Q. 3.22

Miles Run per Week

Find the sample variance and the sample standard deviation for the frequency distribution of the data shown. The data represent the number of miles that 20 runners ran during one week.

 Class Frequency Midpoint 5.5–10.5 1 8 10.5–15.5 2 13 15.5–20.5 3 18 20.5–25.5 5 23 25.5–30.5 4 28 30.5–35.5 3 33 35.5–40.5 2 38

Verified Solution

Step 1 Make a table as shown, and find the midpoint of each class.

Step 2 Multiply the frequency by the midpoint for each class, and place the products in column D.

1 · 8 = 8     2 · 13 = 26     . . .     2 · 38 = 76

Step 3 Multiply the frequency by the square of the midpoint, and place the products in column E.

1 · 8² = 64     2 · 13² = 338     . . .     2 · 38² = 2888

Step 4 Find the sums of columns B, D, and E. The sum of column B is n, the sum of column D is $Σf · X_m$, and the sum of column E is $Σf · X_m^2$. The completed table is shown.

Step 5 Substitute in the formula and solve for $s^2$ to get the variance.

\begin{aligned}s^2 & =\frac{n\left(\Sigma f \cdot X_m^2\right)-\left(\Sigma f \cdot X_m\right)^2}{n(n-1)} \\& =\frac{20(13,310)-490^2}{20(20-1)} \\& =\frac{266,200-240,100}{20(19)} \\& =\frac{26,100}{380} \\& \approx 68.7 \end{aligned}

Step 6 Take the square root to get the standard deviation.

$s \approx \sqrt{68.7} \approx 8.3$

 A Class B Frequency f C Midpoint $X_m$ D $f ·X_m$ E $f · X_m^2$ 5.5–10.5 1 8 10.5–15.5 2 13 15.5–20.5 3 18 20.5–25.5 5 23 25.5–30.5 4 28 30.5–35.5 3 33 35.5–40.5 2 38
 A Class B Frequency C Midpoint D $f · X_m$ E $f · X^2_m$ 5.5–10.5 1 8 8 64 10.5–15.5 2 13 26 338 15.5–20.5 3 18 54 972 20.5–25.5 5 23 115 2,645 25.5–30.5 4 28 112 3,136 30.5–35.5 3 33 99 3,267 35.5–40.5 2 38 76 2,888 n = 20 $Σ f · X_m = 490$ $Σ f · X_m = 13,310$