It is an unfortunate historical error that caused people to think that negative (excess) kurtosis implies a "flat-topped" distribution. Pearson stated it in 1905, Fisher repeated the statement in his classic text, and everyone else repeated the error without even thinking about it. After all, who wanted to disagree with Pearson and Fisher?

But Pearson and Fisher were wrong. You can have any shape of peak whatsoever when the kurtosis is negative (or positive) - infinitely pointy, flat, bimodal, trimodal, sharply peaked, reverse peaked, or anything else. Examples and three theorems are given in my paper Westfall, PH. "Kurtosis as Peakedness, 1905 - 2014. R.I.P.". Am Stat. 68: 191–195. doi:10.1080/00031305.2014.917055. PMC 4321753 , PMID 25678714.

Rather, as correctly indicated by Adriana Santos-Caballero, kurtosis measures the tail (potential outlier) character of the distribution. With negative excess kurtosis the outlier character (as measured by large |Z|-values) of the distribution is less extreme than that of a normal distribution.

Dear Srikanth,

A negative kurtosis means that your distribution is flatter than a normal curve with the same mean and standard deviation.

The easiest way to visualise this is to plot a histogram with a fitted normal curve.

A negative kurtosis implies platykurtosis. For the normal distribution the moment measure is equal to 3.

This means your distribution is platykurtic or flatter as compared with normal distribution with the same M and SD. The curve would have very light tails. It is not unusual. This does not necessarily mean that your distribution differs significantly from a normal distribution. You could run a test to see if it differs significantly.

Measures of Skewness and Kurtosis

Skewness and Kurtosis:

A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis.

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. A uniform distribution would be the extreme case.

The histogram is an effective graphical technique for showing both the skewness and kurtosis of data set.

Definition of Skewness For univariate data Y1, Y2, ..., YN, the formula for skewness is:

g1=∑Ni=1(Yi−Y¯)3/Ns3

where Y¯ is the mean, s is the standard deviation, and N is the number of data points. Note that in computing the skewness, the s is computed with N in the denominator rather than N - 1.

The above formula for skewness is referred to as the Fisher-Pearson coefficient of skewness. Many software programs actually compute the adjusted Fisher-Pearson coefficient of skewness

G1=N(N−1)−−−−−−−−√N−2∑Ni=1(Yi−Y¯)3/Ns3

This is an adjustment for sample size. The adjustment approaches 1 as N gets large. For reference, the adjustment factor is 1.49 for N = 5, 1.19 for N = 10, 1.08 for N = 20, 1.05 for N = 30, and 1.02 for N = 100.

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. For example, the Galton skewness (also known as Bowley's skewness) is defined as

Galton skewness=Q1+Q3−2Q2Q3−Q1

where Q1 is the lower quartile, Q3 is the upper quartile, and Q2 is the median.

The Pearson 2 skewness coefficient is defined as

Sk2=3(Y¯−Y~)s

where Y~ is the sample median.

There are many other definitions for skewness that will not be discussed here.

Definition of Kurtosis For univariate data Y1, Y2, ..., YN, the formula for kurtosis is:

kurtosis=∑Ni=1(Yi−Y¯)4/Ns4

where Y¯ is the mean, s is the standard deviation, and N is the number of data points. Note that in computing the kurtosis, the standard deviation is computed using N in the denominator rather than N - 1.

Alternative Definition of Kurtosis The kurtosis for a standard normal distribution is three. For this reason, some sources use the following definition of kurtosis (often referred to as "excess kurtosis"):

kurtosis=∑Ni=1(Yi−Y¯)4/Ns4−3

This definition is used so that the standard normal distribution has a kurtosis of zero. 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.

Examples The following example shows histograms for 10,000 random numbers generated from a normal, a double exponential, a Cauchy, and a Weibull distribution.

plots showing histograms for 10,000 random numbers generated from

a normal, double exponential, Cauchy, and 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 0.03. The kurtosis of 2.96 is near the expected value of 3. The histogram verifies the symmetry.

Double Exponential Distribution 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 0.06 and the kurtosis is 5.9.

Cauchy Distribution 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 -10 and 10. The full data set for the Cauchy data in fact has a minimum of approximately -29,000 and a maximum of approximately 89,000.

