Key Concepts and Formulas

To accurately solve problems involving the correction of statistical data, we rely on the fundamental definitions of mean, variance, and standard deviation. For a set of $n$ observations $x_1, x_2, \ldots, x_n$:

1. Mean ($\bar{x}$): The average value, indicating the central tendency of the data. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ Here, $\sum_{i=1}^{n} x_i$ represents the sum of all observations.

2. Variance ($\sigma^2$): A measure of how spread out the data points are from their mean. A widely used computational formula is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ Here, $\sum_{i=1}^{n} x_i^2$ represents the sum of the squares of all observations.

3. Standard Deviation ($\sigma$): The square root of the variance. It is expressed in the same units as the data, making it easier to interpret the typical deviation from the mean. $\sigma = \sqrt{\sigma^2}$ By convention, standard deviation is always a non-negative value.

Problem-Solving Strategy

When an error in an observation is identified, the strategy to find the correct statistical measures involves a systematic correction process:

1. Extract Incorrect Sums: Use the given incorrect mean and variance to determine the incorrect sum of observations ($\sum x_{i,inc}$) and the incorrect sum of squares of observations ($\sum x_{i,inc}^2$).
2. Adjust for Error: Correct these sums by subtracting the value (or its square) of the erroneous observation and adding the value (or its square) of the correct observation. This yields the correct sum of observations ($\sum x_{i,cor}$) and correct sum of squares of observations ($\sum x_{i,cor}^2$).
3. Calculate Correct Mean: Use the corrected sum of observations to find the correct mean ($\bar{x}_{cor}$).
4. Calculate Correct Variance: Employ the corrected sum of squares and the corrected mean to compute the correct variance ($\sigma^2_{cor}$).
5. Calculate Correct Standard Deviation: Finally, take the square root of the correct variance to find the correct standard deviation ($\sigma_{cor}$).

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Step-by-Step Solution

We are given information about 10 observations, where one observation was mistakenly recorded. Our goal is to find the correct standard deviation.

Given Information:

• Number of observations ($n$) = 10
• Incorrect Mean ($\bar{x}_{inc}$) = 15
• Incorrect Variance ($\sigma^2_{inc}$) = 15
• Observation mistakenly taken ($x_{inc}$) = 25
• Correct observation ($x_{cor}$) = 15

Step 1: Extracting Information from Incorrect Calculations

First, we use the given incorrect mean and variance to find the sums based on the erroneous data.

a. Calculate the Incorrect Sum of Observations ($\sum x_{i,inc}$): The definition of the mean allows us to find the total sum of observations. $\bar{x}_{inc} = \frac{\sum x_{i,inc}}{n}$ We rearrange this to solve for the sum: $\sum x_{i,inc} = n \times \bar{x}_{inc}$ Substituting the given values: $\sum x_{i,inc} = 10 \times 15 = 150$

• Why this step? This sum (150) includes the incorrect observation (25). It's our starting point to adjust the total sum of the dataset correctly.

b. Calculate the Incorrect Sum of Squares of Observations ($\sum x_{i,inc}^2$): We use the computational formula for variance to find the sum of squares. $\sigma^2_{inc} = \frac{\sum x_{i,inc}^2}{n} - (\bar{x}_{inc})^2$ Substituting the given incorrect variance and mean: $15 = \frac{\sum x_{i,inc}^2}{10} - (15)^2$ $15 = \frac{\sum x_{i,inc}^2}{10} - 225$ Now, we isolate the term $\frac{\sum x_{i,inc}^2}{10}$: $\frac{\sum x_{i,inc}^2}{10} = 15 + 225$ $\frac{\sum x_{i,inc}^2}{10} = 240$ Finally, we find the sum of squares: $\sum x_{i,inc}^2 = 240 \times 10 = 2400$

• Why this step? This sum of squares (2400) includes the square of the incorrect observation ($25^2 = 625$). This value is crucial for correctly calculating the new variance after adjusting for the error.

