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Deriving the Shortcut Formula for the Sample Variance

Steps

Result

Given.

s2 = (X X)2

n 1

Square the binomial in the numerator.

s2 = (X2 2X X + X 2 )

n 1

Split up the sum.

s2 = (X2 ) (2X X) + (X 2 )

n 1

Factor out the constants 2 and X .

s2 = (X2 ) 2X (X) + X 2 (1)

n 1

Substitute using the facts that X = nX and 1 = n .

s2 = (X2 ) 2X (nX) + X 2 n

n 1

Simplify the last two terms in the numerator.

s2 = (X2 ) 2nX 2 + nX 2

n 1

Combine the last two terms in the numerator.

s2 = (X2 ) nX 2

n 1

Multiply numerator and denominator by n.

s2 = n(X2 ) n2X 2

n(n 1)

Modify the second term in the numerator.

s2 = n(X2 ) (nX)2

n(n 1)

Substitute using the fact that nX = X .

s2 = n(X2 ) (X)2

n(n 1)

Thus s2 = (X X)2

n 1=n(X 2 ) (X)2

n(n 1).

by Patrick Quigley

Deriving the Shortcut Formula for the Population Variance

Steps

Result

Given.

2 = (X )2

N

Square the binomial in the numerator.

2 = (X2 2 X + 2 )

N

Split up the sum.

2 = (X2 ) (2 X) + (2 )

N

Factor out the constants 2 and .

2 = (X2 ) 2 (X) + 2 (1)

N

Substitute using the facts that X = N and 1 = N .

2 = (X2 ) 2 (N) + 2 N

N

Simplify the last two terms in the numerator.

2 = (X2 ) 2N2 + N2

N

Combine the last two terms in the numerator.

2 = (X2 ) N2

N

Multiply numerator and denominator by N.

2 = N(X2 ) N 22

N 2

Modify the second term in the numerator.

2 = N(X2 ) (N)2

N 2

Substitute using the fact that N = X .

2 = N(X2 ) (X)2

N 2

Thus 2 = (X )2

N=N(X 2 ) (X)2

N 2.

by Patrick Quigley