Kurtosis is the degree of "peakedness" of a distribution. That of the normal distribution is the benchmark. A distribution that is more peaked (i.e. whose peak is more pointed) is said to be leptokurtic. A distribution which is less peaked (i.e. whose peak is less pointed) is said to be mesokurtic. Outliers are found in the tails of the distribution. They are bound to affect the kurtosis of the distribution as they tend to exert an influence on the rest of the data.

This should do...

https://climatebysurly.files.wordpress.com/2017/10/kurtosis-student1927.gif?w=685

Original Source: Student, 1927. “Errors of Routine Analysis”, Biometrika 19 (1/2), 151-164.

high kurtosis,= outliers; heavy tails.

Kurtosis is the degree of "peakedness" of a distribution. That of the normal distribution is the benchmark. A distribution that is more peaked (i.e. whose peak is more pointed) is said to be leptokurtic. A distribution which is less peaked (i.e. whose peak is less pointed) is said to be mesokurtic. Outliers are found in the tails of the distribution. They are bound to affect the kurtosis of the distribution as they tend to exert an influence on the rest of the data.

There are some graphs in the Wikipedia article.

https://en.wikipedia.org/wiki/Kurtosis, though Juris' answer is more fun.

Dear Srikanth

I think (like Timothy) the best educational resource in this field is wikipedia which has some nice pictures to better understanding kurtosis notion.

In addition presence of outliers in your data lead to reduction in kurtosis and usually make it negative.

Well done @ Ette Etuk

Kurtosis is the index reflecting the absence or abundance of typical mid-range scores on a measure. Many scores in the mid-range (the absence of outliers, thus) will result in a frequency distribution curve with a very high peak in the middle (leptokurtic). Many scores in the very high and very low ranges (outliers, thus) will result in a frequency distribution curve with almost no real peak in the middle (platikurtic).

An outlier is normally defined as an observation that seem to be inconsistent with the rest of the data. If a point is considered as an outlier then it may be relevant not to include it in the calculation of descriptors the kurtosis. For many models it may be perfectly reasoanble to a high value of the kurtosis without declaring any of the observations to be outliers. I high value of the kurtosis may just be an indications that a Gaussian model is not a good model..

No matter the sample size, there is always the possibility from a single set of values to get an unusual result. The outlier is not necessarily an error, but simply an outcome of sampling.

Kurtosis is used to see whether data is normal or not. It helps to identify whether a curve is normal or abnormally shaped. It is a measure that describes the shape of a distribution's tails in relation to its overall shape.

Kurtosis is used as a measure of the tailedness of the probability distribution. Means that high kurtosis give description outliers heavy tails, which we use the normal distribution to provide the tailedness of the distribution.

In non technical term, kurtosis is prepared to show whether the observations made are normal based on the shape of the curve

In general, kurtosis tells you nothing about the "peak" of a distribution, and also tells you nothing about its "shoulders." It measures outliers (tails) only.

To understand this concept, you first need to understand what is a "Z-score." A Z-score is given by

Z = (Data value - Mean of data values)/(Standard Deviation of data values)

For a normal distribution, only 0.3% of absolute Z-scores are more than 3.0. For an outlier-prone (heavy tailed) distribution, this percentage is typically higher, like 2.0%. In other words, with a heavy tailed (outlier-prone) distribution, you will have data values far from the mean (in terms of number of standard deviations) more often than the normal distribution predicts.

A Z-score of 5.0 tells you that the data value is five standard deviations above the mean. A Z-score of -0.2 tells you that the data value is 0.2 standard deviations below the mean.

Now, the logic for why the kurtosis statistic measures outliers (rare, extreme observations in the case of data; potential rare, extreme observations in the case of a pdf), rather than the peak, is actually quite simple. Kurtosis is the average (or expected value in the case of the pdf) of the Z-scores, each taken to the 4th power. In the case where there are (potential) outliers, there will be some extremely large Z^4 values (such as 5^4 = 625), which, when averaged with all the other Z^4 values, gives you a high kurtosis. If there are less outliers than, say, predicted by a normal pdf, then the most extreme Z^4 values will not be particularly large, giving smaller kurtosis.

Near the peak, the data values are close to the mean, and hence the absolute Z-scores are typically less than 1.0 (such as 0.2). For these data values, the Z^4 values are *extremely* small (such as 0.2^4 = 0.0016), and contribute very little to their overall average (which again, is the kurtosis). That is why kurtosis tells you virtually nothing about the shape of the peak. I give mathematical bounds on the contribution of the data near the peak to the kurtosis measure in the following article:

Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195.

Dear Prof. Peter H Westfall

you are really great.

you have demystified the insights of Kurtosis in very lucid manner.

to the best of my knowledge no book on statistics has this type of explaination for kurtosis value.

you are an erudite Statistician having enormous knowledge and experience in the subject.

I once again thank you for your valuable response.

Hi Srikanth I would like to comment that:

kur·to·sis /kərˈtōsəs/ noun Statistics (noun) : kurtosis is

"The sharpness of the peak of a frequency-distribution curve".

"The peakedness or flatness of the graph of a frequency distribution especially with respect to the concentration of values near the mean as compared with the normal distribution"

"Kurtosis is defined as the measure of thickness or heaviness of the given distribution for the random variable along its tail. In other words, it can be defined as the measure of “tailedness” of the distribution. Hence, it is clear that it is considered as a common measure of shape. The outliers in the given data have more effect on this measure. Moreover, it does not have any unit".

