S2 = 1/(n – 1) ∑i (xi – Xbar)2



Why We Use n – 1 to Calculate Sample Variance

Sample variance =

s2 = 1/(n – 1) ∑i (xi – Xbar)2 (1)

Xbar = 1/n (x1 + x2 + ... + xn) (2)

Xbar = (x1/n + x2/n + ... + xn/n) (3)

Substitute (3) for Xbar in (1):

s2 = 1/(n – 1) ∑i (xi – x1/n – x2/n – ... – xn /n)2 (4)

Multiply times n/n:

= 1/[n(n – 1)] ∑i (nxi – x1 – x2 – ... – xn)2 (5)

Expand nxi:

= 1/[n(n – 1)] ∑i [(xi – x1) + (xi – x2) + ... + (xi – xn)]2 (6)

But one of the differences in this series is (xi – xi):

= 1/[n(n – 1)] ∑i [(xi – x1) + ... (xi – xi) + ... + (xi – xn)]2 (7)

And (xi – xi) = 0 of necessity. It can never contribute to an estimate of variability among observations.

Also note from above we get this result:

s2 = 1/[n(n – 1)] ∑i ∑j (xi – xj)2 (8)

showing that sample variance can alternatively be calculated based on a sum of squared distances between all data points.

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