A dataset S consists of 12 numbers listed in increasing order: x_1, x_2, \dots, x_{12}. Th… — GRE General Test

A dataset $S$ consists of 12 numbers listed in increasing order: $x_1, x_2, \dots, x_{12}$ . The median of dataset $S$ is 40. The arithmetic mean of the 6 smallest numbers in $S$ is 28, and the arithmetic mean of the 6 largest numbers in $S$ is 56. A new dataset $T$ is formed by subtracting 4 from each of the 6 smallest numbers in $S$ and adding 8 to each of the 6 largest numbers in $S$ . What is the positive difference between the arithmetic mean of dataset $T$ and the median of dataset $T$ ?

Answer: 2

Answer

The total sum of dataset S is 504, giving a mean of 42. Transforming the elements adds a net total of 24 to the overall sum, so the mean of dataset T becomes 44. Because decreasing the lower half and increasing the upper half preserves the relative sorted order of all 12 numbers, the middle two elements of dataset T are x_6 - 4 and x_7 + 8. Thus, the new median is (x_6 + x_7)/2 + 2 = 40 + 2 = 42. The positive difference between the mean of 44 and the median of 42 is 2.

Step-by-Step Solution

Calculate the arithmetic mean of the original dataset

S

The sum of the 6 smallest numbers is

6 \times 28 = 168

, and the sum of the 6 largest numbers is

6 \times 56 = 336

. The total sum of dataset

S

168 + 336 = 504

. Thus, the mean of

S

\frac{504}{12} = 42

The mean of a dataset is the sum of all elements divided by the total number of elements.

Calculate the arithmetic mean of the new dataset

T

The sum of dataset

T

504 + 6(-4) + 6(8) = 504 - 24 + 48 = 528

. The mean of dataset

T

\frac{528}{12} = 44

Modifying each of the 12 elements changes the overall sum by the sum of individual changes.

Determine the median of the new dataset

T

Since

x_6 < x_7

, after transformations

x_6 - 4 < x_7 + 8

. The relative order of all elements is preserved. The median of

T

\frac{(x_6 - 4) + (x_7 + 8)}{2} = \frac{x_6 + x_7}{2} + 2 = 40 + 2 = 42

The median of an even number of ordered elements is the average of the two middle elements.

Calculate the positive difference between the mean and median of dataset

T

|44 - 42| = 2.

Subtract the median from the mean and take the absolute value.

Key Concept

Effect of linear transformations and subgroup operations on the mean and median of ordered datasets

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