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Mauchly’s test of sphericity is the check SPSS runs automatically inside a repeated-measures ANOVA to see whether the variances of the differences between every pair of within-subject conditions are equal; when the test is significant (Sig. below .05), the fix is to read a corrected result from the same output table — the Greenhouse-Geisser correction if epsilon is below .75, the Huynh-Feldt correction if it is above. This guide walks through exactly where SPSS prints the test, what each column in it means, and which row of the follow-up table to report once you’ve made the call.
Sphericity is easy to misread as a general ANOVA assumption. It isn’t — it only exists for a repeated-measures (within-subjects) design, where the same cases are measured under three or more related conditions or time points. Because those measurements come from the same people, the differences between conditions are correlated in a way a between-subjects design never has to worry about, and that correlation structure is exactly what sphericity constrains.
What sphericity actually means
For a within-subjects factor with three or more levels, compute every possible pairwise difference between conditions (condition A minus B, A minus C, B minus C, and so on). Sphericity is the assumption that the variance of each of those difference scores is the same across all the pairs. It says nothing about the means of the conditions — that’s what the ANOVA’s F-test itself is checking — it’s a condition on the variance-covariance structure underneath the test.
When sphericity is violated, the standard repeated-measures F-test becomes anti-conservative: it produces a smaller p-value than it should, so a real risk of a false positive gets built into the uncorrected result. That’s the whole reason SPSS runs the check automatically and offers a correction rather than leaving it to the analyst to remember.
Sphericity is a non-issue with exactly two levels of the within-subjects factor. With only two conditions there is only one pairwise difference, so there’s nothing for the variances to differ from — the assumption is trivially met, and SPSS won’t print a Mauchly’s Test table for that effect (it shows a dot or omits the row, since the test has zero degrees of freedom). If your output has no Mauchly’s table at all, this is almost always why.
Where SPSS reports the test
Run Analyze → General Linear Model → Repeated Measures, define your within-subjects factor and its levels, add the dependent variables, and run the procedure with default options. SPSS produces several output tables; the one that matters here is titled Mauchly’s Test of Sphericity, and it appears automatically — there’s no separate menu option to request it, and no way to suppress it short of editing the output.
Reading the Mauchly’s Test of Sphericity table
The table has one row per within-subjects effect (just one, if you have a single within-subjects factor) and these columns:
- Mauchly’s W — the test statistic itself, bounded between 0 and 1. Values closer to 1 indicate the pairwise variances are more similar (closer to sphericity); values closer to 0 indicate more violation. W has no fixed interpretive cutoff on its own — you read its significance, not its raw value, to decide whether to act.
- Approx. Chi-Square and df — Mauchly’s W is converted to an approximate chi-square statistic for significance testing; df is fixed by the number of levels of the within-subjects factor (levels × (levels − 1) / 2 − 1).
- Sig. — the p-value for the null hypothesis that sphericity holds. This is the number that actually drives the decision: Sig. below .05 means sphericity is violated and a correction is needed; Sig. .05 or above means sphericity can be assumed and the uncorrected row of the next table is reportable as-is.
- Epsilon (three columns: Greenhouse-Geisser, Huynh-Feldt, Lower-bound) — each is an estimate, on a 0-to-1 scale, of how much sphericity is violated. An epsilon of 1 means no violation at all; smaller values mean more severe violation. These are the numbers used to shrink the degrees of freedom in the corrected F-test, and the Greenhouse-Geisser value specifically is what the .75 decision rule below is applied to.
What a significant result means, and what it doesn’t
A significant Mauchly’s test (Sig. < .05) means: don’t trust the uncorrected (“Sphericity Assumed”) row of the within-subjects effects table — report a corrected row instead. It does not mean the ANOVA result itself is wrong or that the effect isn’t real; it means the standard F-test’s degrees of freedom need adjusting to keep its false-positive rate accurate.
The reverse direction matters too, and is where analysts most often go wrong: a non-significant Mauchly’s test (Sig. ≥ .05) is grounds to report the uncorrected “Sphericity Assumed” row, not an automatic requirement to apply a correction anyway. Mauchly’s test does have limited statistical power at small sample sizes, which is a real caveat — some methodologists recommend defaulting to a conservative correction regardless of the test’s result when the sample is small, but that’s an explicit, stated analytic choice, not something SPSS’s output forces on you.
Choosing Greenhouse-Geisser vs. Huynh-Feldt by the epsilon value
Both corrections do the same thing mechanically — they multiply the numerator and denominator degrees of freedom of the F-test by epsilon, shrinking both and making the test more conservative in proportion to how badly sphericity is violated. They differ only in how epsilon itself is estimated:
- Greenhouse-Geisser (GG) tends to underestimate the true epsilon, which makes it the more conservative of the two — it over-corrects, especially with a more severe violation or a smaller sample.
- Huynh-Feldt (HF) tends to overestimate epsilon, sometimes producing a value above 1 (SPSS caps the displayed value at 1.000 when this happens) — it under-corrects relative to GG, particularly with a mild violation.
The widely used decision rule, attributed to Girden (1992) and reproduced in most repeated-measures ANOVA textbooks and course materials since: use the Greenhouse-Geisser correction when the Greenhouse-Geisser epsilon is below .75; use the Huynh-Feldt correction when it is .75 or above. The logic is that GG’s conservatism only becomes a real cost (under-powering the test unnecessarily) once the violation is mild enough that HF’s less conservative estimate is trustworthy; below that threshold, GG’s extra caution is doing useful work protecting against the false positive that motivated the correction in the first place.
