Skewness and Kurtosis in Clinical Data

Introduction

Clinical datasets rarely look perfectly symmetric. Laboratory measurements, biomarker concentrations, hospitalization durations, time-to-event variables, healthcare costs, and many pharmacokinetic measurements can have long tails or unusually concentrated distributions.

Two descriptive statistics are commonly used to summarize aspects of this distributional shape: skewness and kurtosis.

Skewness describes the degree and direction of asymmetry in a distribution, while kurtosis is primarily related to the behavior of the tails and the frequency of observations that are unusually far from the center.

Why Distributional Shape Matters in Clinical Research

Consider a clinical trial measuring a biomarker before treatment. Suppose most patients have values between 5 and 15, but a small number have values above 50.

The arithmetic mean can be pulled upward by those extreme observations even though most patients are concentrated much lower.

For example, the mean might be 14 while the median is only 9.

That difference is not merely a mathematical curiosity. It affects how the data should be summarized and may influence the choice of statistical model or transformation.

Distributional shape can affect:

• Whether the mean is an informative measure of the typical patient
• Whether a median and interquartile range are more appropriate
• Whether a transformation such as the logarithm is useful
• Whether normal-theory methods are reasonable
• How strongly individual observations influence a model
• How residuals behave in regression or mixed models
• How reference intervals or percentiles should be interpreted

Symmetry and the Normal Distribution

A normal distribution is symmetric around its mean.

For a perfectly symmetric normal distribution:

The two sides of the distribution mirror one another around the center.

Many statistical methods are built around normality assumptions, especially for small-sample inference or models involving normally distributed residuals.

However, clinical data do not need to be normally distributed simply because a statistical method uses a normal approximation.

What Is Skewness?

Skewness is a standardized measure of the third central moment.

For a population random variable \(X\), population skewness can be expressed as:

where:

• \(\mu\) is the population mean.
• \(\sigma\) is the population standard deviation.
• The cubic term preserves the direction of deviations from the mean.

Because positive deviations are cubed to positive values and negative deviations are cubed to negative values, skewness captures directional asymmetry.

Positive Skewness

A distribution with positive skewness has a relatively long right tail.

This is sometimes called:

• Right-skewed
• Positively skewed
• Skewed to the right

A common clinical example is hospital length of stay.

Most patients may have relatively short stays, while a smaller number remain hospitalized for substantially longer periods.

The resulting distribution may look conceptually like:

Negative Skewness

A distribution with negative skewness has a relatively long left tail.

This is sometimes called:

• Left-skewed
• Negatively skewed
• Skewed to the left

For example, consider a clinical scale where most patients score near the maximum possible value.

If a smaller number of patients have substantially lower scores, the distribution may have a longer left tail.

The Mean–Median Relationship

The mean is sensitive to extreme observations. The median is much less sensitive.

For a strongly right-skewed distribution, it is common for:

For a strongly left-skewed distribution, it is common for:

This relationship is useful as a quick descriptive clue, but it is not a formal definition of skewness.

A Simple Clinical Example

Suppose ten patients have the following hospital lengths of stay in days:

Most observations are between 2 and 8 days, but one patient has a 25-day stay.

The median is:

while the mean is:

The extreme 25-day observation substantially increases the mean.

This is a simple illustration of why the median can sometimes better represent a typical patient when a clinical variable is strongly right-skewed.

How Skewness Is Calculated

For a sample of \(n\) observations, a commonly encountered moment-based sample skewness estimator is based on:

The corresponding standardized third moment is:

Different statistical software packages can use different finite-sample corrections.

One common adjusted estimator is:

Therefore, two software packages can occasionally report slightly different numerical skewness values for the same dataset even though they agree on the overall interpretation.

How to Interpret the Magnitude of Skewness

There is no universal set of cutoffs that determines whether skewness is "small," "moderate," or "large."

Rules of thumb such as \(|\text{skewness}|<1\) indicating relatively mild asymmetry can be useful descriptively, but they should not be treated as universal statistical decision rules.

