## Introduction

- Standard Deviation the most widely used method to measure the Dispersion of a set of data. It measures the deviation of elements of a population or dataset from its mean.
- Standard Deviation is calculated as the positive square root of the mean of squared deviations from the mean

### Chebyshev’s Inequality

- According to
**Chebyshev’s inequality**, for any distribution with**finite variance, t**he proportion of the observations within**k standard deviations**of the arithmetic mean is**at least 1 − 1/k**for all k > 1.^{2} **Let us understand the above statement,**- Mean or the average of a given dataset calculated by dividing the sum of all elements of the dataset by the total number of elements in the dataset.
**Mean =**(Sum of dataset elements)/ Total number of elements in the dataset**Standard Deviation**of a dataset is obtained by using the following formula:**Population Standard Deviation =**- Where
**D**is the difference between the elements of a population and its mean - And
**n**is the population size **Let us take k = 2, then,**- As per Chebyshev’s inequality, 1- ¼ = ¾ or 75% of elements should lie in the range of k standard deviation i.e. 2 standard deviations

**Illustration 1:**or a dataset A if Mean is = 10 and Standard Deviation is 2, then 75% of the elements of the dataset A should lie between 10 ± 2 i.e. between 8 and 12.

- For different values of k, we can get different data that how much percent element of a dataset should lie in the range of k Standard Deviations.

#### Example. 1:

**Let us take a sample: 2,5,8,10,12,15,18, verify Chebyshev’s inequality for k=2.****Solution:**For k =2, around 75% of dataset elements should be in the range of Mean ± 2* Standard Deviation i.e. 10 ± 11.134 i.e. -1.134 to 21.134- In our sample dataset, all the elements lie in the above range
- Hence, Chebyshev’s Inequality that
**at least 75% of elements should fall in this range is verified.**