Separate Changes in Mean and Median

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Data 2,3,4,5,6 change only by replacing the largest value with 21. Which statement is true? Candidate answers: (A) Both mean and median stay the same (B) The mean decreases (C) The mean increases; the median stays the same (D) The median increases by 15.
• Independent check inputs: Data 3,4,5,6,7 change only by replacing the largest value with 27. Which statement is true? Candidate answers: (A) Both mean and median stay the same (B) The mean decreases (C) The mean increases; the median stays the same (D) The median increases by 20.

Learn the idea

Data summaries describe center and spread using stated definitions. Mean, median, interquartile range and mean absolute deviation respond differently to extreme values, so the chosen measure should match the question and convention. Replacing one endpoint of a sorted data set can increase the total without changing the middle value. The mean changes by the total change divided by the observation count.

Here is the method on the supplied teaching case. The sum increases by 15, so the mean increases by 15/5=3. The middle sorted value remains 4. The outlier affects these centers differently.

An unchanged median does not mean the data did not change. Identify exactly which observation moved and whether it crossed the median position before claiming how either center responds.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Data 9,10,11,12,13 change only by replacing the largest value with 63. Which statement is true? Candidate answers: (A) Both mean and median stay the same (B) The mean decreases (C) The mean increases; the median stays the same (D) The median increases by 50. Show the mathematical reason, not only the selected answer. Expected result: The mean increases; the median stays the same. The sum increases by 50, so the mean increases by 50/5=10. The middle sorted value remains 11. The outlier affects these centers differently.

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