Decision and conditions
Use sorting, extremes and spread transformations to constrain missing values. A stated median refers to sorted observations; standard deviation is nonnegative.
Worked example
The median of the five numbers 3, 5, 7, 11, x is 6. What is x? None of the four known values equals 6. Therefore the missing value must be the median. x=6, with two known values below it and two above. Method choice: The median of an odd list is one of its observations after sorting.
Contrast: change the reasoning
A data set contains 3, 6, 10, and x. Its range is 13, and x is the largest value. What is x? The minimum is 3. x−3=13. x=16. Method choice: The range is largest minus smallest, and the smallest known value is fixed.
Transfer with solution
A data set has standard deviation 3. Every value is transformed to 4 times its original value minus 9. What is the resulting standard deviation? Multiplication by four multiplies all deviations by four. Subtracting nine shifts values and the mean equally. The new standard deviation is 12. Method choice: A translation does not change distances from the mean; a scale does.
Boundary check
Group A has 5 values with mean 10 and group B has 7 values with mean 20. How much greater is the combined mean than 10? Group A contributes no excess above 10; group B contributes 70. There are 12 observations. The excess in the mean is 35/6. Method choice: Find the excess total above a baseline of ten for every observation.
Misconception to reject
The range fixes only the extremes, not the interior observations. Uses the third displayed value without sorting the unknown observation.
The median of the five numbers 3, 5, 7, 11, x is 6. What is x?
- None of the four known values equals 6.
- Therefore the missing value must be the median.
- x=6, with two known values below it and two above.
Compare the other answer choices
Confuses the requested median with an average of the smallest pair.
Uses the second displayed value as the middle of the known list.
Uses the third displayed value without sorting the unknown observation.
Averages the two largest known values without positioning x.
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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