The mean of 100 observations is 50. If one of the observations which was 50 is replaced by 150, the resulting mean will be

Options

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

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the sum of the original observations.
The mean of the observations is given by the formula:

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

Given that the mean is 50 and the number of observations is 100, we can rearrange the formula to find the sum of the observations:

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

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

\[ \text{New Sum} = \text{Old Sum} - \text{Old Observation} + \text{New Observation} \]

Substituting the values:

\[ \text{New Sum} = 5000 - 50 + 150 = 5000 + 100 = 5100 \]

### Step 3: Calculate the new mean.
The number of observations remains the same at 100. We can now calculate the new mean using the new sum:

\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of observations}} = \frac{5100}{100} = 51 \]

### Final Answer:
The resulting mean after replacing the observation is **51**.

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