For a group of 200 candidates the mean and standard deviation were found to be 40 and 15 repectively . Later on it was found that the score 43 was misread as 34. Find the correct mean and standard deviation.

Answer

To find the correct mean and standard deviation after correcting the misread score, we can follow these steps:

### Step 1: Calculate the original total score
Given:
- Mean (M) = 40
- Number of candidates (N) = 200

The formula for mean is:
\[ M = \frac{\Sigma x_i}{N} \]
Where \(\Sigma x_i\) is the total score.

Substituting the values:
\[ 40 = \frac{\Sigma x_i}{200} \]

Multiplying both sides by 200:
\[ \Sigma x_i = 40 \times 200 = 8000 \]

### Step 2: Correct the total score
The misread score was 34 instead of 43. Therefore, we need to adjust the total score:
\[ \text{Corrected total score} = \Sigma x_i - \text{misread score} + \text{correct score} \]
\[ \text{Corrected total score} = 8000 - 34 + 43 = 8009 \]

### Step 3: Calculate the corrected mean
Using the corrected total score:
\[ \text{Corrected Mean} = \frac{\text{Corrected total score}}{N} = \frac{8009}{200} = 40.045 \]

### Step 4: Calculate the original sum of squares
Given:
- Standard deviation (SD) = 15

The formula for standard deviation is:
\[ \sigma = \sqrt{\frac{\Sigma x_i^2}{N} - M^2} \]
Squaring both sides:
\[ \sigma^2 = \frac{\Sigma x_i^2}{N} - M^2 \]
Substituting the known values:
\[ 15^2 = \frac{\Sigma x_i^2}{200} - 40^2 \]
\[ 225 = \frac{\Sigma x_i^2}{200} - 1600 \]

Rearranging gives:
\[ \frac{\Sigma x_i^2}{200} = 225 + 1600 = 1825 \]
Multiplying by 200:
\[ \Sigma x_i^2 = 1825 \times 200 = 365000 \]

### Step 5: Correct the sum of squares
We need to adjust the sum of squares for the misread score:
\[ \text{Corrected } \Sigma x_i^2 = \Sigma x_i^2 - (\text{misread score})^2 + (\text{correct score})^2 \]
\[ \text{Corrected } \Sigma x_i^2 = 365000 - 34^2 + 43^2 \]
Calculating the squares:
\[ 34^2 = 1156 \quad \text{and} \quad 43^2 = 1849 \]
Thus,
\[ \text{Corrected } \Sigma x_i^2 = 365000 - 1156 + 1849 = 365693 \]

### Step 6: Calculate the corrected standard deviation
Using the corrected sum of squares:
\[ \text{Corrected } \sigma^2 = \frac{\text{Corrected } \Sigma x_i^2}{N} - (\text{Corrected Mean})^2 \]
Substituting the values:
\[ \text{Corrected } \sigma^2 = \frac{365693}{200} - (40.045)^2 \]
Calculating:
\[ \text{Corrected } \sigma^2 = 1828.465 - 1603.604 = 224.861 \]
Taking the square root:
\[ \text{Corrected } \sigma = \sqrt{224.861} \approx 14.995 \]

### Final Results
- Corrected Mean = 40.045
- Corrected Standard Deviation = 14.995