Measures of Shape: Skewness and Kurtosis
Copyright © 2008–2023 by Stan Brown, BrownMath.com
Copyright © 2008–2023 by Stan Brown, BrownMath.com
Why do we care? One application is testing for normality: many statistics inferences require that a distribution be normal or nearly normal. A normal distribution has skewness and excess kurtosis of 0, so if your distribution is close to those values then it is probably close to normal.
The first thing you usually notice about a distribution’s shape is whether it has one mode (peak) or more than one. If it’s unimodal (has just one peak), like most data sets, the next thing you notice is whether it’s symmetric or skewed to one side. If the bulk of the data is at the left and the right tail is longer, we say that the distribution is skewed right or positively skewed; if the peak is toward the right and the left tail is longer, we say that the distribution is skewed left or negatively skewed.
Look at the two graphs below. They both have μ = 0.6923 and σ = 0.1685, but their shapes are different.
The beta distribution is one of the many skewed distributions that are used in mathematical modeling.
1.3846 − Beta(α=4.5, β=2)
The first one is moderately skewed left: the left tail is longer and most of the distribution is at the right. By contrast, the second distribution is moderately skewed right: its right tail is longer and most of the distribution is at the left.
You can get a general impression of skewness by drawing a histogram (MATH200A part 1), but there are also some common numerical measures of skewness. Some authors favor one, some favor another. This Web page presents one of them. In fact, these are the same formulas that Excel uses in its “Descriptive Statistics” tool in Analysis Toolpak, and in the SKEW( ) function.
You may remember that the mean and standard deviation have the same units as the original data, and the variance has the square of those units. However, the skewness has no units: it’s a pure number, like a z-score.
The moment coefficient of skewness of a data set is
skewness: g1 = m3 / m23/2
m3 = ∑(x−x̅)3 / n and m2 = ∑(x−x̅)2 / n
x̅ is the mean and n is the sample size, as usual. m3 is called the third moment of the data set. m2 is the variance, the square of the standard deviation.
The skewness can also be computed as g1 = the average value of z3, where z is the familiar z-score, z = (x−x̅)/σ. Of course the average value of z is always zero, but what about the average of z3? Suppose you have a few points far to the left of the mean, and a lot of points less far to the right of the mean. Since cubing the deviations gives the big ones even greater weight, you’ll have negative skewness. It works just the opposite if you have big deviations to the right of the mean.
You’ll remember that you have to compute the variance and standard deviation slightly differently, depending on whether you have data for the whole population or just a sample. The same is true of skewness. If you have the whole population, then g1 above is the measure of skewness. But if you have just a sample, you need the sample skewness:
(2) sample skewness:
(The formula comes from Joanes and Gill 1998 [full citation in “References”, below].)
Excel doesn’t concern itself with whether you have a sample or a population: its measure of skewness is always G1, the sample skewness.
Here are grouped data for heights of 100 randomly selected male students, adapted from Spiegel and Stephens (1999, 68) [full citation in “References”, below].
A histogram shows that the data are skewed left, not symmetric.
But how highly skewed are they, compared to other data sets? To answer this question, you have to compute the skewness.
Begin with the sample size and sample mean. (The sample size was given, but it never hurts to check.)
n = 5+18+42+27+8 = 100
x̅ = (61×5 + 64×18 + 67×42 + 70×27 + 73×8) ÷ 100
x̅ = 9305 + 1152 + 2814 + 1890 + 584) ÷ 100
x̅ = 6745÷100 = 67.45
Now, with the mean in hand, you can compute the skewness. (Of course in real life you’d probably use Excel or a statistics package, but it’s good to know where the numbers come from.)
|Class Mark, x||Frequency, f||xf||(x−x̅)||(x−x̅)²f||(x−x̅)³f|
|x̅, m2, m3||67.45||n/a||8.5275||−2.6933|
Finally, the skewness is
g1 = m3 / m23/2 = −2.6933 / 8.52753/2 = −0.1082
But wait, there’s more! That would be the skewness if you had data for the whole population. But obviously there are more than 100 male students in the world, or even in almost any school, so what you have here is a sample, not the population. You must compute the sample skewness:
= [√100×99 / 98] [−2.6933 / 8.52753/2] = −0.1098
If skewness is positive, the data are positively skewed or skewed right, meaning that the right tail of the distribution is longer than the left. If skewness is negative, the data are negatively skewed or skewed left, meaning that the left tail is longer.
