A data set consists of 25 different numbers. The median of... | SAT One-variable data: distributions and measures of center and spread Question, Answer Explained

Question bank / One-variable data: distributions and measures of center and spread

Math · Problem-Solving and Data AnalysisHard

A data set consists of $25$ different numbers. The median of the data set is $40$ , and the mean is $46$ . Two numbers, $15$ and $90$ , are added to the data set. Which of the following statements about the new data set of $27$ numbers is true?

A

The median is $40$ , and the mean is greater than $46$ .

B

The median is $40$ , and the mean is less than $46$ .

C

The median is greater than $40$ , and the mean is greater than $46$ .

D

The median is less than $40$ , and the mean is $46$ .

Show answer and explanation

Answer: A

One new number is below $40$ and the other is above $40$ , so $40$ moves from the $13$ th position to the $14$ th position, which is the middle of $27$ values. The median stays $40$ . The original sum is $25 \times 46 = 1{,}150$ , and the new sum is $1{,}150 + 15 + 90 = 1{,}255$ . The new mean is $\frac{1{,}255}{27} \approx 46.5$ , which is greater than $46$ .

Choice C assumes the large value $90$ pulls the median up, but adding one value on each side of the median leaves the median unchanged.

Why the other choices are wrong

B. The added values average 52.5, which is more than 46, so the mean goes up, not down.
C. One value is added below 40 and one above, so 40 stays in the middle and the median doesn't change.
D. The median stays 40, and the mean changes because the added values don't average 46.

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