Theory question 110
Measures of DispersionThe table shows the distribution of ages of a number of children in a school.
Age (years): 3, 4, 5, 6, 7, 8, 9, 10
Number of Children: 2, 6, 5, x, 6, 9, 8, 5
If the mean of the distribiution is 7, find the: (a) value of x; (b) standard deviation of their ages.
Show the model answer
Model answer
(a) Σf = 41 + x; Σfx = 6 + 24 + 25 + 6x + 42 + 72 + 72 + 50 = 291 + 6x
(291 + 6x)/(41 + x) = 7 → 287 + 7x = 291 + 6x → x = 4
(b) Σf = 45; deviations from 7: −4, −3, −2, −1, 0, 1, 2, 3
Σf(x − 7)2 = 32 + 54 + 20 + 4 + 0 + 9 + 32 + 45 = 196
Variance = 196/45 ≈ 4.36; standard deviation ≈ 2.09
Explanation
Use the mean of 7 to find x first, then the deviations from 7.