If skewness = 0, the data are perfectly symmetrical. But a skewness of exactly zero is quite unlikely for real-world data, so how can you interpret the skewness number? There’s no one agreed interpretation, but for what it’s worth Bulmer (1979) [full citation at https://BrownMath.com/swt/sources.htm#so_Bulmer1979] — a classic — suggests this rule of thumb:
With a skewness of −0.1098, the sample data for student heights are approximately symmetric.
Caution: This is an interpretation of the data you actually have. When you have data for the whole population, that’s fine. But when you have a sample, the sample skewness doesn’t necessarily apply to the whole population. In that case the question is, from the sample skewness, can you conclude anything about the population skewness? To answer that question, see the next section.
Your data set is just one sample drawn from a population. Maybe, from ordinary sample variability, your sample is skewed even though the population is symmetric. But if the sample is skewed too much for random chance to be the explanation, then you can conclude that there is skewness in the population.
But what do I mean by “too much for random chance to be the explanation”? To answer that, you need to divide the sample skewness G1 by the standard error of skewness (SES) to get the test statistic, which measures how many standard errors separate the sample skewness from zero:
(3) test statistic: Zg1 = G1/SES where
This formula is adapted from page 85 of Cramer (1997) [full citation in “References”, below]. (Some authors suggest √6/n, but for small samples that’s a poor approximation. And anyway, we’ve all got calculators, so you may as well do it right.)
The critical value of Zg1 is approximately 2. (This is a two-tailed test of skewness ≠ 0 at roughly the 0.05 significance level.)
Don’t mix up the meanings of this test statistic and the amount of skewness. The amount of skewness tells you how highly skewed your sample is: the bigger the number, the bigger the skew. The test statistic tells you whether the whole population is probably skewed, but not by how much: the bigger the number, the higher the probability.
GraphPad suggests a confidence interval for skewness:
(4) 95% confidence interval of population skewness = G1 ± 2 SES
I’m not so sure about that. Joanes and Gill [full citation in “References”, below] point out that sample skewness is an unbiased estimator of population skewness for normal distributions, but not others. So I would say, compute that confidence interval, but take it with several grains of salt — and the further the sample skewness is from zero, the more skeptical you should be.
For the college men’s heights, recall that the sample skewness was G1 = −0.1098. The sample size was n = 100 and therefore the standard error of skewness is
SES = √(600×99) / (98×101×103) = 0.2414
The test statistic is
Zg1 = G1/SES = −0.1098 / 0.2414 = −0.45
This is between −2 and +2 (see above), so from this sample it’s impossible to say whether the population is symmetric or skewed. Since the sample skewness is small, a confidence interval is probably reasonable:
G1 ± 2 SES = −0.1098 ± 2×0.2414 = −0.1098±0.4828 = −0.5926 to +0.3730.
You can give a 95% confidence interval of skewness as about −0.59 to +0.37, more or less.
The other common measure of shape is called the kurtosis. As skewness involves the third moment of the distribution, kurtosis involves the fourth moment. The outliers in a sample, therefore, have even more effect on the kurtosis than they do on the skewness and in a symmetric distribution both tails increase the kurtosis, unlike skewness where they offset each other.
You may remember that the mean and standard deviation have the same units as the original data, and the variance has the square of those units. However, the kurtosis, like skewness, has no units: it’s a pure number, like a z-score.
Traditionally, kurtosis has been explained in terms of the central peak. You’ll see statements like this one: Higher values indicate a higher, sharper peak; lower values indicate a lower, less distinct peak. Balanda and MacGillivray (1988) [full citation in “References”, below] also mention the tails: increasing kurtosis is associated with the “movement of probability mass from the shoulders of a distribution into its center and tails.”
However, Peter Westfall (2014 [full citation in “References”, below]) has been on a bit of a crusade to change this perception, and I think he makes a good case. We might say, following Wikipedia’s article on kurtosis (accessed 15 May 2016), that “higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations.” In other words, it’s the tails that mostly account for kurtosis, not the central peak.
The reference standard is a normal distribution, which has a kurtosis of 3. In token of this, often the excess kurtosis is presented: excess kurtosis is simply kurtosis−3. For example, the “kurtosis” reported by Excel is actually the excess kurtosis.
Note that word “often” in describing changes in the central peak due to changes in the tails. Westfall 2014 [full citation in “References”, below] gives several illustrations of counterexamples.
Kurtosis is unfortunately harder to picture than skewness, but these illustrations, suggested by Wikipedia, should help. All three of these distributions have mean of 0, standard deviation of 1, and skewness of 0, and all are plotted on the same horizontal and vertical scale. Look at the progression from left to right, as kurtosis increases.
The normal distribution will probably be the subject of roughly the second half of your course; the logistic distribution is another one used in mathematical modeling. Don’t worry at this stage about what these distributions mean; they’re just handy examples that illustrate what I want to illustrate.
