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

A dataset SS consists of 55 positive integers. The dataset has a unique mode of 1515, a median of 1515, and an arithmetic mean of 1212.

Which of the following could be the range of dataset SS? Indicate all such values.

  1. A
    8
  2. 10Answer
  3. 18Answer
  4. 26Answer
  5. E
    28

Answer

The values 10, 18, and 26 are all possible values for the range of dataset SS.
The range of dataset S is defined as x_5 - x_1. Based on the mean, median, and unique mode conditions, the 5 positive integers must take the form x_1 < x_2 < 15 = 15 < x_5 with x_1 + x_2 + x_5 = 30. Testing valid integer assignments shows the range can be any integer from 10 to 26 inclusive. Therefore, 10, 18, and 26 are all valid possible range values.

Step-by-Step Solution

1
Express the sum of elements using the mean
Sum of 5 elements = 5 × 12 = 60
Since the mean of 5 numbers is 12, their total sum must equal 60.
2
Set up the ordered elements and apply the median and unique mode conditions
Elements in ascending order: x_1 ≤ x_2 ≤ x_3 ≤ x_4 ≤ x_5, with x_3 = 15 and x_4 = 15.
The median (middle element x_3) is 15. For 15 to be the unique mode, 15 must appear at least twice. Since x_3 = 15, either x_2 = 15 or x_4 = 15. If x_2 = x_3 = x_4 = 15, then x_1 + x_5 = 15, which forces x_1 < 1 and violates positive integer constraints. Thus, 15 appears exactly twice: x_3 = 15 and x_4 = 15 (with x_1 < x_2 < 15 and x_5 > 15).
3
Determine the constraints on x_1, x_2, and x_5
x_1 + x_2 + x_5 = 30, where 1 ≤ x_1 < x_2 ≤ 14 and x_5 > 15.
Substituting x_3 = 15 and x_4 = 15 into the sum gives x_1 + x_2 + 15 + 15 + x_5 = 60, simplifying to x_1 + x_2 + x_5 = 30.
4
Calculate the bounds for the Range R = x_5 - x_1
Minimum range = 10, Maximum range = 26
To maximize R = x_5 - x_1 = 30 - 2x_1 - x_2, choose minimum x_1 = 1 and minimum x_2 = 2, yielding x_5 = 27 and R = 26. To minimize R, maximize x_1 = 6 and x_2 = 8, yielding x_5 = 16 and R = 10.

Key Concept

Measures of Central Tendency (Mean, Median, Mode) and Data Range Constraints
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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