Question

Foot lengths Here are the foot lengths (in centimeters) for a random sample of seven 14-year-olds from the United Kingdom:

Calculate the standard deviation. Interpret this value.

Short answer

The standard deviation is 2.5166 cm.

Step-by-step solution

Step 1. Given information.

The given random sample is:

25, 22, 20, 25, 24, 24, 28

Step 2. Find the mean.

$$\begin{gathered} \overline{x}=\frac{\sum _{i=1}^n{x}_i}{n} \\ =\frac{25+22+20+25+24+24+28}{7} \\ =\frac{168}{7} \\ =24 \end{gathered}$$

Step 3. Find the standard deviation.

$$\begin{gathered} s^2=\frac{\sum {(x-\overline{x})}^2}{n-1} \\ =\frac{{(25-24)}^2+{\left(22-24\right)}^2+{\left(20-24\right)}^2+{\left(25-24\right)}^2+{\left(24-24\right)}^2+{\left(24-24\right)}^2+{\left(28-24\right)}^2}{7-1} \\ =\frac{1+4+16+1+0+0+16}{6} \\ =\frac{38}{6}\approx 6.3333 \end{gathered}$$

The standard deviation is:

role="math" localid="1653665598603" $$\begin{gathered} s=\sqrt{s^2} \\ =\sqrt{6.333} \\ =2.516cm \end{gathered}$$

The standard deviation is roughly 2.5166 cm since the units of the standard deviation are the same as the units of the data values.