Question

the total number of the observations in the following distribution table is 120 and their mean is 50. find the values of missing frequencies f1 and f2.​

Answer 1

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

Mean using Short Cut Method is given by

$\: \: \: \: \: \: \: \: \: \: \: \: \: \pink{\boxed{ \bf \overline{x} = A + \dfrac{\sum f_id_i}{\sum f_i}}}$

Let Construct the frequency distribution table,

$\begin{gathered}\begin{gathered} \boxed{\begin{array}{|c|c|c|c|c|}\bf Class&\bf Frequency (f_i)&\bf x_i & \bf d_i = (x_i - 30)& \bf d_ix_i \\ \sf 0-20&\sf 17&\sf10 & \sf -20 & \sf -340\\\sf 20-40&\sf f_1&\sf 30& \sf 0 & \sf 0\\\sf 40-60&\sf 32 &\sf 50 & \sf 20 & \sf 640\\\sf 60-80&\sf f_2& \sf 70 & \sf 40 & \sf 40f_2\\\sf 80 - 100&\sf 19&\sf 90 & \sf 60 & \sf 1140 \\\end{array}}\end{gathered} \end{gathered}$

Given that,

Sum of frequency = 120

$\rm :\longmapsto\:17 + f_1 + 32 + f_2 + 19 = 120$

$\rm :\longmapsto\:68 + f_1 + f_2 = 120$

$\rm :\longmapsto\:f_1 + f_2 = 120 - 68$

$\red{\bf :\longmapsto\:f_1 + f_2 = 52} - - - (1)$

Now,

From frequency distribution table, we have

$\blue{\bf :\longmapsto\:A \: = \: 30}$

$\blue{\bf :\longmapsto \: \sum \: f_i\: = \: 120}$

$\blue{\bf :\longmapsto\:h \: = \: 20}$

$\blue{\bf :\longmapsto \: \sum \: f_id_i\: = \: 1440 + 40f_2}$

We know,

Mean using Short Cut Method is given by

$\dashrightarrow \: \: \bf \overline{x} = A + \dfrac{\sum f_id_i}{\sum f_i}$

$\rm :\longmapsto\:50 = 30 + \dfrac{1440 + 40f_2}{120}$

$\rm :\longmapsto\:50 - 30 = \dfrac{1440 + 40f_2}{120}$

$\rm :\longmapsto\:20 = \dfrac{1440 + 40f_2}{120}$

$\rm :\longmapsto\:2400 = 1440 + 40f_2$

$\rm :\longmapsto\:2400 - 1440 = 40f_2$

$\rm :\longmapsto\:960 = 40f_2$

$\bf\implies \:f_2 = 24 - - (2)$

On substituting the value in equation (1), we get

${\bf :\longmapsto\:f_1 + 24 = 52}$

${\bf :\longmapsto\:f_1 = 52 - 24}$

$\bf\implies \:f_1 = 28$

Additional Information :-

Mean using Direct Method :-

$\boxed{ \bf \overline{x} = \dfrac{\sum f_ix_i}{\sum f_i}}$

Mean using Step Deviation Method :-

$\boxed{ \bf \overline{x} = A + \dfrac{\sum f_iu_i}{\sum f_i} \times h}$