Difficulty in calculating skewness

Question

I'm trying to quantify the skewness of the distribution of random integer variable, generated in the interval from 1 to 15, with a function that I wrote in C++.

Here are the generated values:

Tested for 5000 elements, with results:

level 1: 2561  level 6: 70   level 11: 4
level 2: 1225  level 7: 44   level 12: 1
level 3: 607   level 8: 17   level 13: 0
level 4: 312   level 9: 9    level 14: 0
level 5: 147   level 10: 3   level 15: 0

From what I observe I can qualify that the distribution has positive skewness, as most of the generated elements (97%) are within the interval 1 to 5.

To quantify the skewness, I'm trying to calculate the Pearson's moment coefficient of skewness using this relation:

where X - random variable, mu - mean, sigma - standard deviation and E - expectation operator.

I understand that I have to subtract the mean from each random variable, divide it by the standard deviation and raise to the 3-rd power, however, I'm having difficulties understanding what is the meaning of the E operator?

Does it mean that I need to simply divide by the total number of values or something else?

Edit:

Is there an easier way to quantify skewness?

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P.S. apologies for the lengthy post I just wanted to show research effort.

Answer 1

You should understand the difference between the parameters and properties of a distribution and the estimators for these parameters and properties. For instance

• The true mean, $\mu = E[X]$ is the the expected value of a stochastic variable $X$ and can not be calculated exactly.
• The sample mean, $m = \sum{x_i/n}$ with $x_i$ your observations of $X$ is the usual estimator for $\mu$.

Chapters in text books and whole scientific articles discuss the quality of estimators. For variance

• The true variance is $\sigma^2 = E[(X-\mu)^2]$
• The sample variance actually is $\frac{1}{n} \sum{(x_i - m)^2}$, but it has a tendency to be smaller than $\sigma^2$, therefore it is said to be biassed. This is related to the fact that $m$ itself is estimated from the same sample.
• The usual estimator, $s^2 = \frac{1}{n-1} \sum{(x_i - m)^2}$, does not have this disadvantage.

For skewness

• The true skewness of the stochastic variable is $\gamma_1 = E[(\frac{X-\mu}{\sigma})^3]$

• The sample skewness is $\frac{1}{n} \sum(\frac{x_i-m}{s})^3$, but again, it is biassed.

• The usual estimator is $\frac{n}{(n-1)(n-2)} \sum(\frac{x_i-m}{s})^3$, in which $s$ is of course the square root of the estimator for the variance.

Have fun implementing this. For further discussion, you might consult

• Wikipedia's article on Bias of an estimator
• this discussion of SAS on descriptive statistics or
• this Cross Validate question on skewness and kurtosis.