Question

Calculate mean, variance, and standard deviation for the following distribution.
$\begin{array}{cccccccc}\text{Classes}& 30-40& 40-50& 50-60& 60-70& 70-80& 80-90& 90-100\\ \text{Frequency}& 3& 7& 12& 15& 8& 3& 2\end{array}$

Answer 1

Step 1: Finding Step-Deviation
Let, $a=$ assumed mean $=65$
$h=$ common factor $=10$
$d_i$ = step deviation
For Marks $(30-40),f_i=3$
$x_i=\frac{30+40}{2}=35$
$d_i=\frac{x_i-a}{h}=\frac{35-65}{10}=-3$
$d_i^2=(-3{)}^2=9$
$f_i{d}_i=3\times -3=-9$
$f_i{d}_i^2=3\times 9=27$

$\begin{array}{ccccccc}\text{Marks}& \text{Number of}& \text{Mid - point}\left(x_i\right)& d_i=\frac{x_i-a}{h}& d_i^2& f_i{d}_i& f_i{d}_i^2\\ \text{obtained}& \text{students}\left(f_i\right)& & & & & \\ 30-40& 3& \frac{30+40}{2}=35& \frac{35-65}{10}=-3& 9& -9& 27\\ 40-50& 7& \frac{40+50}{2}=45& \frac{45-65}{10}=-2& 4& -14& 28\\ 50-60& 12& \frac{50+60}{2}=55& \frac{55-65}{10}=-1& 1& -12& 12\\ 60-70& 15& \frac{60+70}{2}=65& \frac{65-65}{10}=0& 0& 0& 0\\ 70-80& 8& \frac{70+80}{2}=75& \frac{75-65}{10}=1& 1& 8& 8\\ 80-90& 3& \frac{80+90}{2}=85& \frac{85-65}{10}=2& 4& 6& 12\\ 90-100& 2& \frac{90+100}{2}=95& \frac{95-65}{10}=3& 9& 6& 18\\ & \sum f_i=50& & & & \sum f_i{d}_i=-15& \sum f_i{d}_i^2=105\end{array}$

Step 2: Finding mean
Mean $\left(\overline{x}\right)=a+\frac{\sum f_i{d}_i}{\sum f_i}\times h$
Mean $\left(\overline{x}\right)=65-\frac{15}{50}\times 10=62$

Step 3: Finding variance and standard deviation
Variance $\left({\sigma }^2\right)=\frac{h^2}{(\sum f_i{)}^2}\left[\right(\sum f_i).(\sum f_i{d}_i^2)-(\sum f_i{d}_i{)}^2]$
$=\frac{(10{)}^2}{(50{)}^2}[50\times (105)-(-15{)}^2]$
$=\frac{100}{2500}[5250-225]$
$=\frac{1}{25}\times 5025$
$=201$

Standard deviation $\left(\sigma \right)=\sqrt{201}=14.18$
Hence, the mean is $62$, variance is $201$ and the standard deviation is $14.18$