Application and Uses of Standard deviation

Application and Uses of Standard deviation

For any data set, as per "Chebyshev's Inequality" it can be proved mathematically that

1. At least 75% of all data points will lie within 2 standard deviations of the mean.
2. At least 89% of all data points will lie within 3 standard deviations of the mean.
3. At least 95% of the data is within 4.5 standard deviations of the mean.

Application of Standard deviation

1. Standard deviation used as a measure or risk

Example: You are trying to pick stock for investing in the equity market.

• Stock A has an annual return of 15%, with a standard deviation of 30%
• Stock B has an annual return of 12%, with a standard deviation of 8%

• If you were risk averse, which would you choose?

2. Measures of Shape

• Measures of shape describes the distribution or pattern of the data in a set.
• The distribution shape of the quantitative data can be described as there is a logical order to the values and the low and high end values on the horizontal axis of the histogram.
• The distribution shape of the qualitative data cannot be described.

Measures of shape are as follows: 

1. Degree of Skewness
2. Kurtosis

1. Degree of Skewness: 

Skewness is the tendency for the values to be more frequent around the high or low ends of the x axis
Skewness is a measure of symmetry
Symmetric data – The data is symmetrically distributed on both side of medium

• mean = median = mode

Positively skewed - Tail on the right side is longer than the left side.

• mode < median < mean

Negatively skewed - Tail on the left side is longer than the right side.

• mode > median > mean

Skewness Example 

Skewness Interpretation

• If skewness is less than -1 or greater than 1, the distribution is highly skewed.
• If skewness is between -1 and -0.5 or between 0.5 and 1, the distribution is moderately skewed.
• If skewness is between -0.5 and 0.5, the distribution is approximately symmetric, close to Normal Distribution.

The Skewness is 0.91, Mean is 43.17 and Median is 31 which indicates that the data is Positively skewed.

2. Kurtosis:

• Kurtosis is the sharpness of the peak of a frequency-distribution curve.
• It describes the shape of the distribution of the tail’s in relation to its shape 

Types of Kurtosis:

1. Mesokurtic – It has flatter tail than standard normal distribution and slightly lower peak.
2. Leptokurtic – It has extremely thick tail and a very thin and tall peak.
3. Platykurtic – It has slender tail and a peak that’s smaller than Mesokurtic distribution.

Kurtosis - Measure of the relative peak of a distribution.

• K = 3 indicates a normal “bell-shaped” distribution (mesokurtic).
• K < 3 indicates a platykurtic distribution (flatter than a normal distribution with shorter tails).
• K > 3 indicates a leptokurtic distribution (more peaked than a normal distribution with longer tails).

Lets consider 'Virat Kohli’s inning case, the kurtosis is 0.0094. Hence it is Platykurtic as its values is less than 3. 

Note: There might be certain differences in values when calculate on R.

Author:  Mohit T (Algae Services)