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Difficulty: MediumMeasures of Central Tendency (Mean, Median, Mode)

A dataset consists of 7 integers: 5,8,12,14,16,20,5, 8, 12, 14, 16, 20, and xx. If the median of the dataset is strictly greater than the mean of the dataset, which of the following could be the value of xx? Select all such values.

  1. 44Answer
  2. B
    1010
  3. 1313Answer
  4. 1818Answer
  5. E
    2525

Answer

The values of xx that make the median strictly greater than the mean are 44, 1313, and 1818.
The median of a 7-element dataset is its 4th element when arranged in ascending order. Testing the values shows that for 44, the median (1212) exceeds the mean (11.2911.29); for 1313, the median (1313) exceeds the mean (12.5712.57); and for 1818, the median (1414) exceeds the mean (13.2913.29). All three satisfy the condition.

Step-by-Step Solution

1
Express the mean of the dataset in terms of xx.
The sum of the known 6 numbers is 5+8+12+14+16+20=755 + 8 + 12 + 14 + 16 + 20 = 75. The total sum for 7 numbers is 75+x75 + x, making the mean Mean=75+x7\text{Mean} = \frac{75 + x}{7}.
The mean is calculated as the sum of all elements divided by the total count (7).
2
Analyze the median across different ranges of xx.
For 7 numbers sorted in order, the median is the 4th number. If x≤12x \le 12, the sorted list starts with x,5,8,12x, 5, 8, 12 or similar, so the 4th element is 1212. If 12<x<1412 < x < 14, the 4th element is xx. If x≥14x \ge 14, the 4th element is 1414.
The position of xx relative to the known numbers determines which element falls into the middle (4th) spot.
3
Test the condition Median>Mean\text{Median} > \text{Mean} for each piecewise case.
Case 1 (x≤12x \le 12): 12>75+x7  ⟹  84>75+x  ⟹  x<912 > \frac{75 + x}{7} \implies 84 > 75 + x \implies x < 9. Thus, x=4x = 4 works, but x=10x = 10 does not.
Case 2 (12<x<1412 < x < 14): x>75+x7  ⟹  7x>75+x  ⟹  6x>75  ⟹  x>12.5x > \frac{75 + x}{7} \implies 7x > 75 + x \implies 6x > 75 \implies x > 12.5. Thus, x=13x = 13 works.
Case 3 (x≥14x \ge 14): 14>75+x7  ⟹  98>75+x  ⟹  x<2314 > \frac{75 + x}{7} \implies 98 > 75 + x \implies x < 23. Thus, x=18x = 18 works, but x=25x = 25 does not.
Solving the inequality for each case yields all valid ranges for xx: x<9x < 9, x=13x = 13, and 14≤x<2314 \le x < 23.

Key Concept

Measures of Central Tendency (Mean vs. Median Analysis with Variables)
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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