Question

The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate by step deviation method.
$\begin{array}{cc}\text{Literacy rate (in \%)}& \text{No. of cities}\\ 44-55& 3\\ 55-65& 10\\ 65-75& 11\\ 75-85& 8\\ 85-95& 3\end{array}$

• A. 66.44
• B. 69.43
• C. 67.39
• D. 71.55

Answer 1

The correct option is B

69.43

$\begin{array}{cccccc}\text{Literacy rate}\left(in\%\right)& 45-55& 55-65& 65-75& 75-85& 85-95\\ \text{Number of cities}& 3& 10& 11& 8& 3\end{array}$
Class mark (for $1^{st}$ class interval)
$=\frac{\text{upper class limit + lower class limit}}{2}=\frac{45+55}{2}=50$

Let the assumed mean(a) be 70 (Choose the number in order to get minimum values of d)

The class size(h) is = 55 – 45 = 10 (Upperclass limit- lower class limit)

$\begin{array}{cccccc}\text{Class interval}& \text{Frequency}\left(f_i\right)& \text{Class mark}\left(x_i\right)& d_i=x_i-a& u_i=\frac{d_i}{h}& f_i{u}_i\\ 45-55& 3& 50& -20& -2& -6\\ 55-65& 10& 60& -10& -1& -10\\ 65-75& 11& 70& 0& 0& 0\\ 75-85& 8& 80& 10& 1& 8\\ 85-95& 3& 90& 20& 2& 6\\ Total& \sum f_i=35& & & & \sum f_i{u}_i=-2\end{array}$
So, we have $a=70$
$h=10$
$\sum f_i=35$
$\sum f_i{u}_i=-2$

$Mean=a+\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)\times h$
$=70+\left(\frac{-2}{35}\right)\times 10$
$=(70-0.57)$
$=69.43$