Question

Table below shows the number of students in the Maths Club of school classified according to their height.

HEIGHT (in cm)	NUMBER OF STUDENTS
110-120	4
120-130	24
130-140	20
140-150	32
150-160	20

Calculate the mean height.

Answer 1

Hint:
We are given a class of heights of students of the Maths Club of school. And also the number of students belonging or falling in respective classes. We will find the mid-value(x) of the interval . Since here values of the midpoint of class are too big to calculate we will use the assumed mean method. In this method we will consider an assumed mean (a) from the values given in the first column of class. And also a column of deviation (d) will be added here. So let’s start!

Complete step by step solution:
We will consider the assumed mean to be a=135. There is no specific reason for it. We have to select a value such that it makes our calculations easy!
Now tabulate the data.

HEIGHT(cm)	Number of students(f)	Mid-value(x)	\[{d_i} = x - a\]	\[{f_i}{d_i}\]
110-120	4	115	115-135=-20	\[4 \times \left( { - 20} \right) = - 80\]
120-130	24	125	125-135=-10	\[24 \times \left( { - 10} \right) = - 240\]
130-140	20	135	135-135=0	\[0\]
140-150	32	145	145-135=10	\[32 \times 10 = 320\]
150-160	20	155	155-135=20	\[20 \times 20 = 400\]
TOTAL	100			400

We have tabulated all values well in our data table above. Now we will use the formula to calculate the mean by the method assumed mean.
\[mean = a + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}\]
Now let’s find the terms of the formula.
\[\sum {{f_i}{d_i}} = \left( { - 80} \right) + \left( { - 240} \right) + 0 + 320 + 400 = 400\]
\[\sum {{f_i} = 4 + 24 + 20 + 32 + 20 = 100} \]
Now putting the values in formula we get,
\[
  mean = a + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }} \\
   \Rightarrow 135 + \dfrac{{400}}{{100}} \\
   \Rightarrow 135 + 4 \\
   \Rightarrow 139 \\
 \]
Thus the mean height of students is \[ \Rightarrow 139cm\].

Note:
We can go with our traditional formula of finding the mean but that will make our work too tedious so used this assumed mean method. Also note that when you find the term \[{f_i}{d_i}\] we should be aware of negative signs also. Because that will affect when we find the summation.