Symmetry, Skewness and Kurtosis | Real Statistics Using Excel
Request PDF on ResearchGate | The relationships between skewness and kurtosis | Theoretical considerations of kurtosis, whether of partial orderings of. In my book Simulating Data with SAS, I discuss a relationship between the skewness and kurtosis of probability distributions that might not be. Support 5S and LEAN Goals with Custom KPI Boards In part two I will look at the use of skewness and kurtosis statistics . While there is a correlation between peakedness and kurtosis, the relationship is an indirect and.
Are they useful statistics? 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.
The relationship between skewness and kurtosis - The DO Loop
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. 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. 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.
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 Sabove is larger than Sbelow, 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. Dataset with Positive Skewness 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, Sbelow is larger than Sabove. The left-hand tail will typically be longer than the right-hand tail.
Dataset with Negative Skewness So, when is the skewness too much? The rule of thumb seems to be: If the skewness is between This is really the reason this article was updated. The problem is these definitions are not correct.What are Skewness and Kurtosis? (Read info below for more intuition)
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. It provides an alternative way to characterize the dispersion of the probability model: The skewness and kurtosis are collectively known as the shape parameters for the probability model. The skewness parameter for the probability model is defined to be the third standardized central moment. This means that we begin with the standardized form for the random variable: In a similar manner, the kurtosis parameter for the probability model is defined as the fourth standardized central moment: At this point it should be abundantly clear why you never computed the skewness and kurtosis parameters in your stat class.
Moreover, since you do not routinely evaluate integrals, it is fairly safe to say that you have probably not computed any parameters since you finished or dropped out of your stat class. However, because these parameters characterize various aspects of a probability model, they are useful in organizing the zoo of probability models. 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.
Problems with Skewness and Kurtosis, Part One | Quality Digest
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.
Symmetry, Skewness and Kurtosis
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.