The mean and sample standard deviation of the dataset consisting of 9 observations is 14 and 10 respectively. Later it is noted that one observation 11 is wrongly noted as 6. What is the sample variance of the original dataset? (Correct to 2 decimal place accuracy)

To find the sample variance of the original dataset, we need to correct the mistakenly recorded value and then recalculate the variance.

Given:
- Mean (\(\bar{x}\)) of the dataset: 14
- Sample standard deviation (s): 10
- Number of observations (n): 9
- Incorrect value: 6 (should be 11)

First, let's determine the sum of the original dataset using the mean:
\[
\bar{x} = \frac{\sum x_i}{n} \implies \sum x_i = \bar{x} \times n = 14 \times 9 = 126
\]

Now, let's find the sum of squares of the original dataset using the variance formula:
\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \implies \sum_{i=1}^{n} (x_i - \bar{x})^2 = s^2 \times (n-1) = 10^2 \times 8 = 800
\]

Next, we need to adjust this sum of squares to account for the incorrect value:
\[
\text{Original sum of squares} = \sum_{i=1}^{9} x_i^2
\]

The incorrect dataset sum of squares can be expressed as:
\[
\sum_{i=1}^{9} x_i^2 = 800 + n \cdot \bar{x}^2 = 800 + 9 \cdot 14^2 = 800 + 1764 = 2564
\]

Identify and correct the contribution of the incorrect value:
- Incorrect value: 6
- Correct value: 11

The difference in the squares due to the correction is:
\[
11^2 - 6^2 = 121 - 36 = 85
\]

The corrected sum of squares:
\[
\text{Corrected sum of squares} = 2564 + 85 = 2649
\]

Recalculate the sample variance with the corrected sum of squares:
\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i^2) = \frac{2649 - 126^2 / 9}{8} = \frac{2649 - 1764}{8} = \frac{885}{8} = 110.625
\]

Thus, the sample variance of the original dataset, corrected to two decimal places, is:
\[
110.63
\]

Would you like further details or have any questions? Here are some related questions you might find useful:

1. How do you calculate the sample mean from a dataset?
2. What is the difference between sample variance and population variance?
3. How do you find the sum of squares in a dataset?
4. What steps are involved in correcting a dataset with an error?
5. How do changes in individual data points affect the overall variance?

**Tip:** Always double-check your dataset for accuracy before performing statistical calculations to ensure valid results.

Learn how to calculate the sample variance of a dataset with given mean and standard deviation. This problem involves correcting a wrongly noted observation and recalculating variance. Suitable for advanced high school students studying statistics.

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