There are 15 students whose ages are: 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. The frequency and probability distribution of this set can be represented as follows:

\begin{Bmatrix} \text{X} & 14 &15 & 16 & 17 &18 & 19 & 20&21 \\ \text{Frequency} & 2 & 1& 2 & 3& 1 & 2& 3& 1\\ \text{P(X)} & \large\frac{2}{15}& \large\frac{1}{15}&\large\frac{2}{15}&\large\frac{5}{15}&\large\frac{1}{15}&\large\frac{2}{15}&\large\frac{3}{15}&\large\frac{1}{15} \end{Bmatrix}

Mean of the probability distribution = $\sum (X_i \times P(X_i))$

Mean or $E(X) = \sum (X_i \times P(X_i)) = $$14 \times \large\frac{2}{15} + $$ 15\times\large\frac{1}{15} + $$16\times\large\frac{2}{15} + $$17\times\large\frac{3}{15}+ $$18\times\large\frac{1}{15} + $$19\times \large\frac{2}{15} + $$ 20\times\large\frac{3}{15} + $$21\times\large\frac{1}{15} $

$E (X) = \large \frac{28+15+32+51+18+28+60+21}{15} = \frac{263}{15} =$$ 17.53$

$E(X^2) = \sum ((X_i)^2\times P(X_i)) = $$14^2 \times \large\frac{2}{15} + $$ 15^2\times\large\frac{1}{15} + $$16^2\times\large\frac{2}{15} + $$17^2\times\large\frac{3}{15}+ $$18^2\times\large\frac{1}{15} + $$19^2\times \large\frac{2}{15} + $$ 20^2\times\large\frac{3}{15} + $$21^2\times\large\frac{1}{15} $

$E (X^2) = \large \frac{392+225+512+876+324+722+1200+441}{15} = \frac{4683}{15} = $$312.2$

Therefore Variance = $312.2 - 17.53^2 = 4.89$

Therefore Standard Deviation = $\sqrt{4,89} = 2.21$