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Difficulty: HardDescriptive Statistics and Data Representations

A high school robotics team recorded their scores across 5 qualification matches: 90,78,86,82,90, 78, 86, 82, and 8484. In their 6th qualification match, the team earned a score of xx, which increased their overall 6-match mean score by 33 points compared to their 5-match mean score. What is the positive difference between the median and the mean of all 6 match scores?

  1. 22Answer
  2. B
    33
  3. C
    55
  4. D
    1515
  5. E
    1818

Answer

The positive difference between the mean and median of all 6 match scores is 2.
The mean of the original 5 matches is (90+78+86+82+84)÷5=84(90 + 78 + 86 + 82 + 84) \div 5 = 84. Since the 6th match increases the overall mean by 3, the new mean for 6 matches is 84+3=8784 + 3 = 87. The total score for all 6 matches is 87×6=52287 \times 6 = 522. Subtracting the original total of 420 gives the 6th match score: 522−420=102522 - 420 = 102. Arranging all 6 scores in ascending order gives 78,82,84,86,90,10278, 82, 84, 86, 90, 102. The median is the average of the middle two numbers, 84 and 86, which equals 85. The positive difference between the mean (87) and the median (85) is 87−85=287 - 85 = 2.

Step-by-Step Solution

1
Calculate the mean of the first 5 matches.
Sum = 90+78+86+82+84=42090 + 78 + 86 + 82 + 84 = 420. Mean = 420÷5=84420 \div 5 = 84.
The initial mean is needed to establish the target mean for 6 matches.
2
Determine the new 6-match mean and calculate the 6th score xx.
New Mean = 84+3=8784 + 3 = 87. Total required sum = 87×6=52287 \times 6 = 522. x=522−420=102x = 522 - 420 = 102.
An increase of 3 in the mean score means the new average across all 6 matches is 87.
3
Order all 6 match scores from least to greatest and find the median.
Sorted scores: 78,82,84,86,90,10278, 82, 84, 86, 90, 102. Median = (84+86)÷2=85(84 + 86) \div 2 = 85.
For an even number of values, the median is the average of the two central numbers after sorting.
4
Calculate the positive difference between the 6-match mean and median.
∣87−85∣=2|87 - 85| = 2.
Subtracting the median (85) from the mean (87) yields the required difference.

Key Concept

Descriptive Statistics: Finding Missing Data Points, Mean, and Median
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Topics

Descriptive Statistics and Data Representations

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