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Question
Find the values of x and y if the mean and total frequency of the distribution are 25 and 50 respectively.
| Class Interval | 0 – 10 | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 | 50 – 60 |
| Frequency | 7 | x | 5 | y | 4 | 2 |
Solution
Given: Mean = 25
| Class Interval |
Frequency `(f_i)` |
Class mark `(x_i)` |
`d_i = x_i - 25` | `f_i d_i` |
| 0 – 10 | 7 | 5 | – 20 | – 140 |
| 10 – 20 | x | 15 | – 10 | – 10x |
| 20 – 30 | 5 | 25 = A | 0 | 0 |
| 30 – 40 | y | 35 | 10 | 10y |
| 40 – 50 | 4 | 45 | 20 | 80 |
| 50 – 60 | 2 | 55 | 30 | 60 |
| `sumf_i = 18 + x + y` | `sumf_i d_i = 10y - 10x` |
Mean = 25 .......[Given]
Also, Mean, `bar"X" = "A" + (sumf_i d_i)/(sumf_i)`
25 = `25 + (10y - 10x)/(18 + x + y)`
25 – 25 = `(10y - 10x)/(18 + x + y)`
0 = `(10y - 10x)/(18 + x + y)`
0 (18 + x + y) = 10y – 10x
10y – 10x = 0
y – x = 0 ......(i)
Also, 50 = 18 + x + y
x + y = 50 – 18
x + y = 32 ......(ii)
Adding equations (i) and (ii), we get
2y = 32
y = 16
Putting the value of y in equation (ii), we get
x + 16 = 32
x = 32 – 16 = 16
As a result, x and y have values of 16 and 16, respectively.
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