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 69.99 and the kurtosis is 6,693. These extremely high values can be explained by the heavy tails. Just as the mean and standard deviation can be distorted by extreme values in the tails, so too can the skewness and kurtosis measures.

Weibull Distribution The fourth histogram is a sample from a Weibull distribution with shape parameter 1.5. The Weibull distribution is a skewed distribution with the amount of skewness depending on the value of the shape parameter. The degree of decay as we move away from the center also depends on the value of the shape parameter. For this data set, the skewness is 1.08 and the kurtosis is 4.46, which indicates moderate skewness and kurtosis.

Dealing with Skewness and Kurtosis Many classical statistical tests and intervals depend on normality assumptions. Significant skewness and kurtosis clearly indicate that data are not normal. If a data set exhibits significant skewness or kurtosis (as indicated by a histogram or the numerical measures), what can we do about it?

One approach is to apply some type of transformation to try to make the data normal, or more nearly normal. The Box-Cox transformation is a useful technique for trying to normalize a data set. In particular, taking the log or square root of a data set is often useful for data that exhibit moderate right skewness.

Another approach is to use techniques based on distributions other than the normal. For example, in reliability studies, the exponential, Weibull, and lognormal distributions are typically used as a basis for modeling rather than using the normal distribution. The probability plot correlation coefficient plot and the probability plot are useful tools for determining a good distributional model for the data.

Skew, or skewness, can be mathematically defined as the averaged cubed deviation from the mean divided by the standard deviation cubed. If the result of the computation is greater than zero, the distribution is positively skewed. If it's less than zero, it's negatively skewed and equal to zero means it's symmetric.

flatten distribution, not normal!

in my opinion, it refers to the major part of data trends to be magnitude under the mean or the median. So, mass (or density) of this probabilities will aggregated mainly under mean or median of data.

It is an unfortunate historical error that caused people to think that negative (excess) kurtosis implies a "flat-topped" distribution. Pearson stated it in 1905, Fisher repeated the statement in his classic text, and everyone else repeated the error without even thinking about it. After all, who wanted to disagree with Pearson and Fisher?

But Pearson and Fisher were wrong. You can have any shape of peak whatsoever when the kurtosis is negative (or positive) - infinitely pointy, flat, bimodal, trimodal, sharply peaked, reverse peaked, or anything else. Examples and three theorems are given in my paper Westfall, PH. "Kurtosis as Peakedness, 1905 - 2014. R.I.P.". Am Stat. 68: 191–195. doi:10.1080/00031305.2014.917055. PMC 4321753 , PMID 25678714.

Rather, as correctly indicated by Adriana Santos-Caballero, kurtosis measures the tail (potential outlier) character of the distribution. With negative excess kurtosis the outlier character (as measured by large |Z|-values) of the distribution is less extreme than that of a normal distribution.

thank you prof. Peter H Westfall for your most valuable answer to my question.

You demystified the interpretation of kurtosis.

based on your answer, I further got one more doubt.

does positive excess kurtosis mean outlier character of distribution more extrem than that of a normal distribution

thanks in advance

Yes.

thank you professor for your prompt response

Peter H Westfall About the name kurtosis. As you made it clear in your paper that "kurtosis" is not a measure of "peakedness" but "tailedness". Since the word "kurtosis" itself is a Greek-based word meaning "peakedness", isn't it also time to find another word, maybe also Greek-based, to mean "tailedness"? Otherwise, the statement "Kurtosis is a measure of tailedness" is like "Peakedness is a measure of tailedness" which is a self-contradictory statement.

Yes, Taiping, it is time we retire the word "kurtosis." A good substitute would be "heavy-tailedness." Also, we should change the name "kurtosis" in Pearson's "kurtosis" measure to "standardized fourth central moment." Pearson is to blame for this whole mess, stating that his statistic measured "peakedness" and "flatness." Fisher also has fault for repeating Pearson's mis-characterizations through all 14 editions of his classic text. Myriad other researchers and writers simply repeated these errors. Who could argue with Pearson and Fisher, who are giants and founders of modern statistics? But of course, everyone makes mistakes.