Question

the mean shoe size of the students in a math class is 7.5. most of the shoe sizes fall within 1 standard deviation, or between a size 6 and a size 9. what is the standard deviation of the shoe size data for the math class? 1.5 2.7 3.0 3.8

Explanation:

Step1: Recall the property of standard - deviation

In a normal - like distribution, most data (about 68%) falls within 1 standard deviation of the mean. If the mean is $\mu = 7.5$ and the range within 1 standard deviation is from 6 to 9.

Step2: Calculate the standard deviation

The formula to find the value within 1 standard deviation of the mean is $\mu\pm\sigma$, where $\mu$ is the mean and $\sigma$ is the standard deviation. If $\mu = 7.5$ and the lower limit is $\mu-\sigma = 6$ or the upper limit is $\mu+\sigma = 9$. Using $\mu+\sigma = 9$ and $\mu = 7.5$, we can solve for $\sigma$:
\[

$$\begin{align*} 7.5+\sigma&=9\\ \sigma&=9 - 7.5\\ \sigma&=1.5 \end{align*}$$

\]
Or using $\mu-\sigma = 6$ and $\mu = 7.5$:
\[

$$\begin{align*} 7.5-\sigma&=6\\ -\sigma&=6 - 7.5\\ -\sigma&=- 1.5\\ \sigma&=1.5 \end{align*}$$

\]

Answer:

1.5