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Difficulty: HardData Distributions and Measures

A dataset consists of 1111 positive integers. The mean of the dataset is 1414, and the median is 1212. The range of the dataset is 2020. If the maximum possible value of the largest integer in the dataset is LL, what is the value of LL?

Answer: 32

Answer

The maximum possible value of the largest integer in the dataset is 3232.
The correct answer is 3232. Let the 11 integers in nondecreasing order be x1≤x2≤⋯≤x11x_1 \le x_2 \le \dots \le x_{11}. Since the mean of the 11 integers is 1414, their sum is 11×14=15411 \times 14 = 154. Since the median is 1212, the 6th integer is x6=12x_6 = 12. The range is 2020, so x11−x1=20x_{11} - x_1 = 20, which gives x1=x11−20x_1 = x_{11} - 20. To maximize the largest integer x11x_{11}, we need to observe that x1≤x6=12x_1 \le x_6 = 12. Substituting x1=x11−20x_1 = x_{11} - 20 into this inequality yields x11−20≤12x_{11} - 20 \le 12, which simplifies to x11≤32x_{11} \le 32. We can verify that x11=32x_{11} = 32 is achievable with the dataset: 12,12,12,12,12,12,12,12,12,14,3212, 12, 12, 12, 12, 12, 12, 12, 12, 14, 32. Therefore, the maximum possible value of the largest integer is 3232.

Step-by-Step Solution

1
Calculate the sum of the 11 integers in the dataset.
The sum of the 11 integers is 154154.
Since the mean of 11 values is 14, their sum must be 11×14=15411 \times 14 = 154.
2
Identify the median and establish the order constraint.
The 6th value in the ordered list of 11 integers is 1212.
The median of an odd number of sorted values is the middle value, which is the 6th value (x6=12x_6 = 12).
3
Relate the minimum and maximum values using the range.
The minimum value x1x_1 is related to the maximum value x11x_{11} by x1=x11−20x_1 = x_{11} - 20.
The range of the dataset is 20, which is the difference between the maximum and minimum values.
4
Find the maximum possible value of the largest integer.
The maximum value is 3232.
Since the dataset is sorted, the minimum value x1x_1 cannot exceed the median x6=12x_6 = 12. Therefore, x1≤12  ⟹  x11−20≤12  ⟹  x11≤32x_1 \le 12 \implies x_{11} - 20 \le 12 \implies x_{11} \le 32.

Key Concept

Using measures of center (mean, median) and spread (range) to determine bounds on data values.
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Topics

Data Distributions and Measures

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