The mean and standard deviation of 10 items were found to be 17 and `sqrt(33)`. Later it was detected that an item was tekn wrongly as 26 in place of 12. Find the correct mean and standard deviation.

Answer

To solve the problem, we need to find the correct mean and standard deviation after correcting the wrongly recorded item. Let's go through the solution step by step.

### Step 1: Calculate the original sum of the items
Given:
- Mean (\( \bar{x} \)) = 17
- Number of items (n) = 10

Using the formula for mean:
\[ \bar{x} = \frac{\Sigma x_i}{n} \]
We can find the total sum of the items (\( \Sigma x_i \)):
\[ \Sigma x_i = \bar{x} \times n = 17 \times 10 = 170 \]

### Step 2: Correct the sum of the items
The item was recorded as 26 instead of 12. We need to adjust the sum:
\[ \text{Corrected sum} = \Sigma x_i - 26 + 12 \]
Calculating this:
\[ \text{Corrected sum} = 170 - 26 + 12 = 156 \]

### Step 3: Calculate the correct mean
Now that we have the corrected sum, we can find the correct mean:
\[ \text{Correct mean} = \frac{\text{Corrected sum}}{n} = \frac{156}{10} = 15.6 \]

### Step 4: Calculate the original standard deviation
Given:
- Standard deviation (\( \sigma \)) = \( \sqrt{33} \)

Using the formula for standard deviation:
\[ \sigma = \sqrt{\frac{\Sigma x_i^2}{n} - \bar{x}^2} \]
We can express this as:
\[ \sqrt{33} = \sqrt{\frac{\Sigma x_i^2}{10} - 17^2} \]

Squaring both sides:
\[ 33 = \frac{\Sigma x_i^2}{10} - 289 \]
Thus,
\[ \frac{\Sigma x_i^2}{10} = 33 + 289 = 322 \]
So,
\[ \Sigma x_i^2 = 3220 \]

### Step 5: Correct the sum of squares
We need to adjust the sum of squares for the corrected item:
\[ \text{Corrected } \Sigma x_i^2 = \Sigma x_i^2 - 26^2 + 12^2 \]
Calculating this:
\[ \text{Corrected } \Sigma x_i^2 = 3220 - 676 + 144 = 2688 \]

### Step 6: Calculate the correct standard deviation
Now we can find the correct standard deviation using the corrected sum of squares:
\[ \text{Correct standard deviation} = \sqrt{\frac{\text{Corrected } \Sigma x_i^2}{n} - (\text{Correct mean})^2} \]
Substituting the values:
\[ \text{Correct standard deviation} = \sqrt{\frac{2688}{10} - (15.6)^2} \]
Calculating:
\[ \text{Correct standard deviation} = \sqrt{268.8 - 243.36} = \sqrt{25.44} \approx 5.04 \]

### Final Answers
- Correct Mean = 15.6
- Correct Standard Deviation ≈ 5.04

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