Two illustrative examples (hypothetical output, not a real published study) showing both branches of the rule:
- Epsilon below .75: Mauchly’s W = .68, χ²(2) = 9.86, Sig. = .007 (violated); Greenhouse-Geisser epsilon = .70. Because .70 is below .75, report the Greenhouse-Geisser row.
- Epsilon at or above .75: Mauchly’s W = .81, χ²(2) = 4.92, Sig. = .086 … but suppose a related effect in the same design comes back significant with Greenhouse-Geisser epsilon = .83. Because .83 is at or above .75, report the Huynh-Feldt row for that effect instead.
A simpler, more conservative alternative some methodologists prefer: always apply Greenhouse-Geisser regardless of the epsilon value, accepting the extra conservatism as the price of a single, consistent rule. Either approach is defensible; what matters for credibility is stating which rule you followed, not silently picking whichever row gives the lower p-value.
A separate alternative entirely is to sidestep the correction question by requesting SPSS’s multivariate output (Pillai’s Trace, Wilks’ Lambda) for the same within-subjects effect — the multivariate approach doesn’t assume sphericity at all, at the cost of typically lower statistical power with a small sample. It’s a legitimate option when sphericity is severely violated (a very low epsilon) rather than a routine substitute for the univariate corrected tests.
Reading the corrected Tests of Within-Subjects Effects table
The table titled Tests of Within-Subjects Effects lists four rows for the same effect: Sphericity Assumed, Greenhouse-Geisser, Huynh-Feldt, and Lower-bound. All four rows show the identical F-value — the correction never changes F itself — but the degrees of freedom differ (each corrected row’s df is the uncorrected df multiplied by that row’s epsilon, which is why corrected df are non-integer, e.g. 1.67 rather than 2), and because df changed, the Sig. value differs too. Report the F-value alongside the df and Sig. from whichever single row your epsilon-based decision above pointed to — never the F together with one row’s df and a different row’s p-value.
How to report it in text
A standard APA-style pattern, using the first illustrative example above: “Mauchly’s test indicated that the assumption of sphericity had been violated, χ²(2) = 9.86, p = .007, therefore degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = .70). There was a significant effect of time on the outcome, F(1.40, 25.20) = 5.42, p = .018.” Report the epsilon value used, the corrected (non-integer) degrees of freedom, and which correction was applied — a reader checking your work needs all three to confirm you read the table correctly.
Common mistakes
- Reporting the Sphericity Assumed row when Mauchly’s test was significant. This is the single most common error and the one the correction exists to prevent — it silently understates the true p-value.
- Mixing rows — quoting the F from one row with the df or Sig. from another. All three numbers must come from the same row.
- Applying a correction after a non-significant Mauchly’s test without stating that as a deliberate conservative choice, then reporting it as if the test required it.
- Expecting a Mauchly’s table for a two-level within-subjects factor. It won’t appear, and that’s correct behavior, not a missing-output error.
- Treating Mauchly’s W itself as the decision criterion. The decision runs on Sig. (violated or not) and then on the Greenhouse-Geisser epsilon (which correction) — W is diagnostic context, not a threshold to compare against .05 or any other cutoff.
Frequently asked questions
What is Mauchly’s W, in plain terms?
A single number between 0 and 1 summarizing how equal the variances of the pairwise condition-differences are: 1 means perfectly equal (sphericity fully holds), values further from 1 mean more violation. It’s the raw test statistic; the Sig. column next to it is what determines whether that departure is large enough to act on.
What if Mauchly’s test isn’t significant?
Report the “Sphericity Assumed” row of the Tests of Within-Subjects Effects table — the uncorrected F, df, and Sig. No correction is required. Applying one anyway is a defensible conservative choice at small sample sizes, but should be stated explicitly rather than presented as what the test result required.
What’s the actual difference between Greenhouse-Geisser and Huynh-Feldt?
Both shrink the F-test’s degrees of freedom by an epsilon value to correct for a sphericity violation; they differ only in how that epsilon is estimated. Greenhouse-Geisser tends to underestimate it (more conservative, more likely to over-correct), Huynh-Feldt tends to overestimate it (less conservative, occasionally capped at 1.000 by SPSS). The .75 rule of thumb (Girden, 1992) picks between them based on the Greenhouse-Geisser epsilon value.
Why doesn’t my output have a Mauchly’s Test of Sphericity table?
Almost always because the within-subjects factor has only two levels. With two conditions there is exactly one pairwise difference, so there’s no variance-equality question to test — sphericity is automatically satisfied and SPSS omits the table for that effect.
Can I just always use the Greenhouse-Geisser correction and skip the .75 rule?
Yes — always applying Greenhouse-Geisser is a simpler, more conservative alternative some analysts and courses prefer over the epsilon-based rule, and it’s defensible as long as you say that’s the rule you followed. What isn’t defensible is switching between rows opportunistically to get a lower p-value.
Is there a way to avoid the sphericity issue entirely?
Requesting SPSS’s multivariate test output (Wilks’ Lambda, Pillai’s Trace) for the within-subjects effect sidesteps sphericity altogether, since the multivariate approach doesn’t assume it. The tradeoff is lower statistical power with a small sample, which is why it’s typically used as a deliberate alternative for severe violations rather than a routine substitute.
Related CASRAI guides
For the broader ANOVA framework this sits inside, see the ANOVA guide and the ANOVA dictionary term. For the two-group case where sphericity never arises, see t-test: one-sample, independent, and paired. For SPSS itself as a tool, see what is SPSS and, for a free alternative, jamovi vs JASP. For interpreting how large a corrected effect actually is once you’ve settled on a row to report, see effect size: choosing, reporting and interpreting it, and for reading the Sig. column generally, the p-value dictionary term.