Moving from the illustrated uniform distribution to a normal distribution, you see that the “shoulders” have transferred some of their mass to the center and the tails. In other words, the intermediate values have become less likely and the central and extreme values have become more likely. The kurtosis increases while the standard deviation stays the same, because more of the variation is due to extreme values.
Moving from the normal distribution to the illustrated logistic distribution, the trend continues. There is even less in the shoulders and even more in the tails, and the central peak is higher and narrower.
How far can this go? What are the smallest and largest possible values of kurtosis? The smallest possible kurtosis is 1 (excess kurtosis −2), and the largest is ∞, as shown here:
Discrete: equally likely values
Student’s t (df=4)
A discrete distribution with two equally likely outcomes, such as winning or losing on the flip of a coin, has the lowest possible kurtosis. It has no central peak and no real tails, and you could say that it’s “all shoulder” — it’s as platykurtic as a distribution can be. At the other extreme, Student’s t distribution with four degrees of freedom has infinite kurtosis. A distribution can’t be any more leptokurtic than this.
You might want to look at Westfall’s (2014 [full citation in “References”, below]) Figure 2 for three quite different distributions with identical kurtosis.
The moment coefficient of kurtosis of a data set is computed almost the same way as the coefficient of skewness: just change the exponent 3 to 4 in the formulas:
kurtosis: a4 = m4 / m22 and excess kurtosis: g2 = a4−3
m4 = ∑(x−x̅)4 / n and m2 = ∑(x−x̅)2 / n
Again, the excess kurtosis is generally used because the excess kurtosis of a normal distribution is 0. x̅ is the mean and n is the sample size, as usual. m4 is called the fourth moment of the data set. m2 is the variance, the square of the standard deviation.
The kurtosis can also be computed as a4 = the average value of z4, where z is the familiar z-score, z = (x−x̅)/σ. Of course the average value of z is always zero, but the average value of z4 is always ≥ 1, and is larger when you have a few big deviations on either side of the mean than when you have a lot of small ones.
Just as with variance, standard deviation, and skewness, the above is the final computation of kurtosis if you have data for the whole population. But if you have data for only a sample, you have to compute the sample excess kurtosis using this formula, which comes from Joanes and Gill [full citation in “References”, below]:
(6) sample excess kurtosis:
Excel doesn’t concern itself with whether you have a sample or a population: its measure of kurtosis in the KURT( ) function is always G2, the sample excess kurtosis.
Example: Let’s continue with the example of the college men’s heights, and compute the kurtosis of the data set. n = 100, x̅ = 67.45 inches, and the variance m2 = 8.5275 in² were computed earlier.
|Class Mark, x||Frequency, f||x−x̅||(x−x̅)4f|
Finally, the kurtosis is
a4 = m4 / m2² = 199.3760/8.5275² = 2.7418
and the excess kurtosis is
g2 = 2.7418−3 = −0.2582
But this is a sample, not the population, so you have to compute the sample excess kurtosis:
G2 = [99/(98×97)] [101×(−0.2582)+6)] = −0.2091
This sample is slightly platykurtic: its peak is just a bit shallower than the peak of a normal distribution.
Your data set is just one sample drawn from a population. How far must the excess kurtosis be from 0, before you can say that the population also has nonzero excess kurtosis?
The question is similar to the question about skewness, and the answers are similar too. You divide the sample excess kurtosis by the standard error of kurtosis (SEK) to get the test statistic, which tells you how many standard errors the sample excess kurtosis is from zero:
(7) test statistic: Zg2 = G2 / SEK where
The formula is adapted from page 89 of Cramer (1979) [full citation in “References”, below]. (Some authors suggest √24/n, but for small samples that’s a poor approximation. And anyway, we’ve all got calculators, so you may as well do it right.)
The critical value of Zg2 is approximately 2. (This is a two-tailed test of excess kurtosis ≠ 0 at approximately the 0.05 significance level.)
For the sample college men’s heights (n=100), you found excess kurtosis of G2 = −0.2091. The sample is platykurtic, but is this enough to let you say that the whole population is platykurtic (has lower kurtosis than the bell curve)?
First compute the standard error of kurtosis:
SEK = 2 × SES × √(n²−1) / ((n−3)(n+5))
n = 100, and the SES was previously computed as 0.2414.
SEK = 2 × 0.2414 × √(100²−1) / (97×105) = 0.4784
The test statistic is
Zg2 = G2/SEK = −0.2091 / 0.4784 = −0.44
You can’t say whether the kurtosis of the population is the same as or different from the kurtosis of a normal distribution.
