Points to Remember:
- Arithmetic mean is the average of a set of numbers.
- Errors in data affect the calculated mean.
- Correction involves adjusting the sum of scores and recalculating the mean.
Introduction:
The arithmetic mean, or average, is a fundamental statistical measure representing the central tendency of a dataset. It’s calculated by summing all values and dividing by the number of values. Accuracy in data collection is crucial for obtaining a reliable mean. In this case, we have a scenario where two scores were misread, necessitating a correction to the calculated arithmetic mean. This question requires a factual and analytical approach to correct the error and recalculate the mean.
Body:
1. Initial Calculation:
The initial arithmetic mean was calculated as 40 for a group of 200 students. This implies that the sum of all scores was 40 * 200 = 8000.
2. Identifying the Error:
Two scores were misread: 43 was misread as 34 (a difference of +9), and 35 was misread as 53 (a difference of -18).
3. Correcting the Sum of Scores:
To correct the sum, we need to adjust for the errors:
- The sum was underestimated by 9 (43 – 34 = 9).
- The sum was overestimated by 18 (53 – 35 = 18).
Therefore, the net error in the sum is 9 – 18 = -9. The corrected sum of scores is 8000 – 9 = 7991.
4. Calculating the Corrected Arithmetic Mean:
The corrected arithmetic mean is calculated by dividing the corrected sum of scores by the number of students:
Corrected Mean = 7991 / 200 = 39.955
5. Interpretation:
The corrected arithmetic mean of 39.955 is slightly lower than the initial mean of 40. This demonstrates the impact of even small errors in data on the calculated average. The difference is minimal in this case, but in other situations, with larger datasets or larger errors, the impact could be more significant.
Conclusion:
The initial arithmetic mean of 40 was based on inaccurate data. By identifying and correcting the errors in the two misread scores, we obtained a corrected arithmetic mean of 39.955. This highlights the importance of data accuracy in statistical analysis. While the difference is small in this specific example, the process of correcting errors is crucial for maintaining the integrity and reliability of statistical results. Future data collection should emphasize careful recording and verification to minimize such errors and ensure the accuracy of calculated means and other statistical measures. This approach ensures a more robust and reliable understanding of the data representing the students’ performance.
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