Skewness and Kurtosis: Understanding Deviations from Normal Distribution

The normal distribution is statistics’ most celebrated benchmark – that smooth, symmetrical bell curve where data clusters neatly around the mean and tapers off equally on both sides. But real-world data, especially in psychology, rarely behaves so perfectly. Test scores bunch up at the high end. Reaction times drag out into long right-side tails. Income distributions pile up near the bottom with a few extreme values stretching far to the right. These departures from the ideal are not random noise – they are systematic deviations that carry real meaning. Two key statistics help us measure and describe exactly how a distribution strays from normality: skewness and kurtosis.

Why deviations from normality matter 🔗

Many of the most widely used statistical tests in psychology – t-tests, ANOVA, regression – rest on the assumption that the underlying data is approximately normally distributed. When that assumption is violated, the conclusions drawn from those tests can be misleading. Research published in Restorative Dentistry & Endodontics via PubMed Central points out that assessing skewness and kurtosis is a practical method of evaluating normality that works reliably across both small and large sample sizes – something formal tests like the Shapiro-Wilk test struggle to do consistently. Understanding these two statistics is therefore not just an academic exercise; it directly shapes which analytical tools a researcher should use.

A 2025 tutorial in Psychology & Marketing (Wiley) identifies skewness, kurtosis, and outliers as the three primary ways a distribution can appear non-normal. Researchers who recognize these patterns early can make smarter decisions about data transformations or the use of non-parametric alternatives – rather than forcing unsuitable tests onto data that breaks their assumptions.

Skewness: when a distribution leans to one side 🔗

According to the NIST/SEMATECH e-Handbook of Statistical Methods, skewness is a measure of symmetry – or more precisely, the lack of it. A distribution is symmetric if it looks the same to the left and right of its center point. In a perfectly normal distribution, skewness equals zero. When a distribution is not symmetric, one tail stretches longer than the other, and the skewness statistic captures the direction and degree of that lean.

Skewness is mathematically expressed as the third standardized central moment of the distribution. Positive values indicate a rightward lean; negative values indicate a leftward lean. Standard statistical software packages – SPSS, R, SAS – compute this automatically as part of descriptive statistics output.

Positive skewness 🔗

A positively skewed distribution has a longer tail on the right side. The bulk of scores cluster toward the lower end, while a smaller number of unusually high scores pull the tail rightward. In this case, the mean is greater than the median, which in turn is greater than the mode. DataCamp’s guide on skewness and kurtosis describes positive skew as giving the distribution a “leaned, squished-to-one-side look” with values stretching toward the right. A real-world example in psychology: reaction time data, where most people respond quickly but a few very slow responses drag the tail to the right.

Negative skewness 🔗

A negatively skewed distribution mirrors this pattern in the opposite direction – the longer tail extends to the left. Most values cluster near the high end of the scale, with a few very low scores pulling the distribution leftward. Here, the mean falls below the median, which falls below the mode. An example: scores on a very easy test where most students perform well, but a handful score very low.

How much skewness is too much? 🔗

Kim (2013), as cited on PubMed Central, recommends that an absolute skewness value greater than 2 be treated as a reference threshold for substantial non-normality. Hair et al. (2022) suggest that skewness values between −1 and +1 are excellent, while values between −2 and +2 remain generally acceptable. Beyond that range, the assumption of normality becomes difficult to defend, and alternative analytical approaches should be considered.

Kurtosis: the shape of peaks and tails 🔗

The NIST Handbook defines kurtosis as a measure of whether a dataset has heavy tails or light tails relative to a normal distribution. Datasets with high kurtosis tend to have heavier tails – meaning more extreme values or outliers – while datasets with low kurtosis have lighter tails and a flatter overall shape. DataCamp notes that the term “kurtosis” itself comes from the Greek word meaning curved or arched, and was first introduced by the British mathematician Karl Pearson, who spent his career studying probability distributions.

The standard normal distribution has a kurtosis value of 3. Many statistical packages report excess kurtosis (also called Fisher’s kurtosis), which subtracts 3 from the raw value, so that a normal distribution scores zero. This makes it easier to see at a glance whether a distribution is more or less peaked than normal.

Mesokurtic distributions 🔗

A mesokurtic distribution has kurtosis equal to 3 (or excess kurtosis of 0). It matches the normal distribution in terms of peak height and tail weight. This is the baseline against which the other types are compared. SPC for Excel notes that when kurtosis is close to 0 (on the excess scale), a normal distribution is often assumed.