There are many ways to assess normality, and unfortunately none of them are without problems.
One test is the D'Agostino-Pearson omnibus test (D’Agostino and Stephens [full citation in “References”, below], 390–391; for an online source see Öztuna, Elhan, Tüccar [full citation in “References”, below]). I’ve implemented the D’Agostino-Pearson test in an Excel workbook at Normality Check and Finding Outliers in Excel. It’s called an omnibus test because it uses the test statistics for both skewness and kurtosis to come up with a single p-value assessing whether this data set’s shape is too different from normal. The test statistic is
(8) DP = Zg1² + Zg2² follows χ² with df=2
You can look up the p-value in a table, or use
χ²cdf on a TI-83 or TI-84.
χ²cdf(2, 5.991464546) = 0.95, so if the test statistic is bigger than about 6 you would reject the hypothesis of normality at the 0.05 level.
Caution: The D’Agostino-Pearson test has a tendency to err on the side of rejecting normality, particularly with small sample sizes. David Moriarty, in his StatCat utility, recommends that you don’t use D’Agostino-Pearson for sample sizes below 20.
For college students’ heights you had test statistics Zg1 = −0.45 for skewness and Zg2 = 0.44 for kurtosis. The omnibus test statistic is
DP = Zg1² + Zg2² = 0.45² + 0.44² = 0.3961
and the p-value for χ²(df=2) > 0.3961, from a table or a statistics calculator, is 0.8203. You cannot reject the assumption of normality. (Remember, you never accept the null hypothesis, so you can’t say from this test that the distribution is normal.) The histogram suggests normality, and this test gives you no reason to reject that impression.
There’s no One Right Way to test for normality. One of many alternatives to the D’Agostino-Pearson test is making a normal probability plot; the accompanying workbook does this. (See Technology near the top of this page.)
TI calculator owners can use Normality Check on TI-83/84 or Normality Check on TI-89.
See also: The University of Surrey has a good survey of problems with normality tests, at How do I test the normality of a variable’s distribution? That page recommends using the test statistics Zg1 and Zg2 individually.
For a second illustration of inferences about skewness and kurtosis of a population, I’ll use an example from Bulmer [full citation at https://BrownMath.com/swt/sources.htm#so_Bulmer1979]:
|Frequency distribution of litter size in rats, n=815|
I’ll spare you the detailed calculations, but you should be able to verify them by following equation (1) and equation (2):
n = 815, x̅ = 6.1252, m2 = 5.1721, m3 = 2.0316
skewness g1 = 0.1727 and sample skewness G1 = 0.1730
The sample is roughly symmetric but slightly skewed right, which looks about right from the histogram. The standard error of skewness is
SES = √(6×815×814) / (813×816×818) = 0.0856
Dividing the skewness by the SES, you get the test statistic
Zg1 = 0.1730 / 0.0856 = 2.02
Since this is greater than 2, you can say that there is some positive skewness in the population. Again, “some positive skewness” just means a figure greater than zero; it doesn’t tell us anything more about the magnitude of the skewness.
If you go on to compute a 95% confidence interval of skewness from equation (4), you get 0.1730±2×0.0856 = 0.00 to 0.34.
What about the kurtosis? You should be able to follow equation (5) and compute a fourth moment of m4 = 67.3948. You already have m2 = 5.1721, and therefore
kurtosis a4 = m4 / m2² = 67.3948 / 5.1721² = 2.5194
excess kurtosis g2 = 2.5194−3 = −0.4806
sample excess kurtosis G2 = [814/(813×812)] [816×(−0.4806+6) = −0.4762
So the sample is moderately less peaked than a normal distribution. Again, this matches the histogram, where you can see the higher “shoulders”.
What if anything can you say about the population? For this you need equation (7). Begin by computing the standard error of kurtosis, using n = 815 and the previously computed SES of 0.0.0856:
SEK = 2 × SES × √(n²−1) / ((n−3)(n+5))
SEK = 2 × 0.0856 × √(815²−1) / (812×820) = 0.1711
Zg2 = G2/SEK = −0.4762 / 0.1711 = −2.78
Since Zg2 is comfortably below −2, you can say that the distribution of all litter sizes is platykurtic, less sharply peaked than the normal distribution. But be careful: you know that it is platykurtic, but you don’t know by how much.
You already know the population is not normal, but let’s apply the D’Agostino-Pearson test anyway:
DP = 2.02² + 2.78² = 11.8088
p-value = P( χ²(2) > 11.8088 ) = 0.0027
The test agrees with the separate tests of skewness and kurtosis: sizes of rat litters, for the entire population of rats, are not normally distributed.