The sample size and clinical context matter. A skewness estimate of 0.8 in a dataset of 20 patients may have a very different practical meaning from 0.8 in a dataset of 20,000 patients.

What Is Kurtosis?

Kurtosis is based on the fourth central moment. For a population distribution, kurtosis can be expressed as:

Because deviations are raised to the fourth power, both large positive and large negative deviations contribute positively.

This makes kurtosis particularly sensitive to observations far from the center.

Excess Kurtosis

A normal distribution has kurtosis:

For this reason, many statistical programs report excess kurtosis:

Under this convention, a normal distribution has:

This distinction is important because some software reports kurtosis while other software reports excess kurtosis.

Leptokurtic, Mesokurtic, and Platykurtic

Traditional terminology describes distributions according to their kurtosis relative to the normal distribution.

The terminology is useful, but the practical question is usually more important: Does the dataset have unusually heavy tails or extreme observations that could affect the analysis?

Skewness and Kurtosis Measure Different Things

It is possible for a dataset to have:

• High skewness but relatively ordinary tail behavior
• Low skewness but heavy tails
• Both high skewness and high kurtosis
• Neither substantial skewness nor substantial excess kurtosis

For example, a symmetric distribution can have substantial positive excess kurtosis.

Therefore:

They summarize different aspects of distributional shape.

A Useful Four-Case Framework

Clinical Example: Length of Stay

Hospital length of stay is frequently right-skewed.

Imagine a dataset in which most patients are discharged after a few days, while a smaller number remain hospitalized for several weeks.

The distribution may have:

• Positive skewness
• A mean greater than the median
• An extended right tail
• Potentially positive excess kurtosis

In this situation, reporting only mean and standard deviation may obscure important features of the data.

A more informative summary might include:

along with selected percentiles such as the 25th, 50th, and 75th percentiles.

Clinical Example: Biomarker Concentrations

Many biomarkers have distributions that are strongly right-skewed.

For example, a biomarker might have most observations between 1 and 10 units, while a small number exceed 100.

In such situations, investigators often consider a logarithmic transformation:

The transformation compresses large values more strongly than small values.

This can reduce skewness and make the distribution more suitable for some statistical models.

Clinical Example: Laboratory Measurements

Suppose a laboratory measurement has:

and:

This suggests substantial asymmetry and unusually heavy tail behavior.

The investigator should examine:

• Whether the variable has a natural lower bound
• Whether the high values are clinically plausible
• Whether units or data-entry errors exist
• Whether the extreme values correspond to specific patient subgroups
• Whether treatment or disease severity explains the tail
• Whether a transformation is clinically interpretable

The statistics should prompt investigation rather than automatically trigger data deletion.

Outliers and Kurtosis

Because deviations are raised to the fourth power, kurtosis can be strongly affected by extreme observations.

Consider a deviation of 2 standard deviations:

A deviation of 5 standard deviations contributes:

Thus, extremely distant observations can have enormous influence on the fourth moment.

Skewness Is Also Sensitive to Outliers

Skewness uses a cubic deviation:

Therefore, extreme observations can have a large influence on the sample skewness.

This means skewness and kurtosis should not be treated as completely robust statistics.

A useful diagnostic strategy is to compare:

• Histogram
• Density plot
• Boxplot
• Mean
• Median
• Standard deviation
• IQR
• Skewness
• Kurtosis

Boxplots and Skewness

A boxplot can provide a visual clue about skewness.

For right-skewed data, one might observe:

• A longer upper whisker
• The median positioned closer to the lower quartile
• Several high-value observations beyond the upper whisker

For left-skewed data, the opposite pattern may occur.

However, a boxplot does not directly calculate skewness and should not be treated as a substitute for examining the complete distribution.

Why Histograms Are Still Important

A numerical skewness estimate compresses an entire distribution into one number.

Two distributions can have similar skewness but very different shapes.

