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Sphericity is the assumption, underlying every repeated-measures ANOVA, that the variances of the differences between every pair of within-subject conditions are equal; Mauchly’s test is the formal check for that assumption, and when it is violated (Sig. below .05, or more generally a significant chi-square result), the standard fix is not to abandon the ANOVA but to correct its degrees of freedom by a shrinkage factor called epsilon — using the Greenhouse-Geisser epsilon when it falls below .75, and the Huynh-Feldt epsilon when it is .75 or above. This guide covers the statistical theory behind that rule: what sphericity actually is, why a repeated-measures design creates the problem in the first place, how Mauchly’s W statistic tests for it, and how the two epsilon corrections work mathematically. It is software-agnostic — for the step-by-step walkthrough of where SPSS prints each number, see Mauchly’s Test of Sphericity in SPSS.
What sphericity actually means
Sphericity applies to a within-subjects factor with three or more levels — the same cases measured under three or more related conditions or time points, as in a typical repeated-measures ANOVA design. For any such factor, take 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 those difference scores is equal across every pair. It says nothing about the condition means — that is what the F-test itself evaluates — it is a condition on the variance-covariance structure underneath the test.
Sphericity is a special case of a stricter condition called compound symmetry, which requires equal variances across all conditions and equal covariances between every pair of conditions. Compound symmetry is sufficient for a valid repeated-measures F-test but stronger than necessary; sphericity is the minimum condition the test actually requires, which is why it is the one Mauchly’s test checks rather than the stricter property. In practice, data that are compound symmetric are automatically spherical, but data can be spherical without being compound symmetric.
With exactly two levels of the within-subjects factor, sphericity is a non-issue: there is only one pairwise difference, so there is nothing for its variance to differ from. The assumption is trivially satisfied, and a sphericity test has zero degrees of freedom to test with — this is why statistical software omits the test entirely for a two-level factor rather than reporting it as automatically passed.
Why repeated-measures ANOVA depends on it
A between-subjects ANOVA does not need this assumption because each condition is measured on a different, independent group of cases. A repeated-measures design measures the same cases across conditions, so the scores in different conditions are correlated by construction — a person who scores high in condition A tends to score high in condition B too. The repeated-measures F-test’s reference distribution (an F-distribution with df based on the number of conditions and subjects) is only exact when sphericity holds, because the test implicitly assumes the correlation structure across conditions is uniform.
When sphericity is violated, the true sampling distribution of the F-ratio is more variable than the nominal F-distribution assumes. The practical consequence is that the uncorrected test becomes anti-conservative: it produces a smaller p-value than it should, inflating the real Type I error rate above the nominal alpha (for example, a nominal .05 test can behave closer to a .10 or .15 test under severe violation). That is the entire reason the correction exists — not to make the test more powerful, but to keep its actual error rate honest.
How Mauchly’s test works
Mauchly’s test evaluates the null hypothesis that the population covariance matrix of the within-subjects factor’s orthonormalized contrasts is proportional to an identity matrix — the formal statement of sphericity. The test statistic, Mauchly’s W, is built from the eigenvalues of that contrast covariance matrix: specifically, W is the ratio of the geometric mean of the eigenvalues to their arithmetic mean. W ranges from 0 to 1; a value of exactly 1 means the eigenvalues are all equal, which is sphericity itself, while values further from 1 indicate greater departure from it. A transformation of W is approximately chi-square distributed, with degrees of freedom determined by the number of levels in the within-subjects factor, which is how software attaches a p-value to the test.
A significant result (conventionally p < .05) means the sample data are unlikely to have arisen from a population where sphericity holds, so the assumption is treated as violated for that effect. Two well-documented limitations matter when interpreting it: Mauchly’s test is underpowered with small samples, meaning real violations can go undetected, and it is oversensitive with large samples, where even trivial, practically unimportant departures from sphericity become statistically significant. Neither failure mode is a flaw specific to any one software package — it is a property of the test statistic itself, which is why many methodologists recommend applying a correction by default in designs with more than two conditions and treating Mauchly’s result as one input rather than an automatic switch.
The two epsilon corrections
Both corrections work the same way: rather than discarding the standard F-test, they scale down its numerator and denominator degrees of freedom by a shrinkage factor, epsilon (ε), before looking up the p-value. A smaller epsilon means a greater departure from sphericity and a more aggressively reduced df, which makes the corrected test more conservative than the uncorrected one.
The Greenhouse-Geisser epsilon is estimated directly from the sample covariance matrix’s eigenvalues (the squared sum of the eigenvalues divided by the number of levels-minus-one times the sum of their squares). It is bounded between 1/(k−1) and 1, where k is the number of conditions — the lower bound represents maximal departure from sphericity, and 1 represents perfect sphericity (in which case the correction leaves the df unchanged). Greenhouse-Geisser is known to be conservative: it tends to underestimate the true epsilon in small samples, which over-corrects the df and reduces power more than strictly necessary.
