  Skewness

Skewness is a measure of the degree of asymmetry of a probability density function (probability distribution). The distribution has positive skewness (skewed to the right) if the higher tail is longer and negative skewness (skewed to the left) if the lower tail is longer. The usual estimator of skewness is (see): where Sx is the standard deviation for the sample.

The parameter Skewness will be positive for a density function skewed to the right, and negative for a density function skewed to the left. In principle we could find the skewness of the histogram for a sample, and determine what could be the skewness of the density function for the population. The problem is that even symmetric distributions could give a value for skewness small but different from zero (positive or negative).

I made two experiments using first a completely symmetric probability function (normal distribution) , and using an skewed probability function( binomial with p=0.9)to study the distribution of the skewness in both cases.

SYMMETRIC FUNCTION

Using the generator RANDNORM() for mean value µ=0, and standard deviation σ=1, I created samples with 100 individuals and I determined the skewness using the formula (1). I repeated the experiment 500 times. The results are on the following table:

 X Interval Frequency [-0.8,-0.6] 8 [-0.6,-0.4] 21 [-0.4,-0.2] 77 [-0.2,-0.0] 151 [-0.0,0.2] 158 [0.2,0.4] 59 [0.4,0.6] 22 [0.6,0.8] 4

A graph with these results can be seen in the following fig. The mean value for the Skewness  is 0.0138 (should be ZERO for the population !), and the standard deviation for this sample is Sx= 0.249. If we consider 90% interval, then we can assume that a symmetric distribution could have about 90% probability to get a value for the parameter skewness in the interval [-0.4, +0.4]. That will be the criteria I will use with my students. We will consider a skewed probability function if the absolute value of the skewness parameter is greater than 0.4.

SKEWED FUNCTION

Using the generator RANDBIN(10,0.9,500) for n=10, and probability p=0.9, I created samples with 100 individuals and I determined the skewness using the formula (1). I repeated the experiment 500 times. The results are on the following table:

 Skewness Interval Frequency [-2.2,-2.0] 1 [-2.0,-1.8] 0 [-1.8,-1.6] 2 [-1.6,-1.4] 5 [-1.4,-1.2] 26 [-1.2,-1.0] 60 [-1.0,-0.8] 137 [-0.8,-0.6] 177 [-0.6,-0.4] 77 [-0.4,-0.2] 15

A graph with these results can be seen in the following fig. The mean value for the Skewness  is -0.814 (for the population should be -0.843 !), and the standard deviation  σ= 0.260.

 PROGRAM TO CALCULATE SKEWNESS (FOR TI83-TI84)  How to write the Skewness program? 1-) Click on the button PRGM. Scroll to the right and select NEW. ENTER 2-) Using the ALPHA button write the name SKEW. ENTER 3-) Click on 2nd CATALOG (number 0) and scroll down and find dim(. ENTER 4-) Using 2nd LIST select L1. Close parenthesis. Click STO. Click Alpha N. ENTER 5-) From CATALOG select the function mean(. ENTER 6-) Using 2nd LIST select L1. Close parenthesis. Click STO. Click Alpha M.ENTER 7-) From CATALOG select the function stdDev(. ENTER 8-) Using 2nd LIST select L1. Close parenthesis. Click STO. Click Alpha S. ENTER 9-) Click 0 STO ALPHA T 10-) Click on the button PRGM and select the instruction FOR(. ENTER 11-) Type Alpha I. Type Comma(,). Type number 1. Type Comma(,). Type Alpha N. Close parenthesis. ENTER. (You should see this: For(I,1,N)) 12-) Now, type this in the same order you see it!: ( N / ( ( N – 1 ) * ( N – 2 ) ) * ( L1(I) – M ) ^ 3 / S ^ 3 + T -> T (The instruction ^ is the button located under the button CLEAR,  and the arrow -> you get using the button STO->) 13-) Click PRGM and select the number 7, END. Click ENTER Now, click PRGM and scroll to the right until I/O. Scroll down until “Disp”. Click ENTER 14-) Using the ALPHA key, type : "SKEW", T You should see:  Disp "SKEW", T That's it!....test your program. Type the values: 1,2,2,3,3,3,4,4,4,4,5 in list L1. Run SKEW program. Click Prgm. Select SKEW. Click ENTER, and ENTER again. You should see the result : -0.422465427 Done

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