The sample s.d. always underestimates

Population Mean value is under or overestimated by the Sample Mean. On contrary in what concerns standard deviations (and variances) we have always
____s.d. Sample < = s.d. Population

Let be x(j) with j = 1, 2, … , n the elements of a random sample of size n, and m the sample mean. Let be M the population one.

___ d(j) = x(j) - [( m - M) + M]
___ D(j) = x(j) – M
But, unless m=M we have d(j) < D(j)

In it what concerns the sum of squared deviations (ssd)

Population ssd = sum D(j)^2
Sample ssd = sum d(j)^2

Therefore Population ssd >=Sample ssd.
To have the unbiased estimation of the Population variance (sigma the standard deviation) we must to divide the Sample ssd by n-1, to have the estimation of the Population ssd:

______sigma = Population sd =
_____sqrt (variance) = E [(x(j) – M]^2
_______________________ = E [D(j)^2]
_____= E [Population ssd] >= E[Sample ssd]

______licas (Luis A. Afonso)
.

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