two tosses and heads on the third he records a 3). The data are shown.

$\begin{array}{c|cccccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline f & 384 & 208 & 98 & 56 & 28 & 12 & 8 & 2 & 3 & 1 \end{array}$

1. Find the mean of the data.
2. Find the median of the data.

1. Show that no matter what kind of average is used (mean, median, or mode) it is impossible for all members of a data set to be above average.

1. Begin with the following set of data, call it Data Set I.

$\begin{array}{lllllllllll}5 & -2 & 6 & 14 & -3 & 0 & 1 & 4 & 3 & 2 & 5\end{array}$

1. Compute the mean, median, and mode.
2. Form a new data set, Data Set II, by adding 3 to each number in Data Set I. Calculate the mean, median, and mode of Data Set II.
3. Form a new data set, Data Set III, by subtracting 6 from each number in Data Set I. Calculate the mean, median, and mode of Data Set III.
4. Comparing the answers to parts (a), (b), and (c), can you guess the pattern? State the general principle that you expect to be true.

#### LARGE DATA SET EXERCISES

Note: All of the data sets associated with these questions are missing, but the questions themselves are included here for reference.

1. Large Data Set 1 lists the SAT scores of 1,000 students.
1. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean $\mu$.
2. Regard the first 25 observations as a random sample drawn from this population. Compute the sample mean $\bar{x}$ and compare it to $\mu$.
3. Regard the next 25 observations as a random sample drawn from this population. Compute the sample mean $\bar{x}$ and compare it to $\mu$.

1. Large Data Sets 7, 7A, and 7B list the survival times in days of 140 laboratory mice with thymic leukemia from onset to death.
1. Compute the mean and median survival time for all mice, without regard to gender.
2. Compute the mean and median survival time for the 65 male mice (separately recorded in Large Data Set 7A).
3. Compute the mean and median survival time for the 75 female mice (separately recorded in Large Data Set 7B).