The table shows the distribution of ages of 22 students in a school.
Age (years) | 12-14 | 15-17 | 18-20 | 21-23 | 24-26 |
Frequency | 6 | 10 | 3 | 2 | 1 |
Using an assumed mean of 19, calculate, correct to three significant figures, the :
(a) mean age ; (b) standard deviation ; of the distribution.
Explanation
Assumed mean, A = 19.
Age (years) | Mid-age (x) | Frequency | (d = x – A) | (fd) | (fd^{2}) |
12 – 14 | 13 | 6 | -6 | -36 | 216 |
15 – 17 | 16 | 10 | -3 | -30 | 90 |
18 – 20 | 19 | 3 | 0 | 0 | 0 |
21 – 23 | 22 | 2 | 3 | 6 | 18 |
24 – 26 | 25 | 1 | 6 | 6 | 36 |
22 | -54 | 360 |
(a) (bar{x} = A + frac{sum fd}{sum f})
= (19 + frac{-54}{22})
= (19 – 2.455)
= (16.545 approxeq 16.5) years.
(b) Standard deviation , (sigma = sqrt{frac{sum fd^{2}}{sum f} – (frac{sum fd}{sum f})})
= (sqrt{frac{360}{22} – (frac{-54}{22})})
= (sqrt{16.364 – (2.45)^{2}})
= (sqrt{16.364 – 6.025})
= (sqrt{10.339})
= 3.215 (approxeq) 3.22 years.