Based on the value of kurtosis, the distribution can be classified into three categories.

• The distribution with kurtosis equal to 3 is known as mesokurtic. A random variable which follows normal distribution has kurtosis 3.

• If the kurtosis is less than three, the distribution is called as platykurtic. Here, the distribution has shorter and thinner tails than normal distribution. Moreover, the peak is lower and also broader when compared to normal distribution.

• If the kurtosis is greater than three, the distribution is called as leptykurtic. Here, the distribution has longer and fatter tails than normal distribution. Moreover, the peak is higher and also sharper when compared to normal distribution.

Distributions of data and probability distributions are not all the same shape. Some are asymmetric and skewed to the left or to the right. Other distributions are bimodal and have two peaks. Another feature to consider when talking about a distribution is the shape of the tails of the distribution on the far left and the far right. Kurtosis is the measure of the thickness or heaviness of the tails of a distribution. The kurtosis of a distributions is in one of three categories of classification:

1. Mesokurtic

2. Leptokurtic

3. Platykurtic

We will consider each of these classifications in turn. Our examination of these categories will not be as precise as we could be if we used the technical mathematical definition of kurtosis.

Mesokurtic: Kurtosis is typically measured with respect to the normal distribution. A distribution that has tails shaped in roughly the same way as any normal distribution, not just the standard normal distribution, is said to be mesokurtic. The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. Besides normal distributions, binomial distributions for which p is close to 1/2 are considered to be mesokurtic.

Leptokurtic: A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution. Leptokurtic distributions are sometimes identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy. Leptokurtic distributions are named by the prefix "lepto" meaning "skinny." There are many examples of leptokurtic distributions. One of the most well known leptokurtic distributions is Student's t distribution.

Platykurtic: The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have slender tails. Many times they possess a peak lower than a mesokurtic distribution. The name of these types of distributions come from the meaning of the prefix "platy" meaning "broad." All uniform distributions are platykurtic. In addition to this, the discrete probability distribution from a single flip of a coin is platykurtic.

Calculation of Kurtosis: These classifications of kurtosis are still somewhat subjective and qualitative. While we might be able to see that a distribution has thicker tails than a normal distribution, what if we don’t have the graph of a normal distribution to compare with? What if we want to say that one distribution is more leptokurtic than another?. To answer these kinds of questions we need not just a qualitative description of kurtosis, but a quantitative measure. The formula used is μ4/σ4 where μ4 is Pearson’s fourth moment about the mean and sigma is the standard deviation.

Excess Kurtosis: Now that we have a way to calculate kurtosis, we can compare the values obtained rather than shapes. The normal distribution is found to have a kurtosis of three. This now becomes our basis for mesokurtic distributions. A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic. Since we treat a mesokurtic distribution as a baseline for our other distributions, we can subtract three from our standard calculation for kurtosis. The formula μ4/σ4 - 3 is the formula for excess kurtosis. We could then classify a distribution from its excess kurtosis: 1.Mesokurtic distributions have excess kurtosis of zero.

2.Platykurtic distributions have negative excess kurtosis.

3.Leptokurtic distributions have positive excess kurtosis.

A Note on the Name: The word "kurtosis" seems odd on the first or second reading. It actually makes sense, but we need to know Greek to recognize this. Kurtosis is derived from a transliteration of the Greek word kurtos. This Greek word has the meaning "arched" or "bulging," making it an apt description of the concept known as kurtosis.

I hope it Helps Regards

The trouble is Kurtosis does not have a standard formula let alone interpretation. Various persons above are assuming fundamentally different basic definitions of Kurtosis. I personally subscribe to Ette's definition, the 4th moment of a distribution. Because Kurtosis involves the 4th power, outliers only indirectly impact on Kurtosis. Skewness perhaps more directly relates to outliers.

Kurtosis is a measure of the degree of "peakedness" of a curve or of a distribution. A curve or a distribution that has usual (average) peaked is termed as platikurtic. A curve or a distribution that is more peaked is termed as leptokurtic while a curve or a distribution which is less peaked is termed as mesokurtic. The kurtosis of the normal curve or of the normal distribution has been accepted as usual (or standard) and accordingly it has been regarded as platikurtic curve or platikurtic distribution. Thus, kurtosis can be interpreted as the degree of vdeviation from normality.

Kurtosis is not "peakedness" at all. You can have an infinitely peaked curve with negative (excess) kurtosis and you can have a perfectly flat peak with infinite kurtosis. It has been proven mathematically that kurtosis measures tails, while there is no mathematical connection of kurtosis to the "peak." See here for a clear explanation: https://en.wikipedia.org/wiki/Talk:Kurtosis#Why_kurtosis_should_not_be_interpreted_as_%22peakedness%22

Peter H Westfall has mentioned a point on the definition of kurtosis. This point seems to be a new concept/idea. There is necessity of analysis of this point. This can lead to a new technical definition of kurtosis.

Yes, kurtosis should be defined as "tailedness" rather than "peakedness," as given in the current Wikipedia page: https://en.wikipedia.org/wiki/Kurtosis

The "peakedness" definition is officially dead, as discussed in the following article:

Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195 (2014).

The new definition of kurtosis, mentioned by Peter H Westfall, as measure of "tailedness" rather than as measure of "peakedness," seems to be the latest one. It is no doubt is to be given importance.