Kurtosis

Kurtosis in Health Care Cost Data

Kurtosis summarises how prone a distribution is to rare values lying far from the mean, on a scale where the normal distribution scores 3. In health economics it is met mostly when patient-level costs and resource use are described, because a few very expensive patients can stretch the right tail far beyond anything a normal curve allows. A high value warns that methods built on the normal distribution may perform poorly and that the choice of cost model deserves care.

The fourth standardised moment

Kurtosis is defined through the fourth power of standardised deviations from the mean. The fourth power lets values several standard deviations away dominate the average, which is why the measure responds to the tails. Westfall traces the definition to Karl Pearson in 1905, who also coined the terms leptokurtic, mesokurtic and platykurtic.

$$\kappa = E\left[ \left( \frac{X - \mu}{\sigma} \right)^4 \right] = \frac{E[(X - \mu)^4]}{\sigma^4}$$

where $\kappa$ is the kurtosis, $X$ is the random variable (for example one patient's annual cost), $\mu$ is its mean, $\sigma$ is its standard deviation and $E$ denotes the expected value. Westfall notes that kurtosis is never below 1 and equals 3 for any normal distribution.

Many sources subtract 3 so that the normal distribution scores zero, and call the result excess kurtosis.

$$\gamma_2 = \kappa - 3$$

where $\gamma_2$ is the excess kurtosis. Positive values indicate heavier tails than the normal and negative values lighter tails. The NIST/SEMATECH e-Handbook of Statistical Methods describes kurtosis as a measure of whether data are heavy-tailed or light-tailed relative to a normal distribution, and warns that many sources use the word kurtosis when they are computing excess kurtosis.

Reference values for common distributions

Values for familiar distributions give a scale for judging data. The figures come from Westfall's table and, for the gamma distribution, from the NIST handbook, which gives a kurtosis of $\kappa = 3 + 6/k$ for shape $k$, so a small shape parameter implies very heavy tails.

Tail weight in the t distribution rises as degrees of freedom fall. Some distributions, such as the Cauchy, have no finite kurtosis, yet any sample from them still gives a finite value.

Tails, not peaks

Westfall shows that the only unambiguous reading of kurtosis concerns tail extremity, meaning existing outliers for a sample or the propensity to produce them for a distribution. He gives distributions with identical kurtosis but flat, pointed or twin peaks. For a normal distribution, the part of the kurtosis determined by values within one standard deviation of the mean is 0.037 of the total. In a sample of 1,000 Cauchy draws, values within one standard deviation contributed 0.0073 to a sample kurtosis of 437.39. The centre of a cost distribution therefore says almost nothing about its kurtosis, and the largest few patients say almost everything.

Estimating kurtosis from a sample

The sample version replaces expectations with averages over the observed patients. The NIST handbook computes it with $N$ rather than $N - 1$ in the denominator of the standard deviation.

$$m_r = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^r, \qquad b_2 = \frac{m_4}{m_2^2}, \qquad g_2 = b_2 - 3$$

where $x_i$ is the value for patient $i$, $\bar{x}$ is the sample mean, $n$ is the number of patients, $m_r$ is the $r$th sample central moment, $b_2$ is the sample kurtosis on the scale on which the normal scores 3 and $g_2$ is the sample excess kurtosis. Excel reports an adjusted excess kurtosis instead.

$$G_2 = \frac{n - 1}{(n - 2)(n - 3)} \left[ (n + 1) g_2 + 6 \right]$$

where $G_2$ is the adjusted sample excess kurtosis and the other symbols are as above. The adjustment matters in small samples and fades as $n$ grows. That Excel's KURT function returns $G_2$ can be confirmed by reproducing the example in Microsoft's documentation, where ten values give minus 0.1518. The Stata Base Reference Manual defines the coefficient of kurtosis as the fourth moment divided by the square of the second and states that a normal distribution has a coefficient of 3.

Why health care costs have heavy tails

Jones, reviewing econometric models for health care costs, describes individual-level costs as typically having a spike at zero and a strongly skewed distribution with a heavy right-hand tail, with a small minority of patients often accounting for a high proportion of costs. Mihaylova and colleagues, reviewing methods for trial cost data, note that right tails heavier than those of the normal distribution are often observed.

