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.