Question

The average height of 30 boys was calculated to be $150cm$. It was detected later that one value of $165cm$ was wrongly copied as $135cm$ for the computation of the mean. Find the correct mean.

Answer 1

Hint: In this question, we are given an average height of 30 boys but we are also told that one entry has been wrongly entered as $135cm$ instead of $165cm$. We have to find the correct mean. First, find the sum of heights using the given mean. Then, subtract the wrong observation from the sum and add the correct observation. Then divide the new sum by the number of observations. You will get the answer.

Complete step by step solution:
We are given an average height of 30 boys but we are also told that one entry has been wrongly entered as $135cm$ instead of $165cm$. We have been asked to find the correct mean.
We will start by finding the old sum of heights of 30 boys by putting the values in the formula –
$ \Rightarrow $ Mean = $\dfrac{{{\text{Sum of observations}}}}{{{\text{No}}{\text{. of observations}}}}$
Putting all the values,
$ \Rightarrow $$150cm$= $\dfrac{{{\text{Sum}}}}{{30}}$
Shifting and multiplying to find the sum,
$ \Rightarrow $ Sum = $30 \times 150$
$ \Rightarrow $ Sum = $4500cm$
We will name it ‘Old Sum’.
Therefore, old sum = $4500cm$.
Now, to find the correct mean, we will have to find the correct sum first. So, we will subtract the wrong observation and add the correct observation.
New sum = Old Sum – Wrong observation + Correct observation
Putting all the values,
$ \Rightarrow $ New sum = $4500 - 135 + 165$
$ \Rightarrow $ New sum = $4530cm$
Now, to find the new mean, we will put the values in the formula.
$ \Rightarrow $ New Mean = $\dfrac{{4530}}{{30}}$
$ \Rightarrow $ New Mean = $151cm$

Therefore, our new, correct mean is $151cm$.

Note: Instead of calculating the old and new sum, we can use a shortcut. In this shortcut, we find the net difference between the wrong and the correct observation. Then, we will find the average change and add or subtract it from the given average. It is done in the following way:
Net change = Correct observation – Wrong observation
Net change = $165 - 135 = 30$
Average change = $\dfrac{{{\text{Net change}}}}{{{\text{Total observations}}}}$
Average change =$\dfrac{{30}}{{30}}$$ = 1$
Since the average change is positive, we will add the average change to the given mean.
New mean = given mean + average change
New mean = $150 + 1 = 151cm$
Hence, the answer remains the same.