Leptokurtic distributions 🔗

A leptokurtic distribution has kurtosis greater than 3 (positive excess kurtosis). It features a sharper, taller central peak and heavier tails than a normal distribution. DataCamp explains that this pattern reflects more values concentrated tightly around the mean alongside a greater concentration of extreme values in the tails – increasing the likelihood of outliers. In psychology, IQ score distributions within very homogeneous populations may exhibit mild leptokurtosis. Hair et al. (2022) flag excess kurtosis values greater than +2 as indicating a distribution that is too peaked for the normality assumption to hold comfortably.

Platykurtic distributions 🔗

A platykurtic distribution has kurtosis less than 3 (negative excess kurtosis). It is flatter and broader than a normal distribution, with lighter tails and fewer extreme values. SPC for Excel describes platykurtic distributions as having lighter tails – that is, less probability mass in the extreme regions. A uniform distribution, where every outcome is equally likely, approaches the extreme platykurtic case. Hair et al. (2022) suggest that excess kurtosis values below −2 indicate a distribution that is too flat to satisfy normality assumptions.

Reference thresholds for kurtosis 🔗

The PubMed Central clinical statistics notes recommend using an absolute kurtosis value (proper, not excess) greater than 7 as the threshold for substantial non-normality in larger samples. The MRC Cognition and Brain Sciences Unit at Cambridge notes there are no universally official cut-off rules, but that a kurtosis (using SPSS’s excess definition) of 3 or more – equivalent to a proper kurtosis of 6 or more – indicates a large departure from normality. These thresholds provide practical guidance, but researchers are encouraged to combine them with visual inspection of histograms rather than relying on numbers alone.

Skewness and kurtosis in psychological research 🔗

In psychological measurement, both skewness and kurtosis have direct consequences for the validity of data analysis. A study in Psychological Assessment (PMC) reviewed hundreds of educational and psychological test score distributions and found that non-normal distributions are common – and often tied to specific testing conditions like ceiling effects or restricted score ranges. This confirms that skewness and kurtosis are not rare anomalies but routine features of real psychological data.

When skewness or kurtosis indicates substantial non-normality, researchers have several options. They can apply data transformations – such as logarithmic scaling – to bring the distribution closer to normal. Alternatively, they can switch to non-parametric statistical methods that do not assume normality at all. Kim (2013) recommends a combination of visual inspection, skewness and kurtosis assessment, and formal normality tests for the most robust evaluation. Iacobucci et al. (2025) add an encouraging finding: data that deviates from normality due to moderate skewness or kurtosis does not always require deletion or heavy manipulation – many commonly used statistical models are reasonably robust to mild departures, which also supports greater transparency in research reporting.

Putting it together: reading a distribution’s shape 🔗

When you encounter a dataset in psychology – whether it’s anxiety scores, cognitive test results, or survey responses – skewness and kurtosis give you an immediate, quantitative snapshot of its shape. A skewness near zero and excess kurtosis near zero signal a distribution close to normal. Large positive skewness tells you most people scored low with a few very high scorers. Large negative skewness tells you the opposite. Leptokurtosis warns you about outliers and heavy tails. Platykurtosis tells you scores are spread unusually broadly with few extremes.

These are not just diagnostic tools – they are interpretive ones. Knowing that an anxiety measure is positively skewed, for instance, tells a researcher that most participants reported low anxiety, while a minority experienced much higher levels. That finding itself has clinical and theoretical relevance, entirely apart from which statistical test comes next.

What do you think? If you collected data on a psychological measure and found it was strongly positively skewed, how would that shape your interpretation of the results – and your choice of statistical test? And does it change how you view a dataset to know that its kurtosis signals an unusually high risk of extreme outliers?

References

1. https://pmc.ncbi.nlm.nih.gov/articles/PMC3591587/
2. https://onlinelibrary.wiley.com/doi/full/10.1002/mar.22187
3. https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
4. https://www.datacamp.com/tutorial/understanding-skewness-and-kurtosis
5. https://smartpls.com/documentation/functionalities/excess-kurtosis-and-skewness/
6. https://www.spcforexcel.com/knowledge/basic-statistics/are-skewness-and-kurtosis-useful-statistics/
7. https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/Simon
8. https://pmc.ncbi.nlm.nih.gov/articles/PMC5965643/