Deriving the Shortcut Formula for the Sample Variance ?· Deriving the Shortcut Formula for the Sample… page 1
Deriving the Shortcut Formula for the Sample Variance ?· Deriving the Shortcut Formula for the Sample… page 2

Deriving the Shortcut Formula for the Sample Variance ?· Deriving the Shortcut Formula for the Sample…

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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