What is variance in statistics

Definition:

Variance in statistics of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are .It Measures the variability in the data from the mean value.Variance is defined as

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This definition of variance in statistics can be used for both discrete and continuous random variables.Variance will never be negative, provided it is defined, because the squares are positive or zero . The unit of variance is the square of the unit of observation . Example: The variance of a set of heights measured in centimeters will be given in square centimeters. This is an inconvenient result, and so the standard deviation is generally used . The standard deviation is the square root of the variance .



The variance of random variable X is typically designated as Var(X), , or simply σ2 (pronounced “sigma squared”). If a distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either.


The formula is:


The unbiased formula is (for a sample):



with

x = the mean.

N = the population size.

n = the sample size.



Standard Deviation


The standard deviation formula is very simple: it is the square root of the variance in statistics. It is the most commonly used measure of spread.

An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score. In a normal distribution, about 68% of the scores are within one standard deviation of the mean and about 95% of the scores are within two standard deviations of the mean.



Variance and Standard Deviation: Step by Step

1. Calculate the mean, x.
2. Write a table that subtracts the mean from each observed value.
3. Square each of the differences.
4. Add this column.
5. Divide by n -1 where n is the number of items in the sample  This is the variance in statistics.
6. To get the standard deviation we take the square root of the variance.

Example

The owner of the Ches Tahoe restaurant is interested in how much people spend at the restaurant.  He examines 10 randomly selected receipts for parties of four and writes down the following data.

44,   50,   38,   96,   42,   47,   40,   39,   46,   50

He calculated the mean by adding and dividing by 10 to get

x  =  49.2

Below is the table for getting the standard deviation:

x	x - 49.2	(x - 49.2 )2
44	-5.2	27.04
50	0.8	0.64
38	11.2	125.44
96	46.8	2190.24
42	-7.2	51.84
47	-2.2	4.84
40	-9.2	84.64
39	-10.2	104.04
46	-3.2	10.24
50	0.8	0.64
Total		2600.4

Now

2600.4
------- =  288.7
10 - 1

Hence the variance in statistics is 289 and the standard deviation is the square root of  289 = 17.