Decision and conditions
Convert means to totals before adding, removing or mixing observations. An arithmetic mean is a total divided by a count; group sizes supply weights.
Worked example
5 numbers have mean 10. A further number is added, and the new mean is 12. What number was added? The old total is 50. The new total is 12(6) = 72. The added number is 22. Method choice: Recover both totals; the number of observations increases by one.
Contrast: change the reasoning
6 observations have mean 20. After one observation of value 30 is removed, what is the mean of the remaining observations? Original total: 120. Remaining total: 90 across 5 observations. The remaining mean is 18. Method choice: Remove the observation from the total and one from the count.
Transfer with solution
A list contains 6 copies of 10 and x copies of 20. Its mean is 16. What is x? 10(6)+20x=16(6+x). 4x=36. x=9. Method choice: Weight each value by its frequency and express the total count as 6+x.
Boundary check
A student has completed 3 equally weighted tests with mean 70. What score on one more test produces an overall mean 75? Target total is 75(4). Existing total is 70(3). The required score is 90. Method choice: Include the new test when writing the target total.
Misconception to reject
Adding a value changes both the numerator and denominator. Subtracts the means instead of the totals.
5 numbers have mean 10. A further number is added, and the new mean is 12. What number was added?
- The old total is 50.
- The new total is 12(6) = 72.
- The added number is 22.
Compare the other answer choices
Subtracts the means instead of the totals.
Uses the old count for both totals, omitting the new observation.
Reports the old total rather than the added observation.
Reports the new total rather than the added observation.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
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