Skewness

Skewness in Health Care Cost and Utility Data

Skewness is a single number that says which way a distribution leans and how strongly, built from the cubed distances of each value from the mean. In health economics it is reported mostly for patient-level costs, which bunch at low values and trail off into a thin run of very expensive patients, and for quality of life scores, which bunch near full health and trail off towards poor health. A large value warns that the mean cost or mean utility that an economic evaluation needs will behave less like a normal variable than standard methods assume, and it bears on the choice of cost regression and of the distributions placed on costs in a decision model.

The standardised third moment

Skewness is defined through the third power of deviations from the mean, scaled by the cube of the standard deviation so that the result has no units. Cubing keeps the sign of each deviation and lets the most distant values dominate. Von Hippel points out that extreme values affect skewness more than they affect the mean, because they are cubed in the formula.

$$\gamma_1 = E\left[ \left( \frac{X - \mu}{\sigma} \right)^3 \right] = \frac{E[(X - \mu)^3]}{\sigma^3}$$

where $\gamma_1$ is the skewness, $X$ 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.

The NIST/SEMATECH e-Handbook of Statistical Methods gives the skewness of a normal distribution as zero. Positive values mean that the right tail is long relative to the left, and negative values mean the reverse. The handbook adds that measurements with a lower bound are often skewed right, as costs are. Doane and Seward note that an older coefficient comparing the mean with the median can be calculated from summary data alone, as when a modeller has only a published table.

Sample estimators and software conventions

Two sample versions of the moment coefficient are in common use, and they give different numbers from the same data. The NIST handbook calls the plain version the Fisher-Pearson coefficient, computed with $n$ rather than $n - 1$ in the standard deviation, and notes that many programs report an adjusted version.

$$m_r = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^r, \qquad g_1 = \frac{m_3}{m_2^{3/2}}, \qquad G_1 = \frac{\sqrt{n(n-1)}}{n-2}\, g_1$$

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, $g_1$ is the Fisher-Pearson coefficient and $G_1$ is the adjusted coefficient. NIST gives the adjustment factor as 1.49 for 5 observations, 1.19 for 10 and 1.02 for 100.

Doane and Seward report that Excel's SKEW function returns $G_1$ while Stata reports $g_1$. Microsoft's own example bears this out, since SKEW returns 0.359543 and SKEW.P 0.303193, exactly $G_1$ and $g_1$ for its ten values. A skewness reported for a few dozen patients should therefore name its estimator.

Why costs skew right and utilities skew left

Jones, reviewing econometric models for health care costs, describes individual-level costs as typically showing a spike at zero and a strongly skewed distribution with a heavy right-hand tail. He attributes this to severe patients with complications who attract costly services, and notes that a small minority of patients often accounts for a high proportion of costs. Mihaylova and colleagues add that such data are often multimodal, with a mass at zero for non-users.

In Jones's case study of 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 and the median USD 3,334, less than half the mean. The skewness statistic was 4.1, as was that of the residuals from a linear regression on the level of costs.

Quality of life data usually lean the other way. Basu and Manca describe health-related quality of life scores as typically bounded at both ends and negatively skewed, with most values at the top of the scale and a spike at one on instruments such as the EQ-5D and SF-6D. Right skew occasionally appears in groups such as terminally ill patients. The spike at full health is a ceiling effect, and Basu and Manca note that QALYs share many of these features.

Skewness, sample size and the mean cost

Economic evaluation needs the mean cost and mean effect per patient, because totals and incremental ratios are built from means, and the page on descriptive statistics shows why the median cannot stand in for the mean. What skewness changes is how closely the sample mean follows a normal distribution. For independent observations, the skewness of the sample mean falls with the square root of the sample size.

$$\gamma_1(\bar{X}) = \frac{\gamma_1}{\sqrt{n}}$$

where $\gamma_1(\bar{X})$ is the skewness of the mean of $n$ independent observations and $\gamma_1$ is the skewness of one observation. The central limit theorem is visible here, and so is its slowness. If the population skewness were 4.1, as in Jones's sample, the mean of 100 patients would have a skewness of 0.41 and the mean of 400 patients 0.205.

