How do you test a mean change within pairs?
You will be able to: Carry out and interpret a t test on paired differences.
How do you test a mean change within pairs?
Sixteen randomly selected students have after-minus-before task-time differences with mean −2 minutes and SD 4 minutes. The question is whether the population mean change is negative.
A useful starting point: How do you test an average and conclude in context? →
Words and symbols before equations
- Paired standard error
- sd/√n, using n differences.
- Null change
- Typically μd=0.
- Lower-tail probability
- P(T≤observed t) for a decrease alternative.
- Mean change
- A population average; individual changes can differ.
What this picture assumes
Six synthetic independently sampled students, with after−before differences. For the intact teaching dataset assume N=1000 and a difference distribution without strong skewness or outliers. Reversing after-values is a counterexample showing the effect of false matching, never an allowable way to reanalyze observed pairs. Only intact pairing receives an inferential interpretation.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- After−before: mean 2 min, SD 0.8944, SE 0.3651, n=6, df=5. 95% interval (1.0614, 2.9386); two-sided test of μd=0 gives t=5.4772, p=0.002765 under the stated assumptions.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Define μd as the population mean after-minus-before time. H₀:μd=0 and Hₐ:μd<0. Assume independent sampled students and suitable differences.
SE=4/√16=1 minute, so t=−2 with df=15. The lower-tail p-value is about .0320.
At α=.05 the result supports a negative population mean change. If there was no randomized assignment or other defensible control, time trends and other changes remain possible explanations; the t calculation alone does not establish a causal effect.
A worked example, step by step
Explain the same data using before−after rather than after−before.
- Every difference reverses sign, so d̄ becomes +2.
- sd and SE remain 4 and 1.
- The alternative becomes μd>0 and t=+2.
- The upper-tail p remains .0320, yielding the same substantive conclusion.
Do not substitute independent-group SE for paired sd/√n; within-pair association matters.
If all differences increase by 1, does sd change?
Compare with an explanation
No. A common shift changes the mean, not the spread.
Predict. Change one thing. Explain.
Shift all after times together, keeping identities fixed. Predict the effect on mean difference, its SD and t. Compare the exact table with the 3D connectors.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
After−before: mean 2 min, SD 0.8944, SE 0.3651, n=6, df=5. 95% interval (1.0614, 2.9386); two-sided test of μd=0 gives t=5.4772, p=0.002765 under the stated assumptions.
| Student | Before (min) | After (min) | After−before (min) |
|---|---|---|---|
| P1 | 8 | 9 | 1 |
| P2 | 10 | 12 | 2 |
| P3 | 12 | 15 | 3 |
| P4 | 14 | 15 | 1 |
| P5 | 16 | 18 | 2 |
| P6 | 18 | 21 | 3 |
Six synthetic independently sampled students, with after−before differences. For the intact teaching dataset assume N=1000 and a difference distribution without strong skewness or outliers. Reversing after-values is a counterexample showing the effect of false matching, never an allowable way to reanalyze observed pairs. Only intact pairing receives an inferential interpretation.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the means, standard errors, pairing, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor 25 independent pairs, d̄=3,sd=5 and Hₐ:μd>0, set up and calculate the statistic; suppose p=.0031.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: H₀:μd=0; define the response and order.
- 1 point: SE=5/5=1, df=24.
- 1 point: t=3/1=3.
- 1 point: At α=.05 reject H₀, supporting a positive mean change, subject to design and shape conditions.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Does shifting all differences change their SD?
No.
RECALL 2Does a mean decrease require every person to decrease?
No.
RECALL 3Does pairing itself prove causation?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you test a mean change within pairs?
- t=(d̄−0)/(sd/√n).
- df=n−1 pairs.
- A consistent sign reversal preserves the evidence.
Remember: Do not substitute independent-group SE for paired sd/√n; within-pair association matters.
Conditions: Six synthetic independently sampled students, with after−before differences. For the intact teaching dataset assume N=1000 and a difference distribution without strong skewness or outliers. Reversing after-values is a counterexample showing the effect of false matching, never an allowable way to reanalyze observed pairs. Only intact pairing receives an inferential interpretation.
Refresh Kid · AP Statistics Unit 4 · Objectives 4.5.A, 4.5.B, 4.5.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 4.5, objectives 4.5.A, 4.5.B, 4.5.C. Framework effective Fall 2026, checked September 17, 2026. Unit 4 includes sampling distributions of means, one-sample and paired t inference, and independent two-sample t inference; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Mean inference requires a justified design and suitable shape or sample size. Paired analysis uses one sample of differences. Independent two-sample inference uses separate variance estimates and technology-computed Welch degrees of freedom. Extreme skewness and influential observations need attention even in larger samples. This model conservatively withholds inference when those warnings are selected. Conclusions are limited by random sampling and/or assignment as appropriate.
The Organic Chemistry Tutor companion title and destination were located; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original paired-data display uses self-hosted Three.js with its MIT license. Two measurement columns are connected within each labeled student lane; horizontal position is before/after, vertical position is time, and depth separates identities rather than representing a numerical variable. Rotation can separate overlapping connectors. Exact values and differences always remain in the 2D table and labeled plot. The broken-matching option is an explicit counterexample, not a legitimate alternative analysis. No autoplay; complete teaching remains available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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