Find the mean and median of the data :
35, 48, 92, 76, 64, 52, 51, 63 and 71.
If 51 is replaced by 66, what will be the new median ?

Answer

To solve the problem step by step, we will first find the mean and median of the given data, and then we will determine the new median after replacing 51 with 66.

### Step 1: Find the Mean

1. **List the Data**: The data given is: 35, 48, 92, 76, 64, 52, 51, 63, 71.
2. **Count the Number of Observations**: There are 9 observations in total.
3. **Calculate the Sum of Observations**:
\[ 35 + 48 + 92 + 76 + 64 + 52 + 51 + 63 + 71 = 552 \]
4. **Calculate the Mean**:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} = \frac{552}{9} \approx 61.33 \]

### Step 2: Find the Median

1. **Arrange the Data in Ascending Order**:
The ordered data is: 35, 48, 51, 52, 63, 64, 71, 76, 92.
2. **Identify the Number of Observations**: There are still 9 observations (odd number).
3. **Find the Median Position**:
\[ \text{Median Position} = \frac{n + 1}{2} = \frac{9 + 1}{2} = 5 \]
The 5th term in the ordered data is 63.
4. **Thus, the Median is**: 63.

### Step 3: Replace 51 with 66 and Find the New Median

1. **Update the Data**: Replace 51 with 66. The new data set is: 35, 48, 66, 52, 63, 64, 71, 76, 92.
2. **Rearrange the Updated Data in Ascending Order**:
The ordered data is: 35, 48, 52, 63, 64, 66, 71, 76, 92.
3. **Identify the Number of Observations**: There are still 9 observations (odd number).
4. **Find the New Median Position**:
\[ \text{New Median Position} = \frac{n + 1}{2} = \frac{9 + 1}{2} = 5 \]
The 5th term in the ordered data is now 64.
5. **Thus, the New Median is**: 64.

### Summary of Results
- **Mean**: 61.33
- **Original Median**: 63
- **New Median (after replacing 51 with 66)**: 64