A histogram can reveal:

• Multiple modes
• Gaps
• Clusters
• Ceiling effects
• Floor effects
• Extreme observations
• Long tails

Skewness Does Not Mean "Bad Data"

A skewed clinical variable is not necessarily problematic.

Many clinically important variables are naturally skewed.

For example:

• Length of stay
• Healthcare expenditure
• Some biomarker concentrations
• Some pharmacokinetic parameters
• Time to recovery
• Number of hospitalizations

The objective is not to force every clinical variable to resemble a normal distribution.

The objective is to understand the distribution and select an analysis appropriate for the scientific question and data-generating process.

Does Skewness Violate the Normality Assumption?

This question requires careful interpretation.

If an analysis assumes normally distributed residuals, substantial skewness in the raw outcome does not automatically invalidate the analysis.

For example, consider a linear model:

The relevant assumption for classical inference concerns the error term \(\epsilon_i\), not necessarily the marginal distribution of \(Y_i\).

After fitting the model, investigators should examine residual diagnostics.

Skewness in Regression Residuals

Suppose a linear regression model produces residuals with strong right skew.

This may indicate:

• Nonlinear structure
• Heteroscedasticity
• An inappropriate outcome scale
• Influential observations
• A non-normal error distribution
• Model misspecification

The appropriate response is to investigate the cause rather than simply delete observations until skewness disappears.

Skewness in Clinical Trial Endpoints

Suppose a randomized clinical trial evaluates change from baseline in a continuous endpoint.

The change score is:

The distribution of \(\Delta_i\) may be skewed even if the baseline and post-treatment variables individually appear reasonably symmetric.

Therefore, distributional diagnostics should be applied to the variable that is actually analyzed.

Skewness and Summary Statistics

The appropriate descriptive summary depends on the distribution.

There is no rule that prevents reporting both mean and median.

In fact, when skewness is clinically relevant, reporting both can be especially informative.

Geometric Means and Log-Transformed Data

For strictly positive measurements, a logarithmic scale can sometimes provide a more natural description.

The geometric mean is:

Equivalently:

This can be particularly useful when the data are multiplicative rather than additive in nature.

Pharmacokinetic exposure measures, concentrations, and some laboratory measurements are common examples where logarithmic analysis may be considered.

Kurtosis and Model Diagnostics

Heavy tails can influence statistical procedures even when skewness is close to zero.

For example, suppose a distribution is symmetric but contains more extreme observations than a normal distribution.

Its skewness may be near zero while its excess kurtosis is positive.

This distinction is important because symmetry alone does not guarantee normal-like tail behavior.

A Symmetric Heavy-Tailed Example

Imagine a biomarker distribution centered around zero. Most observations lie between \(-2\) and \(2\), but there are unusually many observations near \(-6\) and \(6\).

Because the tails are balanced:

But the extreme observations can produce:

This is a useful reminder that: zero skewness does not imply normality.

Skewness and Kurtosis Are Not Normality Tests

Skewness and kurtosis provide descriptive information about shape.

They do not by themselves establish whether a dataset is normally distributed.

A normality assessment may incorporate:

• Histograms
• Q-Q plots
• Residual plots
• Skewness and kurtosis
• Formal tests when appropriate
• Knowledge of sample size
• Clinical and scientific context

Q-Q Plots

A normal Q-Q plot compares observed quantiles with theoretical normal quantiles.

If the observations follow a normal distribution approximately, the points should lie reasonably close to a straight line.

Systematic curvature in the tails can reveal departures that a single skewness or kurtosis number cannot describe adequately.

R: Calculating Skewness and Kurtosis

Base R provides functions for many descriptive statistics, but skewness and kurtosis are commonly calculated with packages such as moments or e1071.

For example:

install.packages("moments")

library(moments)

x <- c(
  2, 2, 3, 3, 4,
  4, 5, 6, 8, 25
)

skewness(x)
kurtosis(x)

The exact numerical values depend on the estimator used by the package.

