Question

lesson 12 practice
1 the shoe size for all the pairs of shoes in a persons closet are recorded.
7 7 7 7 7 7 7 7 7 7
a. what is the mean?
b. what is the standard deviation?

Explanation:

Step1: Recall mean formula

The mean $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $x_1=x_2=\cdots=x_{10}=7$ and $n = 10$.
$\sum_{i=1}^{10}x_i=7 + 7+\cdots+7=7\times10 = 70$. Then $\bar{x}=\frac{70}{10}=7$.

Step2: Recall standard - deviation formula

The standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$ for a sample and $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}}$ for a population. Since we have all the data (the entire population of shoe - sizes in the closet), we use the population formula.
For each $i$, $x_i = 7$ and $\bar{x}=7$. So, $(x_i-\bar{x})=7 - 7=0$ for $i = 1,2,\cdots,10$. Then $\sum_{i = 1}^{10}(x_i-\bar{x})^2=0$. And $s=\sqrt{\frac{0}{10}} = 0$.

Answer:

a. 7
b. 0