The mean of 100 observations is 50. If one observation which was 60 is replaced by 160, the resulting mean will be:

Options

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

Answer

To solve the problem step by step, we will follow the calculations as described in the video transcript.

### Step 1: Calculate the initial sum of observations
We know that the mean of 100 observations is 50. The mean is calculated as:

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

Let the sum of observations be \( S \). Therefore, we can express this as:

\[ 50 = \frac{S}{100} \]

To find \( S \), we multiply both sides by 100:

\[ S = 50 \times 100 = 5000 \]

### Step 2: Adjust the sum of observations after replacing the value
One observation, which was 60, is replaced by 160. To find the new sum of observations, we first need to remove the old observation (60) from the total sum (5000) and then add the new observation (160).

First, we subtract 60 from 5000:

\[ \text{New sum} = 5000 - 60 = 4940 \]

Next, we add 160 to this new sum:

\[ \text{New sum} = 4940 + 160 = 5100 \]

### Step 3: Calculate the new mean
Now that we have the new sum of observations (5100), we can calculate the new mean using the same formula:

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

Calculating this gives:

\[ \text{New Mean} = 51 \]

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

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