R: Comparing Mean and Median

mean(x)
median(x)
sd(x)
IQR(x)

For the example data:

mean(x)
# 6.2

median(x)
# 4

sd(x)

IQR(x)

The large difference between the mean and median immediately suggests that the 25-day observation has a substantial effect on the arithmetic mean.

R: Visualizing the Distribution

hist(
  x,
  main = "Clinical Length of Stay",
  xlab = "Days"
)

boxplot(
  x,
  horizontal = TRUE,
  main = "Clinical Length of Stay"
)

Graphical inspection should accompany numerical measures of skewness and kurtosis.

R: A More Complete Diagnostic Workflow

summary(x)

mean(x)
median(x)
sd(x)
IQR(x)

library(moments)

skewness(x)
kurtosis(x)

hist(x)
boxplot(x, horizontal = TRUE)

For a clinical analysis, the next step would usually be to investigate the observations and determine whether the observed distribution makes clinical sense.

R: Using a Log Transformation

For a strictly positive variable:

x_log <- log(x)

hist(
  x_log,
  main = "Log-Transformed Length of Stay",
  xlab = "log(Days)"
)

library(moments)

skewness(x_log)
kurtosis(x_log)

The transformation may reduce right skewness, but the decision to use it should depend on the analysis and scientific interpretation.

Clinical Interpretation of a Log Transformation

Suppose the original measurement is \(X\), while the analysis uses:

A difference on the log scale corresponds approximately to a ratio on the original scale.

For example, if two groups differ by:

then the corresponding ratio is:

which can be interpreted as approximately a 22% multiplicative difference.

Skewness and Sample Size

The effect of skewness on inference depends strongly on sample size and the statistical method.

With a sufficiently large sample, the sampling distribution of a mean can sometimes be approximately normal even when the underlying observations are not.

This is related to the Central Limit Theorem.

under appropriate conditions as \(n\) becomes sufficiently large.

However, severe skewness, heavy tails, dependence, or extreme observations can complicate this approximation.

Why Small Samples Require More Care

With a small clinical study, a few observations can dominate the estimated skewness and kurtosis.

For example, in a sample of 12 patients, one extreme value represents more than 8% of the dataset.

In a sample of 1,200 patients, one observation represents less than 0.1%.

The same clinical phenomenon can therefore have very different statistical implications depending on sample size.

Skewness in Subgroup Analysis

Distributional shape can differ substantially between treatment groups or patient subgroups.

For example:

A treatment group with substantially different distributional shape may require additional investigation.

This does not necessarily invalidate a comparison, but it can affect the interpretation of means, confidence intervals, and model assumptions.

Skewness and Missing Data

Missingness can change the apparent distribution of a clinical endpoint.

Suppose patients with severe disease are more likely to have missing measurements.

The observed dataset may appear less right-skewed than the underlying population.

Therefore, distributional summaries should be interpreted alongside the missing-data mechanism and analysis population.

Skewness and Baseline Characteristics

Clinical trial baseline tables often contain continuous variables such as:

• Age
• Body weight
• Blood pressure
• Laboratory values
• Disease duration
• Biomarker measurements

Some may be approximately symmetric, while others may be substantially skewed.

The descriptive summary should reflect the actual distribution rather than following a universal rule that every variable must be presented as mean ± SD.

Mean ± SD Versus Median [IQR]

A common reporting convention is:

for approximately symmetric continuous data.

For skewed data, a common alternative is:

For example:

The exact reporting convention should follow the statistical analysis plan and the characteristics of the endpoint.

Skewness and Confidence Intervals

When a mean is estimated from highly skewed data, its confidence interval may still be valid under some conditions, particularly with large samples and appropriate methods.

However, skewness can affect small-sample inference and the performance of normal-theory intervals.

Alternative approaches may include:

• Transformations
• Bootstrap confidence intervals
• Robust methods
• Nonparametric methods
• Distribution-specific models

The correct choice depends on the estimand and scientific question.

