Key Concepts and Formulas

This problem involves correcting statistical measures (mean and standard deviation) when an error in data entry is identified. The core idea is to adjust the fundamental sums that define these measures: the sum of observations ($\sum x_i$) and the sum of squares of observations ($\sum x_i^2$).

1. Mean ($\bar{x}$): The average of $n$ observations $x_1, x_2, \dots, x_n$. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ From this, the sum of observations can be derived as: $\sum_{i=1}^{n} x_i = n \bar{x}$

2. Variance ($\sigma^2$): A measure of the spread of data points around the mean. The computational formula, particularly useful for correction problems, is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ From this, the sum of squares of observations can be derived as: $\sum_{i=1}^{n} x_i^2 = n \left( \sigma^2 + (\bar{x})^2 \right)$

3. Standard Deviation ($\sigma$): The positive square root of the variance. It indicates the typical deviation of data points from the mean. $\sigma = \sqrt{\sigma^2}$

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Step-by-Step Solution

1. Extract Given Information and Calculate Initial Incorrect Sums

We are given the following information about the 20 observations:

• Number of observations, $n = 20$.
• Incorrect mean, $\bar{x}_{inc} = 10$.
• Incorrect standard deviation, $\sigma_{inc} = 2$.
• Incorrectly recorded observation, $x_{wrong} = 8$.
• Correct observation, $x_{correct} = 12$.

Our first task is to use the incorrect mean and standard deviation to find the incorrect sum of observations ($\sum x_{inc}$) and the incorrect sum of squares of observations ($\sum x_{inc}^2$). These sums are the basis for all further corrections.

• Calculate the incorrect sum of observations ($\sum x_{inc}$): We use the formula $\sum x_i = n \bar{x}$. $\sum x_{inc} = n \times \bar{x}_{inc} = 20 \times 10 = 200$ Reasoning: The sum of observations is directly proportional to the mean and the number of observations.

• Calculate the incorrect sum of squares of observations ($\sum x_{inc}^2$): First, we determine the incorrect variance. Since $\sigma_{inc} = 2$, the incorrect variance $\sigma_{inc}^2 = 2^2 = 4$. Now, we use the formula $\sum x_i^2 = n (\sigma^2 + \bar{x}^2)$. $\sum x_{inc}^2 = n \left( \sigma_{inc}^2 + (\bar{x}_{inc})^2 \right)$ $\sum x_{inc}^2 = 20 \left( 4 + (10)^2 \right)$ $\sum x_{inc}^2 = 20 (4 + 100)$ $\sum x_{inc}^2 = 20 (104) = 2080$ Reasoning: The variance formula relates the sum of squares, the mean, and the number of observations. We rearrange it to find the incorrect sum of squares, which is essential for correcting the variance.

2. Correct the Sum of Observations ($\sum x_{corr}$)

Now, we adjust the incorrect sum of observations by removing the erroneous value and adding the correct one. $\sum x_{corr} = \sum x_{inc} - x_{wrong} + x_{correct}$ $\sum x_{corr} = 200 - 8 + 12$ $\sum x_{corr} = 192 + 12 = 204$ Reasoning: The sum of observations is a linear sum. To correct it, we simply subtract the wrong observation and add the correct one.

3. Correct the Sum of Squares of Observations ($\sum x_{corr}^2$)

This is a critical step for correcting the variance. We must remove the square of the incorrect value and add the square of the correct value to the sum of squares. $\sum x_{corr}^2 = \sum x_{inc}^2 - (x_{wrong})^2 + (x_{correct})^2$ $\sum x_{corr}^2 = 2080 - (8)^2 + (12)^2$ $\sum x_{corr}^2 = 2080 - 64 + 144$ $\sum x_{corr}^2 = 2016 + 144 = 2160$ Reasoning: Variance depends on the sum of squares of observations. Therefore, to correct the sum of squares, we must subtract the square of the wrong observation and add the square of the correct observation. It's a common mistake to just subtract $x_{wrong}$ and add $x_{correct}$.

4. Calculate the Correct Mean ($\bar{x}_{corr}$)

With the corrected sum of observations, we can now calculate the correct mean. $\bar{x}_{corr} = \frac{\sum x_{corr}}{n}$ $\bar{x}_{corr} = \frac{204}{20} = 10.2$ Reasoning: The correct mean is needed to calculate the correct variance, as the variance formula uses the mean.

5. Calculate the Correct Variance ($\sigma_{corr}^2$)

Using the corrected sum of squares and the newly calculated correct mean, we can find the correct variance. $\sigma_{corr}^2 = \frac{\sum x_{corr}^2}{n} - (\bar{x}_{corr})^2$ $\sigma_{corr}^2 = \frac{2160}{20} - (10.2)^2$ $\sigma_{corr}^2 = 108 - 104.04$ $\sigma_{corr}^2 = 3.96$ Reasoning: Variance is an intermediate step to finding the standard deviation. Calculating variance first helps avoid rounding errors until the very end.

6. Calculate the Correct Standard Deviation ($\sigma_{corr}$)

Finally, we take the positive square root of the correct variance to obtain the correct standard deviation. $\sigma_{corr} = \sqrt{\sigma_{corr}^2}$ $\sigma_{corr} = \sqrt{3.96}$ Reasoning: This is the final quantity requested by the problem.

Comparing this with the given options: (A) 1.94 (B) $\sqrt{3.96}$ (C) $\sqrt{3.86}$ (D) 1.8

Our calculated value $\sqrt{3.96}$ matches option (B) exactly.

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Common Mistakes & Tips

• Squaring for Sum of Squares: A frequent error is to subtract $x_{wrong}$ and add $x_{correct}$ when correcting the sum of squares ($\sum x_i^2$). Always remember to use $(x_{wrong})^2$ and $(x_{correct})^2$.
• Order of Operations: Ensure you calculate the sum of squares and the mean correctly before plugging them into the variance formula.
• Work with Variance First: It's generally advisable to calculate the variance first and then take the square root at the very end to get the standard deviation. This minimizes potential rounding errors in intermediate steps.
• Careful with Decimals: Be meticulous with decimal calculations, especially when squaring numbers.

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Summary

To correct the mean and standard deviation after an observation error, we first use the given incorrect statistics to determine the incorrect sum of observations ($\sum x_{inc}$) and sum of squares ($\sum x_{inc}^2$). Then, we adjust these sums by removing the incorrect observation (and its square) and adding the correct observation (and its square). With the corrected sums, we recalculate the correct mean. Finally, using the corrected sum of squares and the corrected mean, we compute the correct variance and then its square root to find the correct standard deviation. Following these steps systematically leads to a correct standard deviation of $\sqrt{3.96}$.

The final answer is $\boxed{\sqrt{3.96}}$, which corresponds to option (B).