Question

The marks obtained by the students in an examination of400 marks is given in the following frequency distribution table .
Find the mean of the distribution.

Answer 1

it means that 10 students has obtained marks Between 200-240,20students has obtained marks Between 240-280,30 students has obtained marks Between 280-320,24 students Has obtained marks Between 320-360and 16 students Has obtained marks Between 360-400.

Answer 2

$\Large{\underbrace{\underline{\sf{Understanding\: the\: Concept}}}}$

Here, this is a question from statistics, where we have to find the mean of the given distribution table.

We have the formula for mean:

$\displaystyle\sf\;Mean,\:(\overline{X})=\dfrac{\sum X_iF_i}{\sum F_i}$

$\rule{380}{2}$

The required value for formula of mean can be obtained by following table:

$\fbox{\boxed{\begin{array}{c|c|c|c}&&&&\bf{Marks\;}&\bf{Frequency}&\bf{X_1}&\bf{X_iF_i}\\&\sf{(F_i)}&\sf{\dfrac{Upper+lower\: limit}{2}}&&&&\\\cline{1-4}&&&&200-240&10&\dfrac{440}{2}=220&2200&&&&\\\cline{1-4}&&&&240-280&20&\dfrac{520}{2}=260&5200&&&&\\\cline{1-4}&&&&280-320&30&\dfrac{600}{2}=300&900&&&&\\\cline{1-4}&&&&320-360&24&\dfrac{680}{2}=340&3160&&&&\\\cline{1-4}&&&&360-400&16&\dfrac{760}{2}=380&6080&&&&\\\cline{1-4}&&&&\sf{Total}&\sf\sum\;F_i=100&&\sf\sum\;F_i X_i=30,640&&&&\end{array}}}$

$\rule{380}{2}$

From here we have obtained values of:

$:\implies \displaystyle\sf\sum X_iF_i$

$\displaystyle\sf:\implies \sum F_i$

Now put these values in the formula:

$\displaystyle\sf\;Mean,\:(\overline{X})=\dfrac{\sum X_iF_i}{\sum F_i}$

$\sf\;Mean,\:(\overline{X})=\dfrac{30,640}{100}$

$\sf\;Mean,\:(\overline{X})=306.4$

∵ So the required mean is 306.4.

$\rule{380}{2}$

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