## 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

1. a. $\mathrm{9}$.
b. $\mathrm{41}$.
c. $\mathrm{0}$.
d. $\mathrm{14}$.

1. $\bar{x}=2.5, \widetilde{x}=2.5$, mode $=\{1,2,3,4\}$

1. $\bar{x}=3, \widetilde{x}=2$, mode $=2$

1. $\bar{x}=3, \widetilde{x}=2$, mode $=2$

1. $\{0,0,3\}$.

1. $\{0,1,1,2\}$.

1. $\bar{x}=146.9, \widetilde{x}=147.5$

1. $\bar{x}=2.6, \widetilde{x}=2$, mode $=2$

1. $\bar{x}=48.96, \widetilde{x}=49$, mode $=49$

1. a. No, the survival times of the fourth and fifth mice are unknown.
b. Yes, $\widetilde{x}=421$

1. $\bar{x}=48.96, \widetilde{x}=49$, mode $=49$

1. $\bar{x}=2.05, \widetilde{x}=2$, mode $=1$

1. Mean: $n x_{\min } \leq \sum x$ so dividing by $n$ yields $x_{\min } \leq \bar{x}$, so the minimum value is not above average. Median: the middle measurement, or average of the two middle measurements, $\widetilde{x}$, is at least as large as $x_{\min }$, so the minimum value is not above average. Mode: the mode is one of the measurements, and is not greater than itself.

1. a. $\bar{x}=3. \overline{18}, \widetilde{x}=3$, mode $=5$.
b. $\bar{x}=6. \overline{18}, \widetilde{x}=6$, mode $=8$
c. $\bar{x}=-2. \overline{81}, \tilde{x}=-3, \operatorname{mode}=-1$
d. If a number is added to every measurement in a data set, then the mean, median, and mode all change by that number.

1. a. $\mu=1528.74$
b. $\bar{x}=1502.8$
c. $\bar{x}=1535.2$

1. a. $\bar{x}=553.4286$ and $\widetilde{x}=552.5$
b. $\bar{x}=665.9692$ and $\widetilde{x}=667$
c. $\bar{x}=455.8933$ and $\widetilde{x}=448$