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The table shows the distribution of ages of 22 students in a school. Age (years)…

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.