GRE® Math Practice | Statistics & Averages #5

The GRE maths sample question given below is a problem solving question in statistics. Concept covered: Average (arithmetic mean) and median.

Question 5 :If the median of -10, 29, 6, 11, 31, and x is 20. What is the least possible average of the 6 numbers?

1. 16\\frac{1}{3})
2. 16\\frac{2}{3})
3. 16\\frac{1}{6})
4. 16
5. 11\\frac{1}{6})

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Video Explanation

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Explanatory Answer | GRE Averages Practice Question 5

Given Data:

Including x, we have a total of 6 numbers.
So, Median = \\frac{\text{3rd term + 4th term}}{2}) = \\frac{t_3 + t_4}{2})
It is given that median = 20
So, \\frac{t_3 + t_4}{2}) = 20
Or t₃ + t₄ = 40

Compute the least possible value for x:

The only unknown is x. The average of all 6 numbers will be least if the value of x is at its least.
The ascending order of the known values is -10, 6, 11, 29, 31.
x has to be arranged in ascending order along with the other 5 numbers -10, 6, 11, 29, 31.
x can take differet values.
So, check for the different possibilities that satisifes the condition that the median is 20.

Possibility 1: x ≤ 11

If x takes any value that is less than or equal to 11, then t₄ will be 11. So, t₃ ≤ 11
Then, t₃ + t₄ ≤ 22.
Does not satisfy the condition that t₃ + t₄ = 40

Possibility 2: 11 < x < 29

If x takes any value that is greater than 11 but less than 29, then the order of the 6 numbers will be -10, 6, 11, x, 29, 31
The value of t₃ = 11 and t₄ = x
The maximum possible value of t₄ < 29
Therefore, t₃ + t₄ < 40.
The given condition is not satisfied.

Possibility 3: x ≥ 29

If x takes a value that is 29 or upwards, then three different possibilities exist.
-10, 6, 11, 29, 29, 31 (or) -10, 6, 11, 29, x, 31 (or) -10, 6, 11, 29, 31, x
In all 3 possibilities, notice that the value of t₃ = 11 and t₄ = 29.
So, the condition t₃ + t₄ = 40 gets satisfied in all three cases.
Among all three possibilities that work, the least possible value for x is 29 (1st of the 3 possibilities).

Compute the least possible average

Average = \\frac{\text{Sum of all observations}}{\text{Total number of observations}})
Average of the given 6 numbers = \\frac{−10 + 6 + 11 + 29 + 29 + 31}{6}) = \\frac{96}{6}) = 16
Therefore, the least possible average of the 6 numbers = 16

Choice D is the correct answer