The key idea
Work with totals: total = mean × count. Adding or removing values changes the total and the count. The median depends only on the middle position, so extreme values move the mean but often leave the median alone.
The trap
Averaging two means without weighting by group size.
Can Desmos do it?
A calculator adds lists, but these questions rarely give you the list. Totals are the tool.
College Board skill: One-variable data: distributions and measures of center and spread · Problem-Solving and Data Analysis
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
The mean of 7 test scores is 63. A new score of 33 is added. What is the mean of the 8 scores?
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Answer: 237/4The original total is 7×63=441. Adding 33 gives 474, and dividing by 8 gives 59.25.
Level 2 · Test level
Class A has 13 students with a mean test score of 62. Class B has a mean test score of 81. The mean score of all the students in both classes combined is 74.5. How many students are in class B?
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Answer: 25Totals: 13(62)+81b=74.5(13+b). So 6.5b=162.5 and b=25.
Level 3 · Beyond the test
Jordan's mean score on 3 tests is 84. What score does Jordan need on the next test to raise the mean of all 4 tests to 85?
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Answer: 88The current total is 3×84=252. A mean of 85 over 4 tests needs a total of 340. The next score must be 340−252=88.
Practice all 36 questions
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Percent change, chained and reversedChained and reversed percent changes, and the right base.36 questions, 3 levelsRates and unit conversionConversion factors, compound units and square units.36 questions, 3 levelsMixtures and concentrationConcentration problems: mixing, adding water, draining and replacing.36 questions, 3 levelsWork ratesJobs done together, fill-and-drain, and late starts.36 questions, 3 levelsConditional probability“Given that” from tables and percents.36 questions, 3 levelsRatios and proportionsRatios that change when one part changes.36 questions, 3 levels