Build the method
Before-and-after measurements on the same participants are paired. Form one difference per person in a fixed order, then analyze the resulting sample of differences. Conditions concern independence between people and the shape of the differences, not independence of the two measurements within a person.
Worked example
For four independently sampled people, before scores 10,20,30,40 and after scores 8,22,32,42 give differences −2,2,2,2. Their mean is one. Squared deviations from one sum to twelve, so the sample variance is twelve divided by three, or four. The sample SD is two and the mean standard error is two divided by square root of four, or one. Testing zero mean change gives t=1, with three degrees of freedom. Assume approximately normal population differences for this small-sample procedure.
Compare the alternatives
For n=16, sample SD 4 and supplied critical value t*=2.131 for df=15, the estimated standard error is 4/4=1 and the margin is 2.131. Using the normal multiplier 1.960 ignores the extra uncertainty from estimating SD. Using df=16 selects the wrong row.
Avoid a specific trap
A small sample with severe skew and an extreme outlier gives weak support for the usual t procedure. Random selection does not make the measurements normal. Investigate the unusual value on substantive grounds; deleting it only to improve significance is not an acceptable assumption check.
Transfer check with explanation
Transfer: two separate groups of thirteen participants have no natural before-and-after pairing. An independent two-sample method is appropriate instead of pairing rows by spreadsheet position. Explain how the design changes the unit of analysis and standard error, even when the columns contain the same number of entries.
Four independently sampled participants have before scores [10, 20, 30, 40] and corresponding after scores [8, 22, 32, 42]. Assume the population of after-minus-before differences is approximately normal. For a one-sample t test of population mean change 0, what is the t statistic?
- Form each participant difference, then compute its sample mean and sample SD before standardizing the mean change.
- Differences are [-2, 2, 2, 2]; their mean is 1. Squared deviations sum to 12, so sample variance=12/3=4 and SD=2. SE=2/√4=1, giving t=(1−0)/1=1.
- Why an alternative fails: Divide the mean change by the sample SD of differences, omitting the square-root-of-n adjustment.
Compare the other answer choices
Divide the difference SD by n rather than square root of n, making the standard error too small.
Divide the mean change by the sample SD of differences, omitting the square-root-of-n adjustment.
Reverse the required difference order to before-minus-after; the mean changes sign while the sample SD and standard error do not.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
For before scores [10,20,30,40] and after scores [8,22,32,42]: (a) compute the four differences and their sample SD; (b) compute the standard error and t statistic for mean change 0; (c) explain the analysis unit and degrees of freedom.
This draft stays on this page only. Use paper if you want to keep your working.
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.
AP Statistics Course and Exam Description, effective Fall 2026