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 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.
The fourth histogram is a sample from a Weibull distribution with shape parameter 1. 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.
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. Leptokurtic distributions have positive kurtosis values. A leptokurtic distribution has a higher peak thin bell and taller i. An extreme positive kurtosis indicates a distribution where more of the values are located in the tails of the distribution rather than around the mean.
McLeod, S. This article conducts an investigation to determine whether the size of the standard theory tests of covariance matrices is affected by kurtosis. Nonlinear pricing kernels, kurtosis preference, and evidence from the cross section of equity returns , Dittmar, R.
The Journal of Finance , 57 1 , This paper examines the nonlinear pricing kernels that determine the risk factor and in which the definition of the pricing kernel is restricted by the preferences.
The kernels are used to generate models of nonlinear and multifactor empirical performance. Comparing measures of sample skewness and kurtosis , Joanes, D. This article compares the measures of sample skewness and kurtosis adopted by established statistical computing packages, focuses on standard sample's bias and mean-squared error while presenting various comparisons arising from simulation results for non-normal samples.
The spectral kurtosis : a useful tool for characterising non-stationary signals , Antoni, J. Mechanical Systems and Signal Processing , 20 2 , This paper attempts to fill the gap of spectral kurtosis caused by the lack of a formal definition and an understood estimation procedure.
It also precedes another paper presenting spectral kurtosis as having found successful applications in vibration-based condition monitoring. Measuring skewness and kurtosis , Groeneveld, R. The Statistician , This article provides an answer on how to measure the extent of skewness on a continuous random variable and also considers several properties to be satisfied by a measure of skewness.
New information on both skewness and kurtosis has also been added. You have a set of samples. Maybe you took 15 samples from a batch of finished product and measured those samples for density. Now you are armed with data you can analyze. And your software package has a feature that will generate the descriptive statistics for these data. You enter the data into your software package and run the descriptive statistics. You get a lot of numbers — the sample size, average, standard deviation, range, maximum, minimum and a host of other numbers.
You spy two numbers: the skewness and kurtosis. What do these two statistics tell you about your sample? This month's publication covers the skewness and kurtosis statistics. These two statistics are called "shape" statistics, i. What do the skewness and kurtosis really represent?
And do they help you understand your process any better? Are they useful statistics? You may download a pdf copy of this publication at this link. You may also download an Excel workbook containing the impact of sample size on skewness and kurtosis at the end of this publication.
You may also leave a comment at the end of the publication. Many books say that these two statistics give you insights into the shape of the distribution.
Skewness is a measure of the symmetry in a distribution. A symmetrical dataset will have a skewness equal to 0. So, a normal distribution will have a skewness of 0.
Skewness essentially measures the relative size of the two tails. Kurtosis is a measure of the combined sizes of the two tails.
It measures the amount of probability in the tails. The value is often compared to the kurtosis of the normal distribution, which is equal to 3.
If the kurtosis is greater than 3, then the dataset has heavier tails than a normal distribution more in the tails. If the kurtosis is less than 3, then the dataset has lighter tails than a normal distribution less in the tails.
Careful here. This makes the normal distribution kurtosis equal 0. Kurtosis originally was thought to measure the peakedness of a distribution. Though you will still see this as part of the definition in many places, this is a misconception. Skewness and kurtosis involve the tails of the distribution.
These are presented in more detail below. A perfectly symmetrical data set will have a skewness of 0. The normal distribution has a skewness of 0. Donald Wheeler, www. Note the exponent in the summation. This sample size formula is used here. It is also what Microsoft Excel uses. The difference between the two formula results becomes very small as the sample size increases.
Figure 1 is a symmetrical data set. It was created by generating a set of data from 65 to in steps of 5 with the number of each value as shown in Figure 1. For example, there are 3 65's, 6 70's, 9 75's, etc. A truly symmetrical data set has a skewness equal to 0. It is easy to see why this is true from the skewness formula. Look at the term in the numerator after the summation sign. Each individual X value is subtracted from the average.
Consider the value of 65 and value of The average of the data in Figure 1 is So, a truly symmetrical data set will have a skewness of 0.
Then, skewness becomes the following:. If S above is larger than S below , then skewness will be positive. This typically means that the right-hand tail will be longer than the left-hand tail. Figure 2 is an example of this. The skewness for this dataset is 0. A positive skewness indicates that the size of the right-handed tail is larger than the left-handed tail.
Figure 3 is an example of dataset with negative skewness. It is the mirror image essentially of Figure 2. The skewness is In this case, S below is larger than S above. The left-hand tail will typically be longer than the right-hand tail. Figure 3: Dataset with Negative Skewness. How to define kurtosis? This is really the reason this article was updated. For example,. You can find other definitions that include peakedness or flatness when you search the web.
The problem is these definitions are not correct. Peter Westfall published an article that addresses why kurtosis does not measure peakedness link to article. He said:. Westfall includes numerous examples of why you cannot relate the peakedness of the distribution to the kurtosis.
Donald Wheeler also discussed this in his two-part series on skewness and kurtosis. However, since the central portion of the distribution is virtually ignored by this parameter, kurtosis cannot be said to measure peakedness directly.
While there is a correlation between peakedness and kurtosis, the relationship is an indirect and imperfect one at best.
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