# Application and Uses of Standard deviation

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

- At least 75% of all data points will lie within 2 standard deviations of the mean.
- At least 89% of all data points will lie within 3 standard deviations of the mean.
- 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.

- Degree of Skewness
- Kurtosis

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

- 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:

**Mesokurtic**– It has flatter tail than standard normal distribution and slightly lower peak.**Leptokurtic**– It has extremely thick tail and a very thin and tall peak.**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)

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