Approximate the population standard deviation given the following frequency distribution.

$\begin{array} { c c }
\hline \text { Class } & \text { Frequency } \
\hline 0 - 14 & 15 \
15 - 29 & 9 \
30 - 44 & 8 \
45 - 59 & 8 \
60 - 74 & 13 \
\hline
\end{array}$

A) 23.6
B) 556.1
C) 545.6
D) 23.4

Question

Approximate the population standard deviation given the following frequency distribution.

$\begin{array} { c c } 
\hline \text { Class } & \text { Frequency } \
\hline 0 - 14 & 15 \
15 - 29 & 9 \
30 - 44 & 8 \
45 - 59 & 8 \
60 - 74 & 13 \
\hline
\end{array}$

A) 23.6
B) 556.1
C) 545.6
D) 23.4

Quizplus · Accepted Answer

First, we need to find the sample mean, which is:
$$ \bar{x} = \frac{\sum\limits_{i=1}^{k} f_i	imes c_i}{n} $$
where $f_i$ is the frequency of the $i$-th class, $c_i$ is the midpoint of the $i$-th class, and $n$ is the total number of observations. Using the given data, we have:
$$ \bar{x} = \frac{(15	imes7) + (9	imes22) + (8	imes37) + (8	imes52) + (13	imes67)}{53} \approx 44.98 $$
Next, we need to find the sample variance, which is:
$$ s^2 = \frac{\sum\limits_{i=1}^{k} f_i	imes(c_i - \bar{x})^2}{n-1} $$
Using the same formula as above, we get : $ s^2 = 516.54 $
Finally, the population standard deviation can be approximated using the sample standard deviation formula: 
$$ \sigma \approx \sqrt{s^2} = \sqrt{516.54} \approx 23.4 $$
Therefore, the best choice is D.

Approximate the Population Standard Deviation Given the Following Frequency Distribution