The mean and sample standard deviation of the dataset consisting of 10 observations is 15 and 11 respectively. Later it is noted that one observation 14 is wrongly noted as 11. What is the sample variance of the original dataset? (Correct to 2 decimal place accuracy)

We are given:

- Mean (\(\bar{x}\)) of the dataset = 15
- Sample standard deviation (\(s\)) = 11
- Number of observations (\(n\)) = 10
- One observation, 14, was incorrectly recorded as 11.

### Step 1: Correct the mean of the dataset
The sum of the original dataset (\(\sum x_i\)) can be calculated from the given mean:
\[
\bar{x} = \frac{\sum x_i}{n}
\]
\[
\sum x_i = \bar{x} \cdot n = 15 \cdot 10 = 150
\]

Now, the wrongly recorded observation was 11 instead of 14. So, to get the correct sum of the dataset, we add the difference:
\[
\text{Corrected sum} = 150 + (14 - 11) = 150 + 3 = 153
\]

The corrected mean of the dataset is:
\[
\bar{x}_{\text{corrected}} = \frac{153}{10} = 15.3
\]

### Step 2: Correct the sample variance
The sample variance is given by:
\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\]
First, compute the sum of squared deviations using the initial sample variance:
\[
s^2 = 11^2 = 121
\]
\[
\sum (x_i - \bar{x})^2 = s^2 \cdot (n - 1) = 121 \cdot 9 = 1089
\]

Now, we need to correct this sum by adjusting for the wrongly recorded observation. The wrongly recorded deviation from the mean (\(x_i - \bar{x}\)) was:
\[
11 - 15 = -4
\]
The correct deviation should have been:
\[
14 - 15 = -1
\]

So, the correction to the sum of squared deviations is:
\[
\text{Correction} = (-1)^2 - (-4)^2 = 1 - 16 = -15
\]

Thus, the corrected sum of squared deviations is:
\[
\sum (x_i - \bar{x}_{\text{corrected}})^2 = 1089 - 15 = 1074
\]

Finally, the corrected sample variance is:
\[
s^2_{\text{corrected}} = \frac{1074}{9} = 119.33
\]

### Conclusion:
The corrected sample variance of the original dataset is \(119.33\).

Would you like more details or have any questions?

Here are some related questions for further practice:
1. How does a change in the mean affect the variance of a dataset?
2. What is the difference between sample variance and population variance?
3. How is sample standard deviation different from variance?
4. What steps would you take to verify if other data points in the set are also incorrect?
5. How would the variance change if two observations were incorrectly noted instead of one?

**Tip:** Always check if any data entry errors exist before calculating statistical measures for accuracy!

Learn how to calculate the corrected sample variance when an observation in a dataset is wrongly recorded. Understand the steps to adjust the mean, standard deviation, and variance, using detailed examples and formulas. Suitable for high school students in Grades 11-12.

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