Do Not Automatically Use a Nonparametric Test

A common misconception is:

"The data are skewed, therefore I must use a nonparametric test."

That conclusion is too simplistic.

For example, a two-sample comparison of means can sometimes remain appropriate with moderately skewed data, particularly with adequate sample sizes and suitable variance methods.

Likewise, a nonparametric test does not automatically solve every problem created by outliers, heteroscedasticity, or an inappropriate estimand.

Skewness and Robust Statistics

Robust statistical methods are designed to reduce sensitivity to unusual observations or distributional departures.

Examples include:

• Median-based summaries
• Quantiles
• Trimmed means
• Robust regression
• Bootstrap procedures
• Other robust estimators

These approaches can be useful when a small number of observations exert disproportionate influence.

Skewness Versus Transformation

There are several possible responses to a skewed endpoint.

Common Misconceptions About Kurtosis

1. "High kurtosis simply means a tall peak." Kurtosis is more usefully interpreted in terms of tail behavior and extreme observations than as a simple measure of peak height.
2. "Kurtosis of zero means the data are normal." Zero excess kurtosis does not prove normality.
3. "Positive kurtosis means bad data." Heavy tails may be a genuine feature of the clinical population.
4. "One high observation cannot affect kurtosis much." Because deviations are raised to the fourth power, extreme observations can have a substantial effect.
5. "Skewness and kurtosis are normality tests." They are descriptive measures, not definitive normality tests.

Common Misconceptions About Skewness

1. "Any nonzero skewness means the data are unusable." Most real datasets are not perfectly symmetric.
2. "Positive skewness means the data are concentrated on the right." Positive skewness refers to a longer right tail; most observations may still be concentrated toward the lower end.
3. "The mean is always inappropriate for skewed data." The mean can remain a meaningful estimand even when the distribution is skewed.
4. "A skewed variable must be log-transformed." Transformation is a modeling decision, not an automatic response to skewness.
5. "If the raw outcome is skewed, a linear model cannot be used." The relevant assumptions often concern model residuals rather than the raw outcome.

A Practical Clinical Data Workflow

A Compact Interpretation Table

Skewness and Kurtosis in Clinical Reporting

A statistical report should focus on clinically meaningful information rather than simply listing distributional statistics.

For example, instead of writing:

"The variable had skewness of 2.1."

a more informative statement might be:

"Length of stay was strongly right-skewed, with most patients discharged within several days but a small number experiencing substantially longer hospitalizations. The median and interquartile range were therefore used as the primary descriptive summary."

This connects the statistical observation to the clinical interpretation.

What to Put in a Statistical Analysis Plan

If distributional shape is expected to influence the analysis, the Statistical Analysis Plan should describe the approach prospectively.

Depending on the endpoint, this may include:

• The descriptive statistics to be reported
• Graphical summaries
• Transformation rules
• Primary analysis model
• Handling of extreme observations
• Analysis of residuals
• Alternative or sensitivity analyses
• Rules for reporting mean versus median

The SAP should avoid creating analysis decisions solely in response to observed results unless the adaptive or exploratory nature of those decisions is explicitly acknowledged.

Skewness and Sensitivity Analyses

When extreme observations materially affect an analysis, sensitivity analyses can help determine whether conclusions are robust.

For example, investigators might compare:

• Original-scale analysis
• Log-transformed analysis
• Robust analysis
• Analysis using a prespecified influential-observation strategy

The purpose is not to select whichever analysis produces the preferred p-value.

The purpose is to understand whether the scientific conclusion depends strongly on a particular distributional assumption or unusual observation.

Skewness and Clinical Trial Sample Size

Distributional shape can also affect sample-size planning for continuous endpoints.

For example, sample-size calculations based on a normal approximation may use a standard deviation:

where \(\Delta\) is the target effect.

If the underlying population is highly skewed or heavy-tailed, the empirical standard deviation may be strongly influenced by extreme values.

