Normal Distribution, Skewness and Kurtosis

The Normal Distribution is also called as Gaussian Distribution and is a type of probability distribution that is symmetric about mean.

It appears as a Bell Curve as shown in the below figure.

The Skewness and Kurtosis coefficients measure how different a given distribution is from a “Normal Distribution”.

Skewness:

The Skewness measures the symmetry of a distribution. The normal distribution is a symmetric and has a skewness of zero. If the data has a skewness less than zero or negative skewness, then the left tail of distribution is longer (Tail shaped).

If the data has Skewness greater than zero or positive skewness then the right tail is longer.

So, when is the skewness too much?

Symmetrical Skewness: If the skewness is between -0.5 and 0.5, the data is fairly symmetrical.

Moderate Skewness: If the data has the skewness in between -1 and -0.5 (negatively skewed) and between 0.5 and 1 (Positively skewed).

Highly Skewed: In case the skewness is like; 1 < skewness < -1

Kurtosis:

Kurtosis is all about the tails of the distribution. It is actually the measure of outliers present in the distribution.

Mesokurtic (Kurtosis = 3): This distribution has kurtosis statistic similar to that of the perfect Normal Distribution.

Lepokurtic (Kurtosis > 3): Distribution is longer, peak is higher and sharper than Mesokurtic

Platykurtic (Kurtosis < 3): Distribution is shorter, peak is lower and broader than Mesokurtic.

Let’s look at those visually,

Standard Score:

The number of standard deviations from the mean is called “Standard Score” also known as “Z - Score”.

Probability Density Function (PDF):

If a random variable “x” followes the normal distribution then it is denoted as

The distribution is with μ = 0 and σ = 1 is called “Standard Normal Distribution”. => N (0,1)