Step 2: Correcting the Sums

Now we adjust the sums obtained in Step 1 to reflect the correct observation.

a. Calculate the Correct Sum of Observations ($\sum x_{i,cor}$): To correct the sum, we remove the incorrect observation's value and add the correct observation's value. $\sum x_{i,cor} = \sum x_{i,inc} - x_{inc} + x_{cor}$ Substituting the values: $\sum x_{i,cor} = 150 - 25 + 15$ $\sum x_{i,cor} = 125 + 15 = 140$

• Why this step? This gives us the true sum of all 10 observations, having replaced the erroneous value with the correct one.

b. Calculate the Correct Sum of Squares of Observations ($\sum x_{i,cor}^2$): Similarly, to correct the sum of squares, we subtract the square of the incorrect observation and add the square of the correct observation. $\sum x_{i,cor}^2 = \sum x_{i,inc}^2 - (x_{inc})^2 + (x_{cor})^2$ Substituting the values: $\sum x_{i,cor}^2 = 2400 - (25)^2 + (15)^2$ $\sum x_{i,cor}^2 = 2400 - 625 + 225$ $\sum x_{i,cor}^2 = 1775 + 225 = 2000$

• Why this step? It's essential to adjust the sum of squares by subtracting the square of the incorrect value and adding the square of the correct value. This is a common point of error if one were to mistakenly subtract/add the values themselves rather than their squares.

Step 3: Calculating the Correct Mean ($\bar{x}_{cor}$)

With the correct sum of observations, we can now find the correct mean. The number of observations ($n$) remains 10. $\bar{x}_{cor} = \frac{\sum x_{i,cor}}{n}$ $\bar{x}_{cor} = \frac{140}{10} = 14$

• Why this step? The correct mean is a necessary component for calculating the correct variance using its computational formula.

Step 4: Calculating the Correct Variance ($\sigma^2_{cor}$)

Using the corrected sum of squares and the newly calculated correct mean, we can determine the correct variance. $\sigma^2_{cor} = \frac{\sum x_{i,cor}^2}{n} - (\bar{x}_{cor})^2$ Substituting the corrected values: $\sigma^2_{cor} = \frac{2000}{10} - (14)^2$ $\sigma^2_{cor} = 200 - 196$ $\sigma^2_{cor} = 4$

• Why this step? This is the central calculation for the spread of the corrected data. It's vital to use both the correct sum of squares and the correct mean to ensure accuracy.

Step 5: Calculating the Correct Standard Deviation ($\sigma_{cor}$)

Finally, we find the correct standard deviation by taking the positive square root of the correct variance. $\sigma_{cor} = \sqrt{\sigma^2_{cor}}$ $\sigma_{cor} = \sqrt{4} = 2$

• Why this step? The question specifically asks for the standard deviation, which is the final step in this correction process. Standard deviation is always non-negative, representing a magnitude of dispersion.

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Common Mistakes & Tips

• Adjust Sums First: Always work with the sums ($\sum x_i$ and $\sum x_i^2$) as intermediate steps. Do not attempt to directly adjust the mean or variance values, as this often leads to errors.
• Square the Values for Sum of Squares: When correcting the sum of squares ($\sum x_i^2$), remember to subtract the square of the incorrect observation and add the square of the correct observation, i.e., $(x_{inc})^2$ and $(x_{cor})^2$. A common mistake is to use $x_{inc}$ and $x_{cor}$ directly.
• Use Correct Mean for Correct Variance: After calculating the correct sum of observations and sum of squares, ensure you calculate the correct mean before using it in the variance formula. Using the original (incorrect) mean with corrected sums of squares is a frequent error.

Summary

To correct the mean and variance after a data entry error, the most reliable method is to first reverse the calculations to find the incorrect sums of observations and their squares. Then, adjust these sums by removing the erroneous data point's contribution and adding the correct data point's contribution. Finally, recalculate the mean, variance, and standard deviation using these corrected sums. Following these steps systematically ensures an accurate result.

The final answer is $\boxed{2}$.