$\text { Given the following frequency distribution, find the } 50^{\text {th }}$percentile.
$\begin{array}{|l|l|}
\hline \text { Class Boundaries } & \text { Frequencies } \
\hline 13.5-18.5 & 4 \
\hline 18.5-23.5 & 9 \
\hline 23.5-28.5 & 12 \
\hline 28.5-33.5 & 15 \
\hline 33.5-38.5 & 18 \
\hline
\end{array}$

A) 30.2
B) 29.8
C) 30.3
D) 29.5

Question

$\text { Given the following frequency distribution, find the } 50^{\text {th }}$percentile.
$\begin{array}{|l|l|}
\hline \text { Class Boundaries } & \text { Frequencies } \
\hline 13.5-18.5 & 4 \
\hline 18.5-23.5 & 9 \
\hline 23.5-28.5 & 12 \
\hline 28.5-33.5 & 15 \
\hline 33.5-38.5 & 18 \
\hline
\end{array}$

A) 30.2
B) 29.8
C) 30.3
D) 29.5

Quizplus · Accepted Answer

First, find the total frequency: 4 + 9 + 12 + 15 + 18 = 58
Since the 50th percentile is the median, we need to find the middle value of the data. The cumulative frequency at the midpoint would be (58 + 1)/2 = 29.5
To find the corresponding value, we need to look at the cumulative frequency column and find the class interval that contains 29.5. It is the fourth interval: 28.5-33.5, with a cumulative frequency of 25. So, we know that the 50th percentile falls within the 28.5-33.5 class interval.
Next, we need to use the formula for finding the value at the percentile within a class interval:
$$
P = L+\left(\frac{\frac{N}{2}-cf_b}{f}ight) 	imes w
$$
Where:
- P = percentile (in this case, 50)
- L = lower class boundary of the interval (28.5)
- N = total frequency (58)
- cf_b = cumulative frequency of the interval before the one containing the percentile (25)
- f = frequency of the interval containing the percentile (15)
- w = width of the interval (5)

Plugging in the values, we get:
$$
50 = 28.5 + \left(\frac{\frac{58}{2}-25}{15}ight) 	imes 5
$$
Solving for the expression inside the parentheses first, we get:
$$
\left(\frac{\frac{58}{2}-25}{15}ight) = 0.6333
$$
Multiplying by the width and adding to the lower boundary, we get:
$$
28.5 + (0.6333 	imes 5) = 29.9985
$$
Rounding to one decimal place, we get 29.8 as the answer. Therefore, the best choice is B.

Given the following frequency distribution, find the 50th \text { Given the following frequency distribution, find the } 50^{\text {th }} Given the following frequency distribution, find the 50th

$\text { Given the following frequency distribution, find the } 50^{\text {th }}$