Question

if mean of first n whole numbers is 27 then what will be the median?
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Answer 1

SOLUTION

GIVEN

The mean of first n whole numbers is 27

TO DETERMINE

The median

CONCEPT TO BE IMPLEMENTED

Mean :

In measure of central tendency mean is the most common measure.

For a given set of observations mean is the average value of a group of numbers.

It is obtained by adding up all the observations divide by the number of observations

Median :

Median is the middle most value of a set of observations when the samples are arranged in order of magnitudes ( Either in ascending or in descending )

EVALUATION

Here the first n whole numbers are

0 , 1 , 2 , 3 , 4 , .... , ( n - 1 )

Now sum of the first n whole numbers

= 0 + 1 + 2 + 3 + 4 + .... + ( n - 1 )

$\displaystyle \sf{ = \frac{n}{2}(0 + n - 1) }$

$\displaystyle \sf{ = \frac{n(n - 1)}{2}}$

So Mean of the first n whole numbers

$\displaystyle \sf{ = \frac{\frac{n(n - 1)}{2}}{n} }$

$\displaystyle \sf{ = \frac{n - 1}{2} }$

By the given condition

$\displaystyle \sf{ \frac{n - 1}{2} = 27}$

$\displaystyle \sf{ \implies \: n - 1 = 54}$

$\displaystyle \sf{ \implies \: n = 55}$

So total number of observations = 55

Now 55 is odd

So required median

$\displaystyle \sf{ = \bigg( \frac{55 + 1}{2} \bigg)th \: \: observation}$

$\displaystyle \sf{ = \bigg( \frac{56}{2} \bigg)th \: \: observation}$

$\displaystyle \sf{ = 28th \: \: observation}$

$\displaystyle \sf{ = 27}$

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