Jones's case study uses 2,955 people aged 65 and over with positive annual medical expenditure in the US Medical Expenditure Panel Survey. Mean expenditure was USD 7,290, the median was USD 3,334 and the maximum was USD 125,610. The skewness statistic was 4.1 and the kurtosis 25.6, against 3 for normal data. Residuals from a linear regression on the level of costs inherited that shape, with a kurtosis of 26.4. At this size a handful of patients accounts for much of the variability, and whether they appear in a sample moves the estimated mean cost.

Kurtosis and the choice of cost model

Jones notes that linear regression on the level of costs may perform poorly because of the high degree of skewness and excess kurtosis, since ordinary least squares can be sensitive to extreme observations. A log transformation makes the distribution more symmetric and shrinks its range, although predictions then have to be retransformed to the cost scale.

Whether the residuals on the log scale are heavy-tailed, which their kurtosis shows, bears on the choice between ordinary least squares on log costs and a generalised linear model with a log link. Summarising Manning and Mullahy's simulations, Mihaylova and colleagues report that the GLM with a log link suffers substantial efficiency losses when the log-scale error variance is large or the log-scale error is symmetric but heavy-tailed, cases in which least squares on log costs is more efficient. They add, citing later work by Manning and colleagues, that the generalised gamma model, which includes the log-normal model as a special case, performs well on heavy-tailed data even when the log-scale error is asymmetric. No single estimator was best in every condition Manning and Mullahy examined, so the kurtosis of the residuals informs the choice without settling it.

Mean costs, interval estimates and decision models

Economic evaluation needs the mean cost per patient, because totals and incremental cost-effectiveness ratios are built from means, so heavy tails cannot be avoided by switching to another summary. Mihaylova and colleagues report that methods based on the normal distribution, such as the t test and least squares, are sensitive to extreme values and likely to be inefficient in small to medium samples when the underlying distribution is not normal. Inference from the central limit theorem and from bootstrapping is asymptotically valid but has uncertain properties in smaller non-normal samples, and they prefer simple methods in large samples where the near-normality of sample means is assured.

Kurtosis also matters when a parametric distribution is fitted to patient-level data, for example to draw individual costs in an individual patient simulation. The method of moments matches the mean and variance only, and the kurtosis then follows from the chosen family, so a gamma distribution and a log-normal distribution with the same two moments imply different tails. Comparing the implied kurtosis with the data, alongside plots and goodness of fit tests, helps judge whether the fitted tail is plausible.

Worked example with ten patients' annual costs

All numbers in this example are illustrative. A small study records annual costs for ten patients of GBP 200, 300, 300, 400, 400, 500, 500, 600, 800 and 4,000. Kurtosis does not depend on units, so working in units of GBP 100 keeps the fourth powers manageable and the values become 2, 3, 3, 4, 4, 5, 5, 6, 8 and 40.

1. Mean and deviations. The total is 80, so the mean is 80 / 10 = 8, or GBP 800. The deviations from the mean are minus 6, minus 5, minus 5, minus 4, minus 4, minus 3, minus 3, minus 2, 0 and 32.

2. Second and fourth moments. The squared deviations sum to 36 + 25 + 25 + 16 + 16 + 9 + 9 + 4 + 0 + 1,024 = 1,164, so $m_2$ = 1,164 / 10 = 116.4. The fourth powers sum to 1,296 + 625 + 625 + 256 + 256 + 81 + 81 + 16 + 0 + 1,048,576 = 1,051,812, so $m_4$ = 1,051,812 / 10 = 105,181.2.

3. Sample kurtosis. The ratio of the fourth moment to the squared second moment gives the kurtosis.

$$b_2 = \frac{m_4}{m_2^2} = \frac{105181.2}{116.4^2} = \frac{105181.2}{13548.96} = 7.76$$

where $b_2$ is the sample kurtosis, $m_4$ is the fourth sample central moment and $m_2$ is the second. The excess kurtosis is $g_2$ = 7.76 minus 3 = 4.76. For the adjusted value that Excel's KURT would return, the factor is (10 minus 1) / ((10 minus 2) × (10 minus 3)) = 9 / 56, and 9 / 56 × (11 × 4.763 + 6) = 9.38. The same data therefore give 7.76, 4.76 or 9.38 depending on the convention, and a report should say which was used.

