Compare an Arithmetic Mean with a Median

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: For the complete supplied data [2, 3, 4, 5, 16], which pair is (mean, median)? Candidate answers: (A) (16, 4) (B) (4, 6) (C) (6, 4) (D) (6, 5).
• Independent check inputs: For the complete supplied data [3, 4, 5, 6, 17], which pair is (mean, median)? Candidate answers: (A) (17, 5) (B) (5, 7) (C) (7, 5) (D) (7, 6).

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. A mean divides the total by the number of values. A median locates the middle of ordered data; a high outlier can pull the mean upward while leaving the middle position unchanged.

Here is the method on the supplied teaching case. The sum is 30; divide by 5 to get mean 6. The third sorted value is the median 4. The high value pulls the mean upward.

Sort before finding the median and count every value in the mean denominator. These complete supplied data need no population inference; report both summaries without pretending they are interchangeable.

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: For the complete supplied data [9, 10, 11, 12, 23], which pair is (mean, median)? Candidate answers: (A) (11, 13) (B) (13, 11) (C) (13, 12) (D) (23, 11). Show the mathematical reason, not only the selected answer. Expected result: (13, 11). The sum is 65; divide by 5 to get mean 13. The third sorted value is the median 11. The high value pulls the mean upward.

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