Question

Find the value of p if in the given data. The mean is 54.

Class Interval	0-20	20-40	40-60	60-80	80-100
Frequency	7	p	10	9	13

Answer 1

Hint: We will first find the classmark which will be a representative of the whole group. We know the formula of the mean. We will make a new table to calculate the mean from ourselves and then we will equate it to the given value to get the value of p.

Complete step-by-step solution:
We have this data,

Class Interval	0-20	20-40	40-60	60-80	80-100
Frequency	7	p	10	9	13

We will calculate Class Mark= $\dfrac{\text{upper class limit +lower class limit }}{2}$
If I want to calculate Class Mark of interval 0-20 I will upper limit=20 and lower limit=0
I will get class mark =10
We know the formula of mean, given by,
\[\Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]
We will make a new table having columns Class mark(${{x}_{i}}$), ${{f}_{i}}{{x}_{i}}$

Class Interval	Class Mark (${{x}_{i}}$)	Frequency (\[{{f}_{i}}\])	\[{{f}_{i}}{{x}_{i}}\]
0-20	10	7	70
20-40	30	p	30p
40-60	50	10	500
60-80	70	9	630
80-100	90	13	1170

From the table we will calculate \[\sum{{{f}_{i}}{{x}_{i}}}\] and \[\sum{{{f}_{i}}}\]
\[\begin{align}
  & \Rightarrow \sum{{{f}_{i}}{{x}_{i}}}=70+30p+500+630+1170 \\
 & \Rightarrow \sum{{{f}_{i}}{{x}_{i}}}=2370+30p \\
\end{align}\]
\[\begin{align}
  & \Rightarrow \sum{{{f}_{i}}}=7+p+10+9+13 \\
 & \Rightarrow \sum{{{f}_{i}}}=39+p \\
\end{align}\]
We will put the value in the formula
$\begin{align}
  & \Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}} \\
 & \Rightarrow \bar{x}=\dfrac{2370+30p}{39+p} \\
 & \Rightarrow 54=\dfrac{2370+30p}{39+p} \\
 & \Rightarrow 54(39+p)=2370+30p \\
 & \Rightarrow 2106+54p=2370+30p \\
 & \Rightarrow 24p=264 \\
 & \Rightarrow p=11 \\
\end{align}$

Note: It is mandatory to convert the class interval to class mark, then only we can calculate the value of \[\bar{x}\] . Many times, we assume that if mean( \[\bar{x}\] ) is given in the question it means average, then we try to apply this formula \[\dfrac{\sum{{{f}_{i}}}}{\text{Number of elements}}\] and equate to given value. Using this approach will give us the wrong answer. It is mandatory to apply this formula \[\Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\].