4. Where the kurtosis comes from. The standard deviation with $n$ in the denominator is the square root of 116.4, about 10.79. Nine patients lie within one standard deviation of the mean, and the fourth powers of their deviations sum to 3,236, the total less the 1,048,576 from the patient costing GBP 4,000. That is 3,236 / 1,051,812 = 0.0031 of the total, so the one patient supplies the remaining 99.7%.

5. Removing the extreme patient. Without the GBP 4,000 patient, the mean of the other nine is 40 / 9 = 4.44, or GBP 444, their second and fourth moments are 2.914 and 23.37, and their sample kurtosis is $b_2 = 23.37 / 2.914^2 = 2.75$, close to the normal value. The mean cost also falls by 44%, so dropping the patient would make the data look well behaved while understating the budget.

6. A gamma distribution fitted by moments. With $n - 1$ in the denominator, the sample variance is 1,164 / 9 = 129.33. The method of moments, as set out in the NIST handbook, gives a shape of $k = 8^2 / 129.33 = 0.495$ and a scale of $\theta = 129.33 / 8 = 16.17$, or GBP 1,617. The implied kurtosis is 3 + 6 / 0.495 = 15.1, about twice the sample value of 7.76. With ten patients the sample kurtosis rests on one observation, so choosing the tail needs larger data or external evidence.

7. The mean of many patients. The sum of independent gamma variables with a common scale is itself gamma, with the shapes added, so the mean of $n$ patients drawn from this fitted distribution has excess kurtosis $\gamma_2 = 6/(nk)$. For ten patients that is 6 / (10 × 0.495) = 1.21, and for 100 patients 6 / (100 × 0.495) = 0.12. A normal-theory interval for the mean cost of ten such patients rests on a sampling distribution that is still far from normal.

Kurtosis beside skewness and spread

Skewness, the standardised third moment, measures asymmetry, while kurtosis measures tail weight, and in cost data both are often high because the long tail sits on one side. The variance and standard deviation measure spread in the units of the data, whereas kurtosis is unit-free and says how that spread is produced. Among descriptive statistics, Stata's detailed summary reports kurtosis next to the four largest values, which is where a high kurtosis should send the analyst. A kurtosis value on its own is not a test, and formal checks of fit to the normal distribution belong to hypothesis testing.

Misreadings and limits

Most errors come from reading kurtosis as something it is not. Others come from forgetting how unstable the sample value is.

1. Reading kurtosis as peakedness. Westfall concludes that kurtosis tells very little about the peak and should never be defined in those terms, yet Microsoft's KURT documentation and the Stata manual both still describe it in terms of peakedness.
2. Not stating the scale. A reported kurtosis of 3 is normal on one convention and heavy-tailed on the other.
3. Trusting a small-sample value. The NIST handbook notes that skewness and kurtosis, like the mean and standard deviation, can be distorted by extreme values in the tails, and in a small sample one patient can set the value, as in the worked example.
4. Trimming patients to reduce kurtosis. Removing high-cost patients changes the quantity estimated, since the remaining patients' mean no longer reflects the cost of treating everyone.
5. Checking raw costs instead of residuals. For regression, the relevant diagnostic is the kurtosis of the residuals on the scale of estimation.

Sources

• Jones AM. Models for health care. HEDG Working Paper 10/01. York: University of York; January 2010. https://www.york.ac.uk/media/economics/documents/herc/wp/10_01.pdf
• Manning WG, Mullahy J. Estimating log models: to transform or not to transform? Journal of Health Economics. 2001;20(4):461-494. https://doi.org/10.1016/S0167-6296(01)00086-8
• Microsoft. KURT function. Microsoft Support, accessed 26 October 2026. https://support.microsoft.com/en-us/office/kurt-function-bc3a265c-5da4-4dcb-b7fd-c237789095ab
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• NIST/SEMATECH. e-Handbook of Statistical Methods, section 1.3.5.11, Measures of skewness and kurtosis, and section 1.3.6.6.11, Gamma distribution. National Institute of Standards and Technology; accessed 26 October 2026. https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm and https://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm
• StataCorp. Stata Base Reference Manual, summarize. College Station, TX: Stata Press; 2025. https://www.stata.com/manuals/rsummarize.pdf
• Westfall PH. Kurtosis as peakedness, 1905-2014. R.I.P. The American Statistician. 2014;68(3):191-195. https://doi.org/10.1080/00031305.2014.917055