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?

- 16\\frac{1}{3})
- 16\\frac{2}{3})
- 16\\frac{1}{6})
- 16
- 11\\frac{1}{6})

From INR 2999

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 t_{3} + t_{4} = 40

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.

If x takes any value that is **less than or equal to 11**, then t_{4} will be 11. So, t_{3} ≤ 11

Then, **t _{3} + t_{4} ≤ 22**.

Does not satisfy the condition that t

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 t_{3} = 11 and t_{4} = x

The maximum possible value of t_{4} < 29

Therefore, **t _{3} + t_{4} < 40**.

The given condition is not satisfied.

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 t_{3} = 11 and t_{4} = 29.

So, the condition t_{3} + t_{4} = 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).

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**

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