A researcher recorded six test measurements arranged in ascending order: 1, 2, 8, x, y, 15… — GMAT

A researcher recorded six test measurements arranged in ascending order: $1, 2, 8, x, y, 15$ . If the unique mode of the data set is $8$ and the arithmetic mean of the measurements is equal to their median, what is the value of $y$ ?

Answer: 14

Answer

Because the data set is arranged in ascending order, the elements satisfy

1 \le 2 \le 8 \le x \le y \le 15

. For

8

to be the unique mode,

8

must appear more times than any other number, which forces

x = 8

. The median of a

6

-element set is the average of the

3^{\text{rd}}

and

4^{\text{th}}

elements, which are both

8

, making the median

8

. Setting the mean

\frac{1 + 2 + 8 + 8 + y + 15}{6} = \frac{34 + y}{6}

equal to

8

gives

34 + y = 48

, which yields

y = 14

. Since

8 \le 14 \le 15

, this maintains the ascending order and unique mode condition.

Step-by-Step Solution

Determine the value of x

x = 8

The set is given in ascending order (1 <= 2 <= 8 <= x <= y <= 15). For 8 to be the unique mode, 8 must appear at least twice, which requires x = 8.

Calculate the median of the data set

Median = 8

With 6 elements, the median is the average of the 3rd and 4th elements: (8 + 8) / 2 = 8.

Set up the equation for the arithmetic mean

(34 + y) / 6 = 8

The sum of the six numbers is 1 + 2 + 8 + 8 + y + 15 = 34 + y. Since the mean equals the median, the mean must equal 8.

Solve for y

y = 14

Multiplying both sides by 6 gives 34 + y = 48, so y = 48 - 34 = 14.

Key Concept

Using statistical properties (mean, median, mode) of an ordered set to determine unknown values

Estimated Time:1m 15s

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