Mihaylova and colleagues conclude that where few awkward features such as skewness and excess zeros are present a sample in the hundreds may be large enough, but where all are present thousands may be too small. Nixon, Wonderling and Grieve found that, for incremental net benefit, bootstrapping gave standard errors no more accurate than the central limit theorem, even with highly skewed costs.

Skewness and the choice of cost model

High skewness in costs or in regression residuals is a main reason analysts move away from ordinary least squares on the cost scale. Jones notes that linear regression on the level of costs may perform poorly because of the high degree of skewness and excess kurtosis, and that a log transformation typically reduces skewness and brings the distribution closer to normal. The NIST handbook likewise suggests a log or square root transformation for data with moderate right skewness.

Transformation brings its own problems. Jones explains that exponentiating predictions from a log model does not give predictions on the cost scale, so a smearing factor is needed, and that heteroskedastic errors on the log scale make this harder. In his case study the Duan smearing factor was 2.0. Mihaylova and colleagues report that methods based on an initial transformation perform well only when the transformation suits the data. They describe the generalised linear model as accommodating skewness through variance weighting, with estimates of the mean that are usually robust to a misspecified variance function. Jones also compares the log-normal model, the generalised gamma model and a finite mixture model.

Bounded, left-skewed utilities call for other models. Basu and Manca's simulations show regression estimators based on the beta distribution estimating covariate effects more reliably than ordinary least squares, especially with a large spike at one.

Skewness in the distributions of a decision model

When a parametric distribution is placed on a cost in probabilistic sensitivity analysis, or used to draw individual costs in an individual patient simulation, its skewness follows from the family chosen. Fitting by the method of moments matches the mean and variance and so fixes the coefficient of variation, and the NIST formulas for the gamma distribution and the log-normal distribution then give different skewness.

$$\gamma_{G} = \frac{2}{\sqrt{k}} = 2c, \qquad \gamma_{LN} = \left(e^{\tau^2} + 2\right)\sqrt{e^{\tau^2} - 1} = (c^2 + 3)\,c, \qquad c = \frac{\sigma}{\mu}$$

where $\gamma_G$ and $\gamma_{LN}$ are the skewness of the gamma and log-normal distributions, $k$ is the gamma shape parameter, $\tau$ is the standard deviation of the natural logarithm of the variable, $c$ is the coefficient of variation, and $\sigma$ and $\mu$ are the standard deviation and mean of the costs represented. The second forms follow from $k = 1/c^2$ for the gamma and $e^{\tau^2} = 1 + c^2$ for the log-normal. With $c = 1$ the gamma has skewness 2 and the log-normal 4, so the log-normal puts more weight on very expensive patients, which can matter in a patient-level simulation driven by the upper tail. A distribution for uncertainty about a mean cost, with its variance taken from the standard error, is far less skewed, as the square-root rule implies.

Worked example with one very costly patient

All numbers in this example are illustrative. Ten patients in a small study have annual costs of GBP 100, 200, 200, 300, 300, 400, 500, 600, 900 and 2,500. Skewness has no units, so in units of GBP 100 the values become 1, 2, 2, 3, 3, 4, 5, 6, 9 and 25.

1. Centre. The total is 60, so the mean is 60 / 10 = 6, or GBP 600. The median is (3 + 4) / 2 = 3.5, or GBP 350.

2. Moments. The deviations from the mean are minus 5, minus 4, minus 4, minus 3, minus 3, minus 2, minus 1, 0, 3 and 19. Their squares sum to 450, so $m_2$ = 450 / 10 = 45. Their cubes sum to minus 316 for the seven patients below the mean and 27 + 6,859 = 6,886 for the two above it, a net 6,570, so $m_3$ = 6,570 / 10 = 657.

