Question

find the mean of the following frequency distribution, using the assume mean method,: Class 100-120 120-140 140-160 160-180 180-200. frequency. 10 20 30 15 5

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Answer 1

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

Frequency distribution table for calculations of mean using Step Deviation Method is as below :

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c|c|c}\sf Class\: interval&\sf Frequency\: (f_i)&\sf \: midvalue \: (x_i)&\sf \: u_i&\sf \: f_iu_i\\\frac{\qquad  \qquad}{}&\frac{\qquad  \qquad}{}\\\sf 100 - 120&\sf 10&\sf110&\sf - 2&\sf - 20\\\\\sf 120 - 140 &\sf 20&\sf130&\sf - 1&\sf - 20\\\\\sf 140-160 &\sf 30 &\sf150&\sf0&\sf0\\\\\sf 160 - 180&\sf 15&\sf170&\sf1&\sf15\\\\\sf 180-200&\sf 5&\sf190&\sf2&\sf10\\\\\ \end{array}}\end{gathered}\end{gathered}\end{gathered}$

So, from above calculations, we have

$\rm \: A = 150$

$\rm \: h = 20$

$\rm \: \sum \: f_i = 80$

$\rm \: \sum \: f_iu_i = - 15$

We know, Mean using Step Deviation Method is given by

$\rm \: Mean = A + h \times \dfrac{ \sum \: f_iu_1}{ \sum \: f_i}$

$\rm \: Mean = 150 + 20 \times \dfrac{( - 15)}{80}$

$\rm \: Mean = 150 - \dfrac{15}{4}$

$\rm \: Mean = 150 - 3.75$

$\rm\implies \:\boxed{ \rm{ \:Mean \: = \: 146.25 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

1. Mean using Direct Method

$\rm \: Mean = \dfrac{ \sum \: f_ix_1}{ \sum \: f_i}$

2. Mean using Short Cut Method

$\rm \: Mean = A + \dfrac{ \sum \: f_id_1}{ \sum \: f_i}$