This is one reason why pilot data should be interpreted carefully when used to inform future sample-size assumptions.

Skewness in Pharmacokinetic Data

Pharmacokinetic variables often provide particularly clear examples of distributional asymmetry.

Measures such as:

• C_max
• AUC
• Clearance
• Volume of distribution

can exhibit substantial between-subject variability.

Because PK parameters are generally positive and may vary multiplicatively, log-scale analyses are often scientifically meaningful.

For example:

may be more suitable for some inferential procedures than the original AUC scale.

Skewness in Healthcare Cost Data

Healthcare cost is another classic right-skewed variable.

Most patients may incur relatively modest costs, while a small number of patients require very expensive procedures, prolonged hospitalization, or intensive care.

This can create:

• Large positive skewness
• High variance
• Heavy right tails
• Large differences between mean and median

The mean remains important when estimating average cost, because total resource utilization is often the scientific estimand.

Therefore, replacing the mean with the median solely because the data are skewed can answer a different question.

Skewness Does Not Determine the Estimand

This is one of the most important lessons.

A skewed distribution does not automatically imply that the median should replace the mean.

The appropriate summary depends on the question.

Skewness, Kurtosis, and the Central Limit Theorem

The Central Limit Theorem explains why the sample mean can have an approximately normal sampling distribution under broad conditions.

However, convergence can be slower for highly skewed or heavy-tailed populations.

Consequently, saying:

"The sample is large, so skewness does not matter."

is too simplistic.

The practical impact depends on:

• Degree of skewness
• Tail heaviness
• Sample size
• Dependence structure
• Presence of influential observations
• Statistical method

A Numerical Illustration of Tail Sensitivity

Suppose two observations are 2 standard deviations from the mean. Their fourth-power contribution is:

Now consider one observation 6 standard deviations from the mean:

The fourth-power contribution is dramatically larger.

This illustrates why kurtosis can react strongly to a small number of extreme observations.

Why Data Cleaning Should Come First

Suppose a dataset contains one laboratory value that is 100 times larger than all other values.

Before interpreting a large kurtosis value, verify:

• Decimal placement
• Measurement units
• Data-entry accuracy
• Laboratory reference range
• Patient eligibility
• Clinical plausibility

If the observation is valid, it should not be removed simply because it makes the distribution inconvenient.

Skewness and Kurtosis: A Practical Decision Framework

Key Differences Between Skewness and Kurtosis

What Should a Clinical Biostatistician Look At?

A strong distributional assessment should rarely rely on a single statistic.

A practical checklist is:

• Mean
• Median
• Standard deviation
• IQR
• Selected percentiles
• Histogram
• Boxplot
• Q-Q plot
• Skewness
• Excess kurtosis
• Clinical plausibility of extreme values
• Model residuals where relevant

Summary of the Core Concepts

Skewness describes the asymmetry of a distribution.

Positive skewness generally indicates a longer right tail, while negative skewness generally indicates a longer left tail.

Kurtosis is related primarily to tail behavior and the frequency or influence of extreme observations.

Excess kurtosis is calculated by subtracting 3 from ordinary kurtosis so that a normal distribution has excess kurtosis of zero.

Neither statistic should be interpreted as a standalone normality test.

In clinical research, the most useful interpretation combines the numerical statistics with graphical diagnostics, clinical knowledge, and the intended estimand.

Quick Reference

References

Joanes, D. N. & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 47(1), 183–189.

Westfall, P. H. (2014). Kurtosis as peakedness, 1905–2014. R.I.P. American Statistician, 68(3), 191–195.

DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2(3), 292–307.

Fisher, R. A. (1930). The moments of the distributions for normal samples of measures of departure from normality. Proceedings of the Royal Society of London. Series A, 130, 16–28.

Kokoska, S. & Zwillinger, D. (2000). CRC Standard Probability and Statistics Tables and Formulae. CRC Press.