3. Sample skewness. The Fisher-Pearson coefficient divides the third moment by the second raised to the power 1.5.

$$g_1 = \frac{m_3}{m_2^{3/2}} = \frac{657}{45^{3/2}} = \frac{657}{301.87} = 2.18$$

where $g_1$ is the Fisher-Pearson coefficient, $m_3$ is the third sample central moment and $m_2$ is the second. The adjustment factor for ten patients is the square root of 90 divided by 8, or 1.186, so Excel's SKEW would return $G_1$ = 1.186 × 2.176 = 2.58.

4. One patient's share. The patient costing GBP 2,500 contributes 6,859 to the sum of cubes, more than the net total of 6,570. Without that patient the mean of the other nine is 35 / 9 = 3.89, or GBP 389, and $g_1$ falls to 0.92. The mean cost also falls by 35%, so dropping the patient would understate the budget.

5. Two fitted distributions. With $n - 1$ in the denominator the sample variance is 450 / 9 = 50, so $c^2$ = 50 / 36 = 1.389 and $c$ = 1.179. A gamma distribution fitted by moments has shape 36 / 50 = 0.72, scale 50 / 6 = 8.33 (GBP 833) and skewness 2 × 1.179 = 2.36. A log-normal distribution with the same mean and variance has skewness (1.389 + 3) × 1.179 = 5.17. The sample value of 2.18 lies nearer the gamma, but with ten patients it rests largely on one observation, so the choice of tail needs more data or external evidence.

Skewness beside kurtosis and spread

Kurtosis, the standardised fourth moment, describes how heavy the tails are rather than which way the distribution leans. Doane and Seward note that data with extreme values are generally both skewed and leptokurtic. The variance and standard deviation measure spread in the units of the data.

Doane and Seward tabulate a 90% range for $G_1$ in samples from a normal distribution, plus or minus 0.391 for 100 observations and plus or minus 0.179 for 500, with values outside it suggesting a non-normal population. They stress that this is not a general test of symmetry, and formal tests of fit belong to hypothesis testing.

Misreadings and limits

The first misreading concerns the mean and median. The rest concern how a sample value is computed and used.

1. Reading the direction from the mean and median. Von Hippel shows that the rule placing the mean beyond the median on the side of the longer tail fails surprisingly often, most commonly in discrete distributions and also in multimodal ones. For more than 30% of parameter values of the Poisson distribution, which is always positively skewed, the mean is below the median. The NIST handbook notes that multimodality may affect the sign, and cost data with a mass at zero are multimodal.
2. Trusting a small-sample value. The NIST handbook notes that skewness can be distorted by extreme values, and in a small sample one patient can set it.
3. Not stating the estimator. The same ten patients give 2.18 in Stata and 2.58 in Excel.
4. Checking raw costs instead of residuals. Jones found that residuals from a linear model inherit the shape of the cost distribution. The skewness that bears on a regression is that of its residuals on the scale of estimation.

Sources

• Basu A, Manca A. Regression estimators for generic health-related quality of life and quality-adjusted life years. Medical Decision Making. 2012;32(1):56-69. https://doi.org/10.1177/0272989X11416988
• Doane DP, Seward LE. Measuring skewness: a forgotten statistic? Journal of Statistics Education. 2011;19(2). https://doi.org/10.1080/10691898.2011.11889611
• 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
• Microsoft. SKEW function and SKEW.P function. Microsoft Support, accessed 28 October 2026. https://support.microsoft.com/en-us/office/skew-function-bdf49d86-b1ef-4804-a046-28eaea69c9fa
• 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, sections 1.3.5.11, 1.3.6.6.9 and 1.3.6.6.11. National Institute of Standards and Technology; accessed 28 October 2026. https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
• Nixon RM, Wonderling D, Grieve RD. Non-parametric methods for cost-effectiveness analysis: the central limit theorem and the bootstrap compared. Health Economics. 2010;19(3):316-333. https://doi.org/10.1002/hec.1477
• von Hippel PT. Mean, median, and skew: correcting a textbook rule. Journal of Statistics Education. 2005;13(2). https://doi.org/10.1080/10691898.2005.11910556