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Problem-Solving and Data Analysis / One-variable data: Distributions and measures of center and spread Difficulty: Hard

The dot plot represents the $15$ values in data set A. Data set B is created by adding $56$ to each of the values in data set A. Which of the following correctly compares the medians and the ranges of data sets A and B?

The median of data set B is equal to the median of data set A, and the range of data set B is equal to the range of data set A.

The median of data set B is equal to the median of data set A, and the range of data set B is greater than the range of data set A.

The median of data set B is greater than the median of data set A, and the range of data set B is equal to the range of data set A.

The median of data set B is greater than the median of data set A, and the range of data set B is greater than the range of data set A.

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Explanation

Choice C is correct. The median of a data set with an odd number of values, in ascending or descending order, is the middle value of the data set, and the range of a data set is the positive difference between the maximum and minimum values in the data set. Since the dot plot shown gives the values in data set A in ascending order and there are $15$ values in the data set, the eighth value in data set A, $23$ , is the median. The maximum value in data set A is $26$ and the minimum value is $22$ , so the range of data set A is $26 minus 22$ , or $4$ . It’s given that data set B is created by adding $56$ to each of the values in data set A. Increasing each of the $15$ values in data set A by $56$ will also increase its median value by $56$ making the median of data set B $79$ . Increasing each value of data set A by $56$ does not change the range, since the maximum value of data set B is $26 plus 56$ , or $82$ , and the minimum value is $22 plus 56$ , or $78$ , making the range of data set B $82 minus 78$ , or $4$ . Therefore, the median of data set B is greater than the median of data set A, and the range of data set B is equal to the range of data set A.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.