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Difficulty: MediumData Distributions and Measures

A software developer recorded the number of software bugs resolved each day for 20 days, as shown in the table below.

Bugs ResolvedNumber of Days
02
14
27
35
42

On the 21st day, the developer resolved 10 bugs. Which of the following statements best describes the effect of this new data point on the mean and the median of the dataset?

  1. The mean increases, and the median remains the same.Answer
  2. B
    The median increases, and the mean remains the same.
  3. C
    Both the mean and the median increase.
  4. D
    Both the mean and the median remain the same.

Answer

The mean increases, and the median remains the same.
The mean is sensitive to extreme values, so adding a value of 10, which is significantly larger than the initial mean of 2.05, increases the mean of the dataset. The median is resistant to extreme values; since the 11th value of the sorted 21-element dataset is still 2, the median remains unchanged.

Step-by-Step Solution

1
Calculate the initial mean and median of the dataset.
The initial mean is 2.05 and the initial median is 2.
The sum of the values is (0 * 2) + (1 * 4) + (2 * 7) + (3 * 5) + (4 * 2) = 41. With 20 days, the mean is 41 / 20 = 2.05. The median of 20 values is the average of the 10th and 11th values when sorted. Ordering the frequencies shows that indices 7 through 13 contain the value 2, so both the 10th and 11th values are 2, giving a median of 2.
2
Calculate the new mean and median after adding the 21st data point (10).
The new mean is approximately 2.43 and the new median is 2.
The new sum of values is 41 + 10 = 51. With 21 days, the new mean is 51 / 21 ≈ 2.43. The median of 21 values is the 11th value in the sorted list. Since the 11th value is still within the group of 2s (indices 7 through 13), the new median is 2.
3
Compare the initial values to the new values to determine the overall changes.
The mean increases from 2.05 to 2.43, while the median remains 2.
Comparing the results shows that the mean increases and the median remains the same.

Key Concept

Effect of outliers on measures of center (mean vs. median)
Estimated Time:1m 30s
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Topics

Data Distributions and Measures

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