Question

$\color{red} \sf \: Calculate \: the \: mean \: and \: the \: median \: for \: the \: following \: distribution: - \\ \\ \color{lightblue} \begin{array}{ | c | c|c | c | c |c | c |c |} \hline \\ \bf \: Number \: \: \: \: & \sf \:5 & \sf \:10& \sf \:15& \sf \:20& \sf \:25& \sf \:30 & \sf \:35 \\ \hline \\ \bf \: Frequency& \sf 1& \sf 2& \sf 5& \sf 6& \sf 3& \sf 2& \sf 1 \\ \hline\end{array}$

$\sf Note:\:The\:required\: mean \: is \: 19.5 \: and \: median \: is \: 20.$​

Answer 1

$\large\underline{\sf{Solution-}}$

Calculations of mean using Direct Method

$\begin{array}{ | c | c|c | c | c |c | c |c |} \hline \\ \bf \: Number (x_i) \: & \sf \:5 & \sf \:10& \sf \:15& \sf \:20& \sf \:25& \sf \:30 & \sf \:35 \\ \hline \\ \bf \: Frequency(f_i)& \sf 1& \sf 2& \sf 5& \sf 6& \sf 3& \sf 2& \sf 1 \\ \hline\\ \bf \: f_ix_i& \sf 5& \sf 20& \sf 75& \sf 120& \sf 75& \sf 60& \sf 35 \\ \hline\end{array}$

So, from above calculations, we concluded that

$\rm \: \sum \: f_i \: = \: 20$

$\rm \: \sum \: f_i x_i\: = \: 390$

We know, Mean using Direct Method is given by

$\rm \: Mean \: = \: \dfrac{ \sum \: f_ix_i}{ \sum \: f_i}$

So, on substituting the values, we get

$\rm \: Mean = \dfrac{390}{20}$

$\rm\implies \:\boxed{ \rm{ \:Mean \: = \: 19.5 \: \: }}$

Now, Calculations of Median

$\begin{array}{ | c | c|c | c | c |c | c |c |} \hline \\ \bf \: Number (x_i) \: & \sf \:5 & \sf \:10& \sf \:15& \sf \:20& \sf \:25& \sf \:30 & \sf \:35 \\ \hline \\ \bf \: Frequency(f_i)& \sf 1& \sf 2& \sf 5& \sf 6& \sf 3& \sf 2& \sf 1 \\ \hline\\ \bf \: c.f.& \sf 1& \sf 3& \sf 8& \sf 14& \sf 17& \sf 19& \sf 20 \\ \hline\end{array}$

$\rm \: N \: = \: \sum \: f_i \: = \: 20$

$\rm\implies \:\dfrac{N}{2} = \dfrac{20}{2} = 10$

Now, In cumulative frequency, the frequency just greater than 10 is 14 and corresponding value of x is 20.

$\rm\implies \:\boxed{ \rm{ \:Median \: = \: 20 \: \: }}$

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Additional Information :-

1. Mean using Short Cut Method

$\boxed{ \rm{ \:\rm \: Mean \: = \:A + \dfrac{ \sum \: f_id_i}{ \sum \: f_i} \: \: }}$

2. Mean using Step Deviation Method

$\boxed{ \rm{ \:\rm \: Mean \: = \:A + \dfrac{ \sum \: f_iu_i}{ \sum \: f_i} \times h \: \: }}$