Question

The mean and median of 100 observations are 50 and 52 respectively. The value of the largest observation is 100. It was later found that it is 110 not 100. Find the true mean and median of the observations?

Answer 1

Hint: We start solving the problem by assuming that the given 100 observations are in ascending order. We then assume a variable the sum of the observations other than the largest observation 100. We then use the fact that the mean of the ‘n’ observations ${{a}_{1}}$, ${{a}_{2}}$,……, ${{a}_{n}}$ is $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$ to find the value of the assumed sum. We then add 110 to the obtained sum and use $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$ to get the actual mean. We then recall the fact that the median of the ‘n’ observations (where n is even) written in ascending or descending order is the average of the ${{\dfrac{n}{2}}^{th}}$ and ${{\left( \dfrac{n}{2}+1 \right)}^{th}}$ observation to get the actual median.

Complete step by step answer:
According to the problem, we are given that the mean and median of 100 observations are 50 and 52 respectively. The value of the largest observation is 100 and later it was found that it is 110 not 100. We need to find the true mean and median of the 100 observations.
Let us assume that the given 100 observations are written in ascending order.
Since we have the largest observation as 100, we assume the sum of remaining 99 observations as S.
We know that the mean of the ‘n’ observations ${{a}_{1}}$, ${{a}_{2}}$,……, ${{a}_{n}}$ is $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$.
So, we have $\dfrac{S+100}{100}=50$.
$\Rightarrow S+100=5000$.
$\Rightarrow S=4900$ ---(1).
According to the problem, we are given that the largest observation is 110 not 100. Now, let us find the mean of the updated 100 observations.
$\Rightarrow mean=\dfrac{S+110}{100}$.
From equation (1), we get
$\Rightarrow mean=\dfrac{4900+110}{100}$.
$\Rightarrow mean=\dfrac{5010}{100}$.
$\Rightarrow mean=50.1$.

∴ The actual mean of the observations is 50.1.

We know that median of the ‘n’ observations (where n is even) written in ascending or descending order is the average of the ${{\dfrac{n}{2}}^{th}}$ and ${{\left( \dfrac{n}{2}+1 \right)}^{th}}$ observation.
So, the median of the given 100 observations will be average of $\dfrac{100}{2}={{50}^{th}}$ and ${{\left( \dfrac{100}{2}+1 \right)}^{th}}={{51}^{st}}$ observations.
We know that the ${{50}^{th}}$ and ${{51}^{st}}$ observations are not changed in the given 100 observations. So, the median will not be changed. So, the actual median is equal to the given median 52.

∴ The actual median is 52.

Note: Whenever we get this type of problem, we first assign variables for the unknowns present in the problem. We should know that the data of observations should be sorted in either ascending or in descending order to find the median of the observations. We can also find the value of the mode before and after making the correction of the largest observation of the data. Similarly, we can expect problems to find the difference of the mode before and after making corrections of the largest observation.