Using a paired test
Independent rows do not identify the same record before and after.
Compare independent groups without pooling their variances
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Task requirements
Enter separate formulas in G2:G9. A2:A21 and B2:B21 are independent groups; their lengths may differ and their row positions do not form pairs. D2 is alpha. Report each count, mean A minus mean B, its unequal-variance standard error, fractional Welch degrees of freedom, signed t for a zero population-mean difference, two-sided unequal-variance p, and the Boolean result p<=alpha. Use T.TEST tails=2/type=3; do not pool variances or round the displayed degrees of freedom.
Edit the sample report
Apply one numeric value to the actual worksheet. Observations and alpha retain their separate meanings. The complete sample design is validated before an edit replaces a valid input.
Applied value:
Follow the highlighted cells. Switch to Your turn to edit. Answer cells: G2, G3, G4, G5, G6, G7, G8, G9. Browser calculations may differ from your Excel version.
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Practice stays on this device. No worksheet data is uploaded.
Notes & examples
About this lesson
Keep unequal group sizes and sample variances separate, retain fractional Welch degrees of freedom and report a two-sided comparison.
THE QUICK ANSWER01
Keep unequal group sizes and sample variances separate, retain fractional Welch degrees of freedom and report a two-sided comparison.
G2
=COUNT($A$2:$A$21)G3
=COUNT($B$2:$B$21)G4
=AVERAGE($A$2:$A$21)-AVERAGE($B$2:$B$21)G5
=SQRT(VAR.S($A$2:$A$21)/COUNT($A$2:$A$21)+VAR.S($B$2:$B$21)/COUNT($B$2:$B$21))G6
=LET(u,VAR.S($A$2:$A$21)/COUNT($A$2:$A$21),v,VAR.S($B$2:$B$21)/COUNT($B$2:$B$21),(u+v)^2/(u^2/(COUNT($A$2:$A$21)-1)+v^2/(COUNT($B$2:$B$21)-1)))G7
=G4/G5G8
=T.TEST($A$2:$A$21,$B$2:$B$21,2,3)G9
=G8<=$D$2For the example data
G2 5G3 7G4 3.2857142857142865G5 1.028505288878818G6 9.784705654237516G7 3.1946498683502838G8 0.009838693428628293G9 true
Enter each formula in its labeled answer cell. These report formulas perform different jobs; do not fill the first formula down.
How it works
The design selects the calculation
Enter separate formulas in G2:G9. A2:A21 and B2:B21 are independent groups; their lengths may differ and their row positions do not form pairs. D2 is alpha. Report each count, mean A minus mean B, its unequal-variance standard error, fractional Welch degrees of freedom, signed t for a zero population-mean difference, two-sided unequal-variance p, and the Boolean result p<=alpha. Use T.TEST tails=2/type=3; do not pool variances or round the displayed degrees of freedom.
Keep the model assumptions visible
The teaching inputs assume independent randomly sampled records and approximately normal outcomes (normal within-record differences for the paired task). The supplied numbers do not establish randomization, independence or distribution shape. A confidence procedure has a long-run coverage interpretation; p is not the probability that its null hypothesis is true. These are model-based sample calculations, not causal or practical-impact conclusions.
Declared numeric scope
Use 3–20 numeric observations per group, magnitudes at most 10000, positive relevant sample variance at least 1e-10, and alpha 0.01–0.20. Paired identities stay fixed and complete. Independent samples are contiguous with trailing blanks. Absolute paired/Welch t must be at most 20. Unsupported types, holes, zero spread and numerically ambiguous p-versus-alpha thresholds withhold grading instead of changing the data. Keep full precision; nonzero p values cannot pass as zero.
Read the full step-by-step walkthrough
- Count the independent groups separately
Trailing blanks identify unused positions, not zero observations. Group sizes can differ. Row alignment does not assert matched records.
- Retain both variance contributions
The squared standard error is sA^2/nA + sB^2/nB. A pooled variance assumes a different model and can change the answer materially.
- Keep the fractional degrees of freedom
Welch degrees of freedom come from both variance contributions and their sample counts. Retain the fractional result. Manually substituting it in functions that truncate degrees of freedom changes the requested p.
- Use the unequal-variance test
T.TEST type=3 calculates the two-sided p without an equal-variance assumption. Compare it with the prespecified alpha, retaining the result as a Boolean rather than text or 1/0. A threshold outcome does not measure the practical size of the mean difference.
- Separate the threshold from the effect size
The Boolean p<=alpha compares tail evidence with the chosen threshold. Read it beside the signed mean difference and standard error. A false outcome does not prove equality or practical equivalence.
Things to watch
Pooling unequal variances
Type=2 is a separate equal-variance assumption. The requested test uses type=3.
Concluding that a large p proves equality
Failure to cross alpha is not proof of identical population means or practical equivalence.
About this example
Original bounded source reports checked with real formulas and independent quadrature/incomplete-beta targets. Native recalculation and edit/save remain separate.
Content version 1. Practice checks compare calculated results across different datasets; they do not certify every possible formula.