Symmetry, Skewness and Kurtosis

 

Symmetric distribution:

If the left side of distribution of data (continuous or discrete) is mirror image of the right side of the distribution then this type of distribution is known as symmetric distribution.

Fig1. Symmetric distribution

Above distribution is symmetric about point x.

The probability distribution is said to be symmetric if:

                                    p(x-δ) = p(x+δ)

where, p is probability density function (pdf) or probability mass function (pmf)

            x is a point about which distribution is symmetric

            δ is a real number

Properties of symmetric distribution:

• The mean and median of symmetric distribution is the point (x) about which the distribution is symmetric.
• If the distribution is unimodal (distribution has only one mode) then,    mean = median = mode.
• Skewness of symmetric distribution is zero.

 Skewness:

Skewness is the measure of asymmetry of the probability distribution. If the skewness of the distribution is zero then the distribution is symmetric.

There are two types of skewness:

1. Left skewed (negative skew)
2. Right skewed (positive skew)
   

 

Fig2. Positive and negative skewed distribution

If the tail is on the left side of the hump then it is left skewed and if the tail is on the right side of the hump then the distribution is right skewed.

We cannot find the relation between mean and median using skewness. But there is general relationship between mean, median and mode which can fail in some cases. The relationship is as follows:

Fig3. General relationship between mean, median and mode of skewness

Kurtosis:

Kurtosis is the measure of tailedness of the probability distribution. Measuring kurtosis on the basis of peakedness of distribution is incorrect. It describes the shape of the tail of the distribution.

Fig4. Kurtosis

Distribution with fat tail have high kurtosis, and they have the large number of outliers and distribution with low kurtosis and thin tail have less number of outliers.

There are three types of kurtosis:

1. Mesokurtic - Distribution with kurtosis similar to normal distribution is called mesokurtic. Kurtosis of normal distribution is 3.
   
2. Leptokurtic - Distribution with kurtosis > 3 is called leptokurtic.
3. Platykurtic - Distribution with kurtosis <3 is called platykurtic. It has short tail means less number of outliers.