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Question
Find the values of a and b, if the sum of all the frequencies is 120 and the median of the following data is 55.
| Marks | 30 – 40 | 40 – 50 | 50 –60 | 60 – 70 | 70 –80 | 80 – 90 |
| Frequency | a | 40 | 27 | b | 15 | 24 |
Solution
Given: Median of observations = 55
And sum of frequencies = 120
| Marks `(f_i)` |
Frequency `(c.f.)` |
Cumulative frequency |
| 30 – 40 | a | a |
| 40 – 50 | 40 | a + 40 |
| 50 – 60 | 27 | a + 67 |
| 60 – 70 | b | a + b + 67 |
| 70 – 80 | 15 | a + b + 82 |
| 80 – 90 | 24 | a + b + 106 |
Now, `sumf_i` = 120 ......(Given)
⇒ a + b + 106 = 120
⇒ a + b = 14 ......(i)
Also, Median = 55
∴ Median class = 50 – 60
So, L = 50, h = 10, f = 27, c.f. = a + 40
Median = `L + ((N/2 - c.f.)/f) xx h`
55 = `50 + ((120/2 - (a + 40))/27) xx 10`
55 – 50 = `((60 - (a + 40))/27) xx 10`
5 = `((20 - a)/27) xx 10`
5 × 27 = 200 – 10a
135 = 200 – 10a
10a = 65
a = 6.5
Putting the value of a in equation (i), we get
6.5 + b = 14
b = 14 – 6.5 = 7.5
Hence, a = 6.5 and b = 7.5.
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