Consider the following distribution.
| Class Interval | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
| Frequency | 17 | 28 | 32 | f | 19 |
If the mean of the above distribution is 50, find the value of f.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 24
In a grouped distribution each class is represented by its class mark (the mid-point), and mean \(=\frac{\sum fx}{\sum f}\). Setting this equal to 50 gives an equation in \(f\).
Class marks (mid-points): 10, 30, 50, 70, 90
Total frequency: \(\sum f=17+28+32+f+19=96+f\)
Sum of frequency × class mark: \(\sum fx=170+840+1600+70f+1710=4320+70f\)
Mean \(=50\): \(\displaystyle \frac{4320+70f}{96+f}=50\), so \(4320+70f=4800+50f\)
\(20f=480\), so \(f=24\)
Shortcut: The deviations from the mean add up to zero, so take each class mark’s distance from 50.
Deviations of the class marks from 50: \(-40, -20, 0, +20, +40\)
Frequency × deviation, added and set to zero: \(17(-40)+28(-20)+32(0)+20f+19(40)=0\)
\(-680-560+760+20f=0\), so \(20f=480\) and \(f=24\)