Question

Find the mean, variance and standard deviation using the short cut method

$\begin{array}{cc}Heightincm& No.ofchildren\\ 70-75& 3\\ 75-80& 4\\ 80-85& 7\\ 85-90& 7\\ 90-95& 15\\ 95-100& 9\\ 100-105& 6\\ 105-110& 6\\ 110-115& 3\end{array}$

Answer 1

$\begin{array}{cccccc}Heightincm& x_i& f_i& u=\frac{x-92.5}{5}& fu& f{u}^2\\ 70-75& 72.5& 3& -4& -12& 48\\ 75-80& 77.5& 4& -3& -12& 36\\ 80-85& 82.5& 7& -2& -14& 28\\ 85-90& 87.5& 7& -1& -7& 7\\ 90-95& 92.5& 15& 0& 0& 0\\ 95-100& 97.5& 9& 1& 9& 9\\ 100-105& 102.5& 6& 2& 12& 24\\ 105-110& 107.5& 6& 3& 18& 54\\ 110-115& 112.5& 3& 4& 12& 48\\ & & 60& & 6& 254\end{array}$

Mean $\left(\overline{x}\right)=A+\frac{\sum fu}{N}\times h=92.5+\frac{6}{60}\times 5=92.5+0.5=93$
Variance (${\sigma }^2\left)\frac{h^2}{N^2}\right[N\sum f{u}^2-(\sum fu{)}^2]=\frac{h^2}{N^2}\times [60\times 254-(6{)}^2]$
$=\frac{25}{3600}[15240-36]=\frac{25}{3600}\times 15204=105.58$
Standard deviation $\left(\sigma \right)=\sqrt{105.58}=10.27$