# Skewness and kurtosis relationship problems

### Problems with Skewness and Kurtosis, Part One | Quality Digest

In part two I will look at the use of skewness and kurtosis statistics . is a correlation between peakedness and kurtosis, the relationship is an. JB=n/6(SK2+KU2/4) where sk is skewness and ku is kurtosis NO, there is no relationship between skew and kurtosis. Similar questions and discussions. June , Volume , Issue 3, pp – | Cite as. The Relationship between Skewness and Kurtosis of A Diffusing Scalar. Authors; Authors and affiliations.

To illustrate how the skewness and kurtosis parameters characterize the shape of a probability model, we shall use a simple probability model for which the integrals above will be easy to illustrate and evaluate. This probability model is the standardized right triangular distribution. It has a probability density function f x of: This probability model has a mean of zero, a standard deviation of 1. The Standardized Right Triangular Distribution Since this is a standardized distribution, the standardized form for the random variable reduces down to simply [x].

Thus, the formulas for the skewness and kurtosis parameters reduce to the following: Thus, we see that in this case, the skewness is the integral of the product of the cubic curve and the density function, while the kurtosis is the integral of the product between the quartic curve and the density function. Figure 2 shows the density function along with the cubic and quartic curves. Figures 3 and 4 show the resulting product curves. Cubic and Quartic Curves with f x Figure 3: The Areas That Define the Skewness Parameter Interpreting the integral as the area between the product curve and the X axis, we find that the skewness parameter for this probability model may be interpreted as: Figure 4 shows the curve that results when we multiply the probability model by the quartic curve.

The kurtosis parameter for this probability model may be interpreted as the area under the curve in figure 4. The Areas That Define the Kurtosis Parameter The fact that all four regions in figures 3 and 4 pinch down near zero suggests that the central region of the probability model contributes very little to either of these two parameters.

Since the distribution in this example is already in its standardized form, the units on the horizontal axis in figures 3 and 4 represent the standardized distance from the mean. Thus, the contribution of the central portion of the probability model can be seen by considering how much of the total area under the curves corresponds to X values which fall between —1. While the central portion of this probability model contributes 63 percent of the total area, only 11 percent of the combined areas in figure 3, and only 5 percent of the area in figure 4, correspond to the central portion of the probability model.

Therefore, we must conclude that both skewness and kurtosis are primarily concerned with characteristics of the tails of the probability model. Skewness and Kurtosis Characterize the Tails of a Probability Model The skewness parameter measures the relative sizes of the two tails. Distributions that have tails of equal weight will have a skewness parameter of zero. If the right-hand tail is more massive, then the skewness parameter will be positive.

If the left-hand tail is more massive, the skewness parameter will be negative. Moreover, the greater the difference between the two tails, the greater the magnitude of the skewness parameter. The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution. As the tails of a distribution become heavier, the kurtosis will increase.

As the tails become lighter, the kurtosis will decrease. As defined here kurtosis cannot be less than 1. Probability models with kurtosis values between 1. Probability models with kurtosis values in excess of 3. Kurtosis was originally thought to measure the "peakedness" of a distribution. 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. Thus, the shape parameters of skewness and kurtosis actually tell us more about the tails of a probability model than they do about the central portion of that model.

At the beginning of the 20th century the shape parameters were used simply because Karl Pearson had developed seven families of probability models that were fully characterized by the first four moments.

By plotting the values of the shape parameters on Cartesian coordinates, Pearson was able to show how these families of probability models were related to each other. This plot is known as the shape characterization plane.

In this plane a probability model is represented by a single point, while families of probability models will sometimes fall on a line or fall within in a region of the plane. For example, all normal distributions will have a skewness of zero and a kurtosis of 3. In the shape characterization plane, the skewness squared defines the X-coordinate, while the kurtosis defines the Y-coordinate. Thus, the family of all normal distributions will be shown on the shape characterization plane by a single point at 0, 3.

