# Sample Variance Formula

The sample variance formula is an important formula in mathematics. It has the vast use in statistics and probability theories. This formula is used to calculate the deviation of any value or result from any test or experiment from the real or actual value. The deviation is determined as the mean root square deviation from thee expected value or result.

The variance is the measure of the distribution in the probability theory. The deviation has the range from the real value; between these two values, the probability theory has the consideration to have the result. Usually we prefer to get the average result as (9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) / 9 = 5. So, the absolute deviation can be determined as ((5-1) + (5-2) + (5-3) + (5-4) + (5-5) + (6-5) + (7-5) + (8-5) + (9-5)) / 9 = 2.22. And the mean root square of the deviations can be calculated as √ (((5-1)2 + (5-2)2 + (5-3)2 + (5-4)2 + (5-5) 2+ (6-5) 2+ (7-5) 2+ (8-5) 2+ (9-5)2) / 9) = 2.582. The sample variance formula can be formulated for continuous, discrete or mixed variables for functions. For the random variable x and the mean for m=F(x).

So, Var(x) = F[(x-m)2]

= F[x2-2xm+m2]

=F[x2] - 2mF[x] + m2

= F[x2] - 2m2 + m2

= F[x2] – m2

= F[x2] - F[x]2; which is simply denoted by σ2, pronounced square of sigma.

For the continuous case the formula can be written as

Var(x) = ʃ(y-m)2 f(y) dy

For discrete functions:

Var(x) = ∑i=1n ai . (yi - m)2, where y1

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