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The variance of a set of observation is defined as mean of the squares of deviations of all observations from their mean. When it is calculated from entire population, the variance is called the population variance, traditionally denoted by (σ is the Greek lower case “sigma”). If instead, the data from the sample are used to calculate the variance, it is referred to as the sample variance and is denoted by S² in order to distinguish between the two. The symbol definition of variance is
σ² =∑(xi-µ)²/N, for population sample
S²=∑(xi−Ẍ)²/n, for sample data
The variance is also denoted by Var(X). Variance of sample is calculated by dividing the sum of squared deviations by n. the statistics computed by dividing the sum of squared deviations by n-1 is called the sample variance corrected for the mean. Because this statistics is used extensively, we will shorten its name to sample variance. One of the objectives of statistical inference is to estimate the parameter from statistics. For example, we estimate the populations mean µ from the sample mean x. statistics created by dividing ∑(xi- Ẍ)² by n-1 is a better estimator than is the one created by dividing by n.
The term variance was introduced in 1918 by R.A Fisher 1890-1962). It should be noted that the variance is in square of units in which the observations are expressed and the variance is a larger number compared to observations themselves. The variance because of its some nice mathematical properties, assumes an extremely important role in statistical theory. Expectation of a random variable X is often called the mean and is denoted by µ. Another quantity of great important in probability and statistics is the variance of a random variable denoted by var(X) or σ². The variance is a non –negative number and its unit of measurement is squared units on the origin scale.
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