The gamma distributions are represented by the line defined by the normal and exponential distributions. All of the chi-square distributions fall on this line.

The beta distributions occupy the whole region of the plane below the gamma distribution line. The shape characterization plane can be divided as shown into regions according to whether the probability models are mound-shaped, J-shaped, or bimodal.

At the apex of the dividing lines between these three divisions, we find the family of uniform distributions, which are neither mound-shaped, J-shaped, nor bimodal. Above this line we find the family of Burr distributions effectively covering the rest of the region of mound-shaped probability models. Thus, skewness and kurtosis parameters are useful because of their ability to characterize and organize the zoo of probability models. Moreover, as seen in figures 6 and 7, the families of the betas and Burrs, plus their limiting families of the gammas and the Weibulls will effectively cover the whole shape characterization plane.

Does this mean that these are the only probability models? If the skewness is between This is really the reason this article was updated. The problem is these definitions are not correct. Peter Westfall published an article that addresses why kurtosis does not measure peakedness link to article.

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.

Wheeler defines kurtosis as: It measures the tail-heaviness of the distribution. Kurtosis is defined as: If you use the above equation, the kurtosis for a normal distribution is 3. Most software packages including Microsoft Excel use the formula below.

This formula does two things. It takes into account the sample size and it subtracts 3 from the kurtosis. With this equation, the kurtosis of a normal distribution is 0. This is really the excess kurtosis, but most software packages refer to it as simply kurtosis. The last equation is used here. So, if a dataset has a positive kurtosis, it has more in the tails than the normal distribution.

If a dataset has a negative kurtosis, it has less in the tails than the normal distribution. Since the exponent in the above is 4, the term in the summation will always be positive — regardless of whether Xi is above or below the average.

Xi values close to the average contribute very little to the kurtosis. The tail values of Xi contribute much more to the kurtosis. Look back at Figures 2 and 3. They are essentially mirror images of each other.

### Statistics LESSON 5 MEASURES OF SKEWNESS AND KURTOSIS

The skewness of these datasets is different: But the kurtosis is the same. Both have a kurtosis of This is because kurtosis looks at the combined size of the tails. The kurtosis decreases as the tails become lighter.

It increases as the tails become heavier. Figure 4 shows an extreme case.

In this dataset, each value occurs 10 times. The values are 65 to in increments of 5. The kurtosis of this dataset is It has as much data in each tail as it does in the peak. Note that this is a symmetrical distribution, so the skewness is zero. Negative Kurtosis Example Figure 5 is shows a dataset with more weight in the tails. The kurtosis of this dataset is 1. Positive Kurtosis Example Most often, kurtosis is measured against the normal distribution. If the kurtosis is close to 0, then a normal distribution is often assumed.

### skewness - Relationship between skew and kurtosis in a sample - Cross Validated

These are called mesokurtic distributions. If the kurtosis is less than zero, then the distribution is light tails and is called a platykurtic distribution.

#01 Statistics -MOMENT ABOUT MEAN OF STATISTIC IN HINDI, SKEWNESS & KURTOSIS

If the kurtosis is greater than zero, then the distribution has heavier tails and is called a leptokurtic distribution. The problem with both skewness and kurtosis is the impact of sample size.

## Mathematics

This is described below. Our Population Are the skewness and kurtosis any value to you? You take a sample from your process and look at the calculated values for the skewness and kurtosis. What can you tell from these two results? To explore this, a data set of points was randomly generated. The goal was to have a mean of and a standard deviation of The random generation resulted in a data set with a mean of The histogram for these data is shown in Figure 6 and looks fairly bell-shaped.

Population Histogram The skewness of the data is 0. The kurtosis is 0. Both values are close to 0 as you would expect for a normal distribution. These two numbers represent the "true" value for the skewness and kurtosis since they were calculated from all the data.

In real life, you don't know the real skewness and kurtosis because you have to sample the process. This is where the problem begins for skewness and kurtosis. Sample size has a big impact on the results.