To determine the correct mean and standard deviation, we need to adjust for the errors in the recorded data and then recalculate these values. Here are the steps:

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1. Correct the Errors in the Summation

• Incorrect data: $18, 27, 85$
• Correct data: $180, 72, 185$

The error introduced into the dataset: $\Delta X = (180 - 18) + (72 - 27) + (185 - 85)$ $\Delta X = 162 + 45 + 100 = 307$

Thus, the total sum of the observations should increase by $307$.

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2. Adjust the Mean

The arithmetic mean of $200$ observations was originally $145$. The sum of the data can be calculated using: $\text{Original Sum} = \text{Mean} \times \text{Number of Observations} = 145 \times 200 = 29000$ After adjusting for the error ($\Delta X = 307$): $\text{Corrected Sum} = 29000 + 307 = 29307$ The corrected mean is: $\text{Corrected Mean} = \frac{\text{Corrected Sum}}{\text{Number of Observations}} = \frac{29307}{200} = 146.535$

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3. Adjust the Standard Deviation

The formula for standard deviation is: $\sigma = \sqrt{\frac{\sum X^2}{n} - \left(\bar{X}\right)^2}$ Let us adjust the term $\sum X^2$. Initially, we can calculate: $\sum X^2 = \sigma^2 \cdot n + \left(\text{Mean}\right)^2 \cdot n = 20^2 \cdot 200 + 145^2 \cdot 200$ $\sum X^2 = 80000 + 4205000 = 4285000$ We now adjust $\sum X^2$ for the incorrect and correct values: $\Delta \sum X^2 = (180^2 - 18^2) + (72^2 - 27^2) + (185^2 - 85^2)$ $\Delta \sum X^2 = (32400 - 324) + (5184 - 729) + (34225 - 7225)$ $\Delta \sum X^2 = 32076 + 4455 + 27000 = 63531$ Corrected $\sum X^2$: $\text{Corrected } \sum X^2 = 4285000 + 63531 = 4348531$ The corrected standard deviation is then calculated as: $\sigma = \sqrt{\frac{\text{Corrected } \sum X^2}{n} - \left(\text{Corrected Mean}\right)^2}$ $\sigma = \sqrt{\frac{4348531}{200} - (146.535)^2}$ $\sigma = \sqrt{21742.655 - 21474.07} = \sqrt{268.585} \approx 16.39$

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Final Results

• Correct Mean: $146.535$
• Correct Standard Deviation: $16.39$

Would you like a detailed explanation of any step or additional examples?

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5 Questions for Exploration

1. Why does correcting individual data points have such an impact on the mean and standard deviation?
2. How does the sum of squared deviations contribute to the standard deviation?
3. Could the use of the grouped data approximation impact accuracy when errors are present?
4. How does sample size affect the sensitivity of the mean and standard deviation to data errors?
5. What are other methods to check for errors in summarized data?

Tip: Always verify the raw data against the summarized data to prevent propagation of errors in statistical analyses.