Skewness, Kurtosis and Symmetric Distributions

The first thing is why do we need to study Symmetricity, Skewness and Kurtosis? The simplest answer is because Skewness and Kurtosis help us understand the shape of the Probability Density Functions.

Symmetric Distribution:

The formal definition of symmetric distribution:
$$f(x_{o} – \delta) = f(x_{o} – \delta)$$,
where$$x_{o}$$ is the point of symmetricity, $$\delta \: \in \: R$$ and f(x) is the PDF if distribution is continuous and Prob. Mass Function if dist. is discrete.

Gaussian Distribution is also symmetric distribution. But note, not all symmetric distributions are Gaussian distribution
More Details: https://en.wikipedia.org/wiki/Skewness

Skewness

Skewness is a measure of the asymmetry of the probability dist. of a real-valued random variable about its mean. It can be positive, negative or undefined (e.g. in case of Dirac delta function).

Sample skewness, b is defined as :
$$\frac{\frac{1}{n}\sum_1^n(x_{i} – \overline{x})^{3}}{[\frac{1}{n-1}\sum_1^n(x_{i} – \overline{x})^{2}]^{\frac{3}{2}}}$$,
where $$\overline{x}$$ is the sample mean.

We can easily visualize the shape of the PDF just by knowing whether it is positive or negative skewed.

Mean, Median and Mode in skewed data:

1. For negatively skewed data:
Mean < Median < Mode
2. For positively skewed Distribution:
Mode < Median < Mean
3. For symmetric Distribution:
i. One peak (unimodal)
Mean = Median = Mode
i
i. More than one peak (multimodal)
Mean = Median $$\neq$$ Mode

Kurtosis

It is the measure of “TAILNESS” (NOT PEAKEDNESS) of the PDF. It is a descriptor of the shape of the PDF. However, Skewness only described out the side of the tail i.e. left or right.

Excess Kurtosis = Kurtosis – 3

Sample Excess Kurtosis, g is described as:
$$g = \frac{\frac{1}{n}\sum_1^n(x_{i} – \overline{x})^{4}}{(\frac{1}{n}\sum_1^n(x_{i} – \overline{x})^{2})^{{2}}}$$

Kurtosis of a Gaussian random variable is 3.
Hence, Excess Kurtosis = Kurtosis – 3
If Kurtosis > 3, it is said to have a heavy tail (Leptokurtic).
If kurtosis < 3, it is said to have a thin tail (Platykurtic).

Kurtosis tells us whether there exists a problem of an outlier in the dataset.

It also helps us in distinguishing between two graphs, because there can be multiple functions. with the same mean and standard deviation.

In order to compare the kurtosis of two graphs, they must have the same variance.
More details: https://en.wikipedia.org/wiki/Kurtosis

What do you mean by tailedness in Kurtosis?

To understand, what tailedness mean consider a heavy-tailed distribution which has a tail that’s heavier than an exponential distribution. For instance, probability distributions where tails are not exponentially bounded. Heavy-tailed distributions tend to have many outliers with very high values. Usually the heavier the tail, the larger the probability that you will get one or more disproportionate values in a sample.

For example, if you sample from the income of the people in the US, the bulk of your data will be relatively small – around $50000. However, one or two values in your sample would be ridiculously large (i.e, outliers). You could draw 99 items that are around$50000, then a “Bill Gates” – he earns several billion dollars per year.

Some examples of heavy-tailed distributions are Log-Normal Distribution, Cauchy Distribution, Pareto Distribution.

Thank you for reading this far, I hope this was helpful. Comment below in case of any doubt or feedback.