Question

Well rested? A random sample of 6 students in a first-period statistics class was asked how much sleep (to the nearest hour) they got last night. Their responses were 6, 7, 7, 8, 10, and 10. Calculate the standard deviation. Interpret this value.

Short answer

The standard deviation is 1.6733 hours.

Step-by-step solution

Step 1. Given information.

The given responses are:

6, 7, 7, 8, 10, and 10

Step 2. Find the mean.

$$\begin{gathered} \overline{x}=\frac{\sum _{i=1}^n{x}_i}{n} \\ =\frac{6+7+7+8+10+10}{6} \\ =\frac{48}{6} \\ =8 \end{gathered}$$

Step 3. Find the standard deviation.

$$\begin{gathered} s^2=\frac{\sum {(x-\overline{x})}^2}{n-1} \\ =\frac{{\left(6-8\right)}^2+{\left(7-8\right)}^2+{\left(7-8\right)}^2+{\left(8-8\right)}^2+{\left(10-8\right)}^2+\left(10-8\right)}{6-1} \\ =\frac{4+1+1+0+4+4}{5} \\ =\frac{14}{5} \\ =2.8 \end{gathered}$$

The standard deviation is:

$$\begin{gathered} s=\sqrt{s^2} \\ =\sqrt{2.8}\approx 1.6733 \end{gathered}$$

The standard deviation is roughly 1.6733 hours since the units of the standard deviation are the same as the units of the data values.