The Huynh-Feldt epsilon is an adjusted version of Greenhouse-Geisser designed to correct that small-sample bias. It is calculated as a function of the Greenhouse-Geisser estimate, sample size, and the number of conditions, and it is typically larger than Greenhouse-Geisser — sometimes exceeding 1, in which case software caps it at 1 and the corrected test reduces to the uncorrected one. Because it is less conservative, Huynh-Feldt trades away some of Greenhouse-Geisser’s protection against Type I error inflation in exchange for more statistical power.
Choosing between Greenhouse-Geisser and Huynh-Feldt
The widely reproduced decision rule, traceable to Girden’s 1992 treatment of the topic, is to check the Greenhouse-Geisser epsilon value itself: use the Greenhouse-Geisser correction when epsilon is below .75, and the Huynh-Feldt correction when epsilon is .75 or above. The logic is that Huynh-Feldt’s small-sample bias correction is only reliable when the true epsilon is relatively close to 1 (mild sphericity violation); when epsilon is small (severe violation), Huynh-Feldt tends to overestimate it, so Greenhouse-Geisser’s more conservative estimate is the safer choice.
Some methodologists simplify this further and recommend defaulting to Greenhouse-Geisser in all cases, accepting the modest loss of power as the price of a correction that reliably does not overstate epsilon. Both are defensible published conventions; what matters for a write-up is naming which one was used and why, not silently picking whichever gives the smaller p-value.
A third, more conservative option some texts describe is the lower-bound correction, which sets epsilon to its theoretical minimum of 1/(k−1) regardless of the sample data. It guarantees the corrected test never exceeds the true Type I error rate, but at a real cost in power, and it is rarely used in practice now that Greenhouse-Geisser and Huynh-Feldt are calculated automatically by every major statistics package.
The alternative: a multivariate approach
Because sphericity is a univariate-ANOVA-specific requirement, one way to avoid the correction question altogether is to analyze the within-subjects factor with a multivariate (MANOVA-based) approach instead — treating the repeated measurements as multiple correlated dependent variables rather than levels of one factor. A multivariate test (Wilks’ lambda, Pillai’s trace, and related statistics) makes no sphericity assumption at all, so no correction is needed. The trade-off is that multivariate tests generally have less power than a correctly corrected univariate F-test, particularly with a small number of subjects relative to the number of conditions, which is why the corrected-univariate route remains the default choice for most repeated-measures designs rather than a fallback.
Reporting the corrected result
A complete write-up states four things: that Mauchly’s test was significant (with its own chi-square, df, and p-value, or a note that it was not significant and no correction was needed), which correction was applied and why, the corrected degrees of freedom, and the corrected F and p-values. A typical sentence reads: “Mauchly’s test indicated the assumption of sphericity had been violated, χ²(2) = 8.42, p = .015, therefore degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = .71). The effect of condition was significant, F(1.42, 28.34) = 6.19, p = .011.” Note that the corrected df are not whole numbers — they are the uncorrected df multiplied by epsilon, and both numerator and denominator df get corrected by the same factor. Reporting effect size alongside the corrected F is standard practice and unaffected by which correction was chosen, since epsilon only rescales df, not the underlying sums of squares.
Frequently asked questions
Does sphericity apply to a one-way between-subjects ANOVA?
No. Sphericity is specific to within-subjects (repeated-measures) factors with three or more levels, because it concerns the correlation structure created by measuring the same cases repeatedly. A between-subjects design, where each condition is measured on an independent group, has no such structure to test.
What does it mean if Mauchly’s test is not significant?
A non-significant result (p ≥ .05) means the data do not show evidence against sphericity, so the assumption is treated as tenable and the standard, uncorrected repeated-measures F-test and its usual degrees of freedom are reported as-is — no epsilon correction is needed.
Can Greenhouse-Geisser epsilon be greater than 1?
No, Greenhouse-Geisser epsilon is mathematically bounded at a maximum of 1 (perfect sphericity). Huynh-Feldt epsilon can exceed 1 in its raw calculation because of how its small-sample correction is derived, and in that case software caps the reported value at 1.
Is Mauchly’s test reliable with a small sample?
Not fully. Mauchly’s test is known to be underpowered in small samples, meaning a genuine sphericity violation can fail to reach significance and go uncorrected. This is one reason some methodologists recommend applying a Greenhouse-Geisser correction by default whenever a within-subjects factor has more than two levels, rather than relying on Mauchly’s result alone as the sole trigger.
Do Greenhouse-Geisser and Huynh-Feldt ever change which conclusion you draw?
Yes, occasionally. Because both corrections reduce degrees of freedom, a result that is significant under the uncorrected test can become non-significant after correction, especially with a smaller epsilon (more severe sphericity violation) or a result close to the alpha threshold. This is exactly the scenario the correction exists to guard against — an uncorrected test that looks significant only because its actual Type I error rate is inflated.
