### The mean of 100 items was 46. Later on it was discovered that an item 16 was misread as 61 and another item 43 was misread as 34. It was also found that the number of item was 90 and not 100. Then what is the correct mean ?

A. 50

B. 50.7

C. 52

D. 52.7

$\begin{array}{l}\text{According to the question,}\\ \text{Mean of 100 items is = 46}\\ \text{Sum of 100 items}\\ \text{= 46 × 100 = 4600}\end{array}$ Misread 61 instead of 16 and 34 instead of 43 $\begin{array}{l}\text{∴ Difference}\\ \text{= (61 + 34) – (16 + 43)}\\ \text{= 95 – 59}\\ \text{= 36 (more)}\\ \text{∴ Actual sum}\\ \text{= 4600 – 36}\\ \text{= 4564}\end{array}$ Now total observations are = 90 $\begin{array}{l}\therefore\text{ Actual average }\\ =\Large\frac{4564}{90}\\ =50.7\end{array}$
Alternate Subtract the misread and add the correct from the sum. $\begin{array}{l}\text{Sum = 100 × 46 = 4600}\\ \text{New sum}\\ \text{= 4600 – (61 + 34) + (16 + 43)}\\ \text{= 4564}\\ \text{New number of observations}\\ =90\\ \text{New average}\\ =\frac{4564}{90}\\ =50.7\end{array}$