Question

The mean of \(50\) observations was \(36\). It was found later that an observation \(48\) was wrongly taken as \(23\). The corrected new mean is?

Hint:

When correcting a wrong observation in the mean, adjust the total sum by the difference between the correct and wrong value, then divide by the total number of observations.

Answer 1

Solution:
Step 1 (Find the reported total).
Mean = \(36\), number of observations \(n = 50\). Reported sum of all observations: \[ S_{\text{reported}} = \text{Mean} \times n = 36 \times 50 = 1800. \] Step 2 (Correct the error in the total).
The wrong entry \(= 23\) should have been \(48\). Difference to add: \(48 - 23 = 25\). Corrected sum: \[ S_{\text{correct}} = 1800 + 25 = 1825. \] Step 3 (Compute the corrected mean).
\[ \text{Corrected Mean} = \frac{S_{\text{correct}}}{n} = \frac{1825}{50} = 36.5. \] Step 4 (Verification).
Replacing \(23\) with \(48\) increases the sum by \(25\), which when distributed over \(50\) observations increases the mean by \(25/50 = 0.5\), so \(36 + 0.5 = 36.5\).
\[ {36.5} \]