The mean of 100 observations is 50. If one of the observation which was 50 is replaced by 150, the resulting mean will be
(i) 50.5
(ii) 51
(iii) 51.5
(iv) 52

Options

A. 50.5
B. 51
C. 51.5
D. 52

Answer

To find the new mean after replacing one of the observations, we can follow these steps:

### Step 1: Calculate the original sum of observations
The mean of the 100 observations is given as 50. The formula for mean is:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Using this formula, we can rearrange it to find the sum of observations:

\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} \]

Substituting the given values:

\[ \text{Sum of observations} = 50 \times 100 = 5000 \]

### Step 2: Replace the observation and calculate the new sum
One observation, which was 50, is replaced by 150. To find the new sum of observations, we will subtract the old observation (50) and add the new observation (150):

\[ \text{New sum of observations} = \text{Old sum} - \text{Old observation} + \text{New observation} \]

Substituting the values we have:

\[ \text{New sum of observations} = 5000 - 50 + 150 = 5000 + 100 = 5100 \]

### Step 3: Calculate the new mean
Now that we have the new sum of observations, we can calculate the new mean using the same formula for mean:

\[ \text{New Mean} = \frac{\text{New sum of observations}}{\text{Number of observations}} \]

Substituting the values:

\[ \text{New Mean} = \frac{5100}{100} = 51 \]

### Conclusion
Therefore, the resulting mean after replacing the observation is **51**.