Question

practice standard deviation

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?

1. which of these best estimates the standard deviation of points in a card game? points in a card game

a. 5 points
b. 20 points
c. 50 points
d. 500 points

Explanation:

Step1: Calculate the mean for shoe - size data

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here, $x_{i}=7$ for $i = 1,2,\cdots,10$ and $n = 10$. So, $\bar{x}=\frac{7 + 7+\cdots+7}{10}=\frac{7\times10}{10}=7$.

Step2: Calculate the standard deviation for shoe - size data

The formula for the standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$ (for sample, here we can also use population formula $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$ as we have all data). Since $x_{i}=7$ and $\bar{x}=7$, then $(x_{i}-\bar{x})=0$ for all $i$. So, $s = 0$.

Step3: Estimate the standard deviation for card - game data

The data points in the card - game are spread around 500. The range of the data is $510 - 490=20$. A rough estimate of the standard deviation is $\frac{\text{Range}}{4}$ (a common rule - of - thumb). $\frac{20}{4}=5$.

Answer:

1.
a. 7
b. 0

1. A. 5 points