The mean and standard deviation of 18 observations are found to be 7 and 4 respectively. On rechecking it was found that an observation 12 was misread as 21. Calculate the correct mean and standard deviation.

Answer

To solve the problem step by step, we will first calculate the correct mean and then the correct standard deviation after correcting the misread observation.

### Step 1: Calculate the Incorrect Summation of Observations
Given:
- Incorrect mean (\( \bar{x} \)) = 7
- Number of observations (n) = 18

The formula for mean is:
\[ \bar{x} = \frac{\Sigma x_i}{n} \]
Thus, we can find the incorrect summation of observations (\( \Sigma x_i \)):
\[ \Sigma x_i = \bar{x} \times n = 7 \times 18 = 126 \]

### Step 2: Correct the Summation of Observations
The misread observation was 12, which was read as 21. Therefore, to correct the summation:
\[ \text{Correct } \Sigma x_i = 126 + 12 - 21 = 126 + 12 - 21 = 117 \]

### Step 3: Calculate the Correct Mean
Now, we can find the correct mean (\( \bar{x}_c \)):
\[ \bar{x}_c = \frac{\text{Correct } \Sigma x_i}{n} = \frac{117}{18} = 6.5 \]

### Step 4: Calculate the Incorrect Summation of Squares
To find the standard deviation, we need the summation of squares (\( \Sigma x_i^2 \)). The formula for the standard deviation is:
\[ \sigma = \sqrt{\frac{1}{n} \Sigma x_i^2 - \bar{x}^2} \]
Given the incorrect standard deviation (\( \sigma \)) is 4, we can find \( \Sigma x_i^2 \):
\[ \sigma^2 = \frac{1}{n} \Sigma x_i^2 - \bar{x}^2 \]
Thus,
\[ 16 = \frac{1}{18} \Sigma x_i^2 - 49 \]
Rearranging gives:
\[ \frac{1}{18} \Sigma x_i^2 = 16 + 49 = 65 \]
So,
\[ \Sigma x_i^2 = 65 \times 18 = 1170 \]

### Step 5: Correct the Summation of Squares
Now, we need to correct \( \Sigma x_i^2 \) for the misread values:
\[ \text{Correct } \Sigma x_i^2 = 1170 - 21^2 + 12^2 \]
Calculating the squares:
\[ 21^2 = 441, \quad 12^2 = 144 \]
Thus,
\[ \text{Correct } \Sigma x_i^2 = 1170 - 441 + 144 = 1170 - 297 = 873 \]

### Step 6: Calculate the Correct Standard Deviation
Now we can find the correct standard deviation (\( \sigma_c \)):
\[ \sigma_c^2 = \frac{1}{n} \Sigma x_i^2 - \bar{x}_c^2 \]
Substituting the values:
\[ \sigma_c^2 = \frac{1}{18} \times 873 - (6.5)^2 \]
Calculating:
\[ \sigma_c^2 = \frac{873}{18} - 42.25 = 48.5 - 42.25 = 6.25 \]
Thus,
\[ \sigma_c = \sqrt{6.25} = 2.5 \]

### Final Answers
- Correct Mean (\( \bar{x}_c \)) = 6.5
- Correct Standard Deviation (\( \sigma_c \)) = 2.5