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 :. 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. Since the sample skewness is small, a confidence interval is probably reasonable:. 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. Traditionally, kurtosis has been explained in terms of the central peak. The reference standard is a normal distribution, which has a kurtosis of 3.
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. 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?
A discrete distribution with two equally likely outcomes, such as winning or losing on the flip of a coin, has the lowest possible 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:. Again, the excess kurtosis is generally used because the excess kurtosis of a normal distribution is 0. 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 this is a sample, not the population, so you have to compute the sample excess kurtosis:. This sample is slightly platykurtic : its peak is just a bit shallower than the peak of a normal distribution. 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:. The critical value of Z g2 is approximately 2. If there is a high kurtosis, then, we need to investigate why do we have so many outliers. It indicates a lot of things, maybe wrong data entry or other things. Low kurtosis in a data set is an indicator that data has light tails or lack of outliers.
If we get low kurtosis too good to be true , then also we need to investigate and trim the dataset of unwanted results.
Mesokurtic : This distribution has kurtosis statistic similar to that of the normal distribution. It means that the extreme values of the distribution are similar to that of a normal distribution characteristic. This definition is used so that the standard normal distribution has a kurtosis of three. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. The peak is lower and broader than Mesokurtic, which means that data are light-tailed or lack of outliers.
The reason for this is because the extreme values are less than that of the normal distribution. If you want to get an Introduction to Machine Learning , click here. Thanks for reading! If you liked this article, you can read my other articles here. I write everything related to Python Programming. Bursts of code to power through your day. Web Development articles, tutorials, and news. Sign in. Which definition of kurtosis is used is a matter of convention this handbook uses the original definition.
When using software to compute the sample kurtosis, you need to be aware of which convention is being followed. Many sources use the term kurtosis when they are actually computing "excess kurtosis", so it may not always be clear. The following example shows histograms for 10, random numbers generated from a normal, a double exponential, a Cauchy, and a Weibull distribution.
Normal Distribution. The first histogram is a sample from a normal distribution. The normal distribution is a symmetric distribution with well-behaved tails. This is indicated by the skewness of 0. The kurtosis of 2. The histogram verifies the symmetry. The second histogram is a sample from a double exponential distribution. The double exponential is a symmetric distribution.
Compared to the normal, it has a stronger peak, more rapid decay, and heavier tails. That is, we would expect a skewness near zero and a kurtosis higher than 3. The skewness is 0. The third histogram is a sample from a Cauchy distribution. For better visual comparison with the other data sets, we restricted the histogram of the Cauchy distribution to values between and The full data set for the Cauchy data in fact has a minimum of approximately , and a maximum of approximately 89, The Cauchy distribution is a symmetric distribution with heavy tails and a single peak at the center of the distribution.
Since it is symmetric, we would expect a skewness near zero. Due to the heavier tails, we might expect the kurtosis to be larger than for a normal distribution. In fact the skewness is
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