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Difficulty: MediumDescriptive Statistics and Data Representations

A customer service call center recorded the duration, rounded to the nearest minute, for a sample of 2020 technical support calls. The results are summarized in the frequency table below.

Call Duration (minutes)Frequency (number of calls)
34
56
85
123
152

What is the positive difference between the mean call duration and the median call duration, in minutes?

  1. A
    0.6
  2. 0.9Answer
  3. C
    1.9
  4. D
    2.1
  5. E
    2.4

Answer

The positive difference between the mean call duration and the median call duration is 0.9 minutes.
The mean is found by calculating the total sum of call durations ((3×4)+(5×6)+(8×5)+(12×3)+(15×2)=148(3 \times 4) + (5 \times 6) + (8 \times 5) + (12 \times 3) + (15 \times 2) = 148) and dividing by the sample size (2020), giving 7.47.4 minutes. Since there are 2020 values, the median is the average of the 10th value (55) and 11th value (88), which is 6.56.5 minutes. The positive difference is 7.4−6.5=0.97.4 - 6.5 = 0.9 minutes.

Step-by-Step Solution

1
Calculate the total duration of all calls combined.
(3×4)+(5×6)+(8×5)+(12×3)+(15×2)=12+30+40+36+30=148 minutes(3 \times 4) + (5 \times 6) + (8 \times 5) + (12 \times 3) + (15 \times 2) = 12 + 30 + 40 + 36 + 30 = 148\text{ minutes}.
Each call duration must be multiplied by its corresponding frequency to find the total sum of all call durations.
2
Calculate the mean call duration.
Mean=14820=7.4 minutes\text{Mean} = \frac{148}{20} = 7.4\text{ minutes}.
The mean is equal to the sum of all durations divided by the total number of calls (2020).
3
Determine the median call duration.
Median=5+82=6.5 minutes\text{Median} = \frac{5 + 8}{2} = 6.5\text{ minutes}.
For 2020 sorted data points, the median is the average of the 10th and 11th values. The cumulative frequencies show that the 1st through 4th calls are 33 minutes, the 5th through 10th calls are 55 minutes, and the 11th through 15th calls are 88 minutes. Thus, the 10th value is 55 and the 11th value is 88.
4
Compute the positive difference between the mean and the median.
7.4−6.5=0.9 minutes7.4 - 6.5 = 0.9\text{ minutes}.
Subtracting the median (6.56.5) from the mean (7.47.4) yields the required positive difference.

Key Concept

Calculating mean and median from a frequency table
Estimated Time:1m 15s
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Topics

Descriptive Statistics and Data Representations

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