The skewness and kurtosis statistics appear to be very dependent on the sample size. The table above shows the variation. In fact, even several hundred data points didn't give very good estimates of the true kurtosis and skewness. Smaller sample sizes can give results that are very misleading. Dr Wheeler wrote in his book mentioned above:. Shewhart made this observation in his first book. The statistics for skewness and kurtosis simply do not provide any useful information beyond that already given by the measures of location and dispersion.
So, don't put much emphasis on skewness and kurtosis values you may see. And remember, the more data you have, the better you can describe the shape of the distribution. But, in general, it appears there is little reason to pay much attention to skewness and kurtosis statistics. Just look at the histogram. It often gives you all the information you need.
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We hope you find it informative and useful. Happy charting and may the data always support your position. Below is the e-mail Dr. Westfall sent concerning the describing kurtosis as a measure of peakedness. It is printed with his permission. It did lead to the re-writing of the article to remove the peakedness defintion of kurtosis. Thank you for making your information publically available.
I often point students to the internet for supplemental information, and some of your is valuable. Thus, if you see a large kurtosis statistic, you know you have a quality control problem that warrants further investigation. The average is 2. Subtract 3 if you want excess kurtosis.
Now, replace the last data value with so it becomes an outlier: 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, The average is Clearly, only the outlier s matter. Nothing about the "peak" or the data near the middle matters. Further, it is clear that kurtosis has very positive implications for spc in its detection of outliers. Here is a paper that elaborates: Westfall, P. Kurtosis as Peakedness, — The American Statistician, 68, — May I suggest that you either modify or remove your description of kurtosis.
It does a disservice to consumers and users of statistics, and ultimately harms your own business because it presents information that is completely off the mark as factual. Excellent way of explaining, and nice article to get information on the topic of my presentation topic, which i am going to deliver in institution of higher education.
I have many samples, let us say , with say 50 cases within each sample. I compute for each sample the skewness and kurtosis based on the 50 observations. In the scatter plot of the sample skewness and sample kurtosis data points I observe a curved cloud of data points between the skewness and kurtosis. When I used simulated data sets with 50 simulated measurements generated according to an exponential distribution I again found the curved shaped cloud of scatterpoints.
Theoretically, however, the skewness is equal to 2 and the kurtosis equal to 6. Can youn elaborate about this? My e-mail address is A very informative and insightful article. But one small typo, I think. When defining the figure 3 in the associated description it was mentioned that "Figure 3 is an example of dataset with negative skewness.
The right-hand tail will typically be longer than the left-hand tail. Please correct me if I am wrong. Thanks Pavan. You are correct. I fixed the typo. Shouldn't kurtosis for normal distribution be 3? And skewness is Please see the equation for a4 above. It will give 3 for a normal distribution. But many software packages including Excel use the formula below that which subtracts 3 - and it gives 0 for a normal distribution.
Please, I need your help. I'm doing a project work on skewness and kurtosis and its applications. Could you please help me with some of the areas of applications of skewness and kurtosis and also the scope and delimitations undergone during the study. Hello Anita,. I am not sure what you are asking. You can find applications by searching the internet. For example, they are used by some stock traders to help determine when to sell or buy stocks.
Please e-mail at [email protected] if you need more. Questions: What does the little i mean in the variable Xi 2. Impressive: I thought the overall article was well-written and had good examples.
Needs Improvement: It would be helpful to have simpler problems as a basis of each example and skew and kurtosis topic. Thanks for the comment. The little i is simply denotes the ith result. Your discription of figure 4 and 5 seem backward. Wouldn't that be heavy tailed? Likewise for figure 5, the tail region is short relative to the central region i. Heavy or light as to do with the tails. The uniform distribuiton in Figure 4 has no tails. It is "light" in tails.
The other has long tails - so it is heavy in tails. Maybe broad or tight would be better descriptors as heavy and light imply high and low frequency at least in my mind. I would agree with those descriptors. From figure 8, the kurtosis sees to somewhat converge to its 'true' value as the data points are increased. However, in my empirical tests, the kurtosis is simply increasing in the number of data points, going beyond the 'true' kurtosis as well.
What could be the reason for this? I dont find it intuitive. As it increases, the kurtosis will approach that of the normal distribution, 0 or 3 depending on what equation you use. How are you doing your empirical testing? Thanks for letting me know.
Thanks for revising the information about kurtosis. There are still a couple of small issues that should be addressed, though. The graph showing "high kurtosis" is misleading in the way that it presents "heavy tails". The graph actually looks similar to a. For a better example, consider simulating data from a T 5 distribution and drawing the histogram.
There, the positive kurtosis more correctly appears as the presence of occasional outliers. The "heavy tailedness" of kurtosis is actually hard to see in a histogram, because, despite the fact that the tails are heavy, they are still close to 0 and hence difficult to see.
A better way to demonstrate the tailedness of high kurtosis is to use a normal q-q plot, which makes the heavy tails very easy to see. The argument that the kurtosis is not a good estimate of the "population" or "process" parameters is true, but not a compelling argument against using the statistic for quality control or SPC. A high kurtosis alerts you to the presence of outlier s , commonly known as out-of-control conditions, possibily indicating special causes of variation at work.
Of course, such cases should be followed up by a plot of some sort, but just the fact that the kurtosis indicates such a condition tells you that it is indeed useful and applicable for SPC.
There is no need for the "population" framework here, as Deming would agree, considering that this is an analytic not enumerative study. So the argument that kurtosis is not useful for SPC is overstated at best, and not supportable at worst. The skewness for a normal distribution is zero, and any symmetric data should have a skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right.
By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. If the data are multi-modal, then this may affect the sign of the skewness. Some measurements have a lower bound and are skewed right. For example, in reliability studies, failure times cannot be negative. It should be noted that there are alternative definitions of skewness in the literature.
There are many other definitions for skewness that will not be discussed here. Note that in computing the kurtosis, the standard deviation is computed using N in the denominator rather than N - 1. The kurtosis for a standard normal distribution is three.
In addition, with the second definition positive kurtosis indicates a "heavy-tailed" distribution and negative kurtosis indicates a "light tailed" distribution. 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. 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. 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? There are many ways to assess normality, and unfortunately none of them are without problems. The test statistic is. The omnibus test statistic is.
You cannot reject the assumption of normality. The histogram suggests normality, and this test gives you no reason to reject that impression. See Technology near the top of this page. The sample is roughly symmetric but slightly skewed right, which looks about right from the histogram. The standard error of skewness is.
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