Question

Given the following series, calculate standard deviation by direct method:

$\begin{array}{ccccccc}\text{Size}& 0-2& 2-4& 4-6& 6-8& 8-10& 10-12\\ \text{Frequency}& 2& 4& 6& 4& 2& 6\end{array}$

Answer 1

$\begin{array}{ccccccc}\text{Size}& \text{Mid - value}& \text{Frequency}& \text{Multiple of Mid - value}& \text{Deviation from Mean Value}& \text{Square of Deviation}& f{x}^2\\ \left(X\right)& \left(m\right)& \left(f\right)& \text{and Frequency}& (x=m-\overline{X})& \left(x^2\right)& \\ & & & \left(fm\right)& \overline{X}=6.5& & \\ 0-2& 1& 2& 2& -5.5& 30.25& 60.50\\ 2-4& 3& 4& 12& -3.5& 12.25& 49.00\\ 4-6& 5& 6& 30& -1.5& 2.25& 13.50\\ 6-8& 7& 4& 28& +0.5& 0.25& 1.00\\ 8-10& 9& 2& 18& +2.5& 6.25& 12.50\\ 10-12& 11& 6& 66& +4.5& 20.25& 121.50\\ & & N=24& \sum fm=156& & & \sum f{x}^2=258\end{array}$

$\overline{X}=\frac{\sum fm}{N}=\frac{156}{24}=6.5$

$\sigma =\sqrt{\frac{\sum f{x}^2}{N}}=\sqrt{\frac{258}{24}}=\sqrt{10.75}=3.28$

Standard Deviation ($\sigma$) = 3.28