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Question
The median of the following data is 16. Find the missing frequencies a and b if the total of frequencies is 70.
| Class | 0 – 5 | 5 – 10 | 10 – 15 | 15 – 20 | 20 – 25 | 25 – 30 | 30 – 35 | 35 – 40 |
| Frequency | 12 | a | 12 | 15 | b | 6 | 6 | 4 |
Solution
| Class | Frequency (f) | Cumulative Frequency (cf) |
| 0 – 5 | 12 | 12 |
| 5 – 10 | a | 12+a |
| 10 – 15 | 12 | 24+a |
| 15 – 20 | 15 | 39+a |
| 20 – 25 | b | 39+a+b |
| 25 – 30 | 6 | 45+a+b |
| 30 – 35 | 6 | 51+a+b |
| 35 – 40 | 4 | 55+a+b |
| Total | N = Σ𝑓𝑖 = 70 |
Let a and b be the missing frequencies of class intervals 5 – 10 and 20 – 25 respectively.
Then, 55 + a + b = 70⇒a +b=15 …..(1)
Median is 16, which lies in 15 – 20. So, the median class is 15 – 20.
∴ l = 15, h = 5, N = 70, f = 15 and cf = 24 + a
Now,
Median,` M = l + ((N/2−cf)/f) × h`
`⇒ 16 = 15 + ((70/2 −(24 + a))/ 15)× 5`
`⇒ 16 = 15 + ((35−24−a)/3)`
`⇒ 16 = 15 + ((11−a)/3)`
`⇒ 16 – 15 = (11−a)/3`
⇒ 1 × 3 = 11 – a
⇒ a = 11 – 3
⇒ a = 8
∴ b = 15 – a [From (1)]
⇒ b = 15 – 8
⇒ b = 7
Hence, a = 8 and b = 7.
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