Variance: A Measure of Dispersion

Variance is a measure of dispersion of a distribution of a random variable. The term variance was introduced by R. A. Fisher in 1918. The variance of a set of observations (data set) is defined as the mean of the squares of deviations of all the observations from their mean. When it is computed for entire population, the variance is called the population variance, usually denoted by \sigma^2, while for sample data, it is called sample variance and denoted by S^2 in order to distinguish between population variance and sample variance. Variance is also denoted by Var(X) when we speak about the variance of a random variable. The symbolic definition for population and sample variance is

\sigma^2=\frac{\sum (X_i - \mu)^2}{N}; \quad \text{for population data}

\sigma^2=\frac{\sum (X_i - \overline{X})^2}{n-1}; \quad \text{for sample data}

It should be noted that the variance is in square of units in which the observations are expressed and the variance is a large number compared to observations themselves. The variance because of its some nice mathematical properties, assumes an extremely important role in statistical theory.

Variance can be computed if we have standard deviation as variance is square of standard deviation i.e. \text{Variance} = (\text{Standard Deviation})^2.

Variance can be used to compare dispersion in two or more set of observations. Variance can never be negative since every term in the variance is squared quantity, either positive or zero.
To calculate the standard deviation one have to follow these steps:

  1. First find the mean of the data.
  2. Take difference of each observation from mean of the given data set. The sum of these differences should be zero or near to zero it may be due to rounding of numbers.
  3. Square the values obtained in step 1, which should be greater than or equal to zero, i.e. should be a positive quantity.
  4. Sum all the squared quantities obtained in step 2. We call it sum of squares of differences.
  5. Divide this sum of squares of differences by total number of observation if we have to calculate population standard deviation (\sigma). For sample standard deviation (S) divide the sum of squares of differences by total number of observation minus one i.e. degree of freedom.
    Find the square root of the quantity obtained in step 4. The resultant quantity will be standard deviation for given data set.

The major characteristics of the variances are:
a)    All of the observations are used in the calculations
b)    Variance is not unduly influenced by extreme observations
c)    The variance is not in same units as the observation, the variance is in square of units in which the observations are expressed.

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