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Chapter 1: Problem 27

In a class of 25 students, 24 of them took an exam in class and 1 student took a make-up exam the following day. The professor graded the first batch of 24 exams and found an average score of 74 points with a standard deviation of 8.9 points. The student who took the make-up the following day scored 64 points on the exam. (a) Does the new student's score increase or decrease the average score? (b) What is the new average? (c) Does the new student's score increase or decrease the standard deviation of the scores?

Short Answer

Expert verified
(a) Decrease; (b) 73.6; (c) Increase.

Step by step solution

01

Understand the Problem

You are given the average and standard deviation for an exam taken by 24 students. You also know the score of a 25th student who took a makeup exam. You need to determine how this new score affects the average and standard deviation.
02

Calculate the New Total Score

First, find the total score for the original 24 students and then add the score of the 25th student.The original average is 74, so:\[\text{Total for 24 students} = 24 \times 74 = 1776\]The new student scored 64:\[\text{New total score} = 1776 + 64 = 1840\]
03

Calculate the New Average

To find the new average, divide the new total score by the number of students including the 25th:\[\text{New average} = \frac{1840}{25} = 73.6\]
04

Compare New and Old Average

The original average was 74, and the new average is 73.6. Since 73.6 is less than 74, the new student's score decreases the average.
05

Effect on Standard Deviation Explanation

When a score is added that is below the mean, it tends to increase the standard deviation. This is because it increases the dispersion of scores from the average. However, to precisely calculate the new standard deviation, you would need more detailed formula work involving all individual scores.
06

Summarize the Results

The makeup student's score lowers the average to 73.6 and increases the dispersion, thus likely increasing the standard deviation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Calculation
When calculating the average, or mean, it's essential to understand how each additional data point affects the overall calculation. The average is determined by summing up all individual scores or data points and dividing this total by the number of scores. In our scenario, we initially have 24 students with an average score of 74. Given that this is an arithmetic mean, it means
  • Total sum of scores for 24 students = 24 \times 74 = 1776.
  • After adding the 25th student's score of 64, the new total sum of scores becomes 1840.
  • The new average score is calculated by dividing 1840 by 25, the new total number of students, giving us \(\frac{1840}{25} = 73.6\).
As observed, the introduction of a lower score tends to pull the mean down, making the new average of 73.6, which is slightly less than the initial 74. Remember, averages provide a central value but do not account for outliers or variations in scores directly.
Standard Deviation
Standard deviation (SD) is a statistical measure, reflecting how much individual scores in a dataset differ from the mean. It provides insight into the spread of data points:
  • If scores cluster around the mean, the SD is low.
  • If scores disperse broadly, the SD increases.
In the given example, before the makeup score was added, the SD was 8.9. Introducing the new score that differs significantly from the mean (64 instead of approximately 74) increases the spread of scores. Hence, it intuitively raises the standard deviation. Why? Because the 64 score contributes to spreading the scores further from the average. However, to calculate the exact new SD would require additional computations using the variance formulas and individual score data.
Score Comparison
Comparing scores can help analyze how individual performances affect overall statistics like average and standard deviation. Here, the new score (64) is compared with the existing average score of 74:
  • The new score is clearly lower, suggesting it will pull the average downward when added to the dataset.
  • In practical terms, such a scoring pattern could indicate whether a student needs more assistance or if test conditions varied significantly.
When integrated into the class score data, observing the effects on both average and SD confirms that placing such a divergent score emphasizes variation in performance across the class. Therefore, comparisons like these are vital to understand the broader picture of class results, pinpointing anomalies, or addressing exceptional conditions or supportive insights like needing additional help for certain students.

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Most popular questions from this chapter

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