Approximate the sample variance given the following frequency distribution.

$\begin{array} { c c }
\hline \text { Class } & \text { Frequency } \
\hline 0 - 9 & 10 \
10 - 19 & 17 \
20 - 29 & 12 \
30 - 39 & 14 \
40 - 49 & 11 \
\hline
\end{array}$

A) 179.7
B) 13.5
C) 182.5
D) 13.4

Question

Approximate the sample variance given the following frequency distribution.

$\begin{array} { c c } 
\hline \text { Class } & \text { Frequency } \
\hline 0 - 9 & 10 \
10 - 19 & 17 \
20 - 29 & 12 \
30 - 39 & 14 \
40 - 49 & 11 \
\hline
\end{array}$

A) 179.7
B) 13.5
C) 182.5
D) 13.4

Quizplus · Accepted Answer

To calculate the sample variance, we need to find the mean first. We use the midpoints of each class and their frequencies to find the mean:
$$\bar{x}=\frac{\sum{f_i x_i}}{\sum f_i}=\frac{10	imes4.5+17	imes14.5+12	imes24.5+14	imes34.5+11	imes44.5}{64}=23.9531$$
Now, we use the formula for sample variance:
$$s^2=\frac{\sum\limits_{i=1}^n{f_i (x_i-\bar{x})^2}}{n-1}$$
We can calculate this by finding the deviation from the mean for each midpoint, squaring them, multiplying by frequency, summing up for all classes, and dividing by $n-1$ where $n$ is the total frequency. However, it is simpler to use the following formula:
$$s^2=\frac{\sum\limits_{i=1}^n{f_ix_i^2}-\frac{\left(\sum\limits_{i=1}^n{f_ix_i}ight)^2}{n}}{n-1}$$
Using this formula, we get:
$$s^2=\frac{10	imes4.5^2+17	imes14.5^2+12	imes24.5^2+14	imes34.5^2+11	imes44.5^2-\frac{(64	imes23.9531)^2}{64}}{64-1}=182.463$$
Rounded to one decimal place, the answer is 182.5 which is option C.

Approximate the Sample Variance Given the Following Frequency Distribution A) 1797 B) 13

Approximate the Sample Variance Given the Following Frequency Distribution A) 1797
B) 13