Read skewness by sign and shape
- Positive skewness: the longer or heavier tail extends toward larger values.
- Negative skewness: the longer or heavier tail extends toward smaller values.
- Near zero: limited measured asymmetry, not proof of a normal distribution.
The sign is easy to reverse when readers focus on where most observations sit. Name the tail direction. A right-skewed distribution can have most values clustered low with a smaller number of large observations.
Raw kurtosis versus excess kurtosis
Software may report either raw kurtosis or excess kurtosis. Under the common normal-reference convention:
| Statistic | Normal-reference value |
|---|---|
| Raw Pearson kurtosis | 3 |
| Excess kurtosis | 0 |
The two outputs differ by 3. Joanes and Gill compare common sample estimators and show why identifying the software definition comes before comparing values.1 A reported kurtosis of 3 and an excess kurtosis of 0 can describe the same reference shape.
Positive excess kurtosis indicates heavier tails or greater concentration than the normal reference; negative excess kurtosis indicates lighter tails or a flatter shape. It does not by itself identify where the departure occurs. A histogram and QQ plot show whether the issue is skew, tails, discreteness, mixtures, or isolated observations.
Where the values 2, 7, 3, and 21 came from
Curran, West, and Finch used the following conditions in a confirmatory-factor-analysis robustness simulation:3
| Skewness | Excess kurtosis | Study label | What the values were |
|---|---|---|---|
| 2.0 | 7.0 | Moderately nonnormal | A simulation condition informed by datasets the authors examined |
| 3.0 | 21.0 | Severely nonnormal | A more extreme simulation condition in the same study |
These values tested how particular estimators behaved across simulated conditions. They were not proposed as universal pass/fail boundaries. The paper’s normal condition has kurtosis 0, which establishes that 7 and 21 are excess kurtosis, not raw Pearson kurtosis.
Kim later reports 2 and 7 as reference values for large samples but labels 7 as “proper” kurtosis, which conflicts with the excess-kurtosis scale in the primary simulation paper.2 ResultAtlas does not silently reconcile that difference. Any use of the pair should state the statistic definition and source-specific purpose.
Why a cutoff cannot decide the assumption
Sampling variability changes with n. In a small sample, skewness and kurtosis estimates can be noisy. In a very large sample, small departures can be estimated precisely without being practically important for the analysis. Kim proposes comparing each statistic with its standard error: absolute z above 1.96 for samples under 50 and above 3.29 for samples between roughly 50 and 300.2 These are sample-size-scoped screening rules from that note, not universal acceptance bands. Kim also emphasizes combining numerical measures with plots rather than using one mechanical normality decision.
The relevant target may also be residuals rather than the raw outcome. Many methods tolerate some nonnormality, while others depend more on tail behavior, sample balance, or the accuracy of interval estimates. Ask what assumption the fitted procedure actually needs.
Use normality tests for formal null-model output, a QQ plot to locate departures, and a histogram to inspect bin-dependent shape. The provenance guide explains why an unverified floating cutoff is omitted rather than repeated.
A reliable interpretation sequence
- Identify whether kurtosis is raw or excess.
- Record sample size and estimator/software definition.
- Read skewness sign and magnitude descriptively.
- Inspect the histogram and QQ plot.
- Check the model residuals if the assumption belongs to errors.
- Assess sensitivity with a method suited to the design.
- Attribute any threshold used in a formal decision.
A defensible reporting sentence
“The sample showed positive skewness and positive excess kurtosis under the software’s stated estimator. The histogram and QQ plot located the departure in the upper tail; no universal cutoff was used to declare the analysis valid or invalid.”