Estimating Variance (2 of 4)

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With only one score, you have one squared deviation of a score from μ. In this example, the one squared deviation is: (X - μ)² = (14-12)²= 4.

This single squared deviation from the mean is the best estimate of the average squared deviation and is an unbiased estimate of σ². Since it is based on only one score, the estimate is not a very good estimate although it is still unbiased. It follows that if μ is known and N scores are sampled from the population, then an unbiased estimate of σ² could be computed with the following formula:

Σ(X - μ)²/N.

Now it is time to consider what happens when μ is not known and M is used as an estimate of μ. Which value is going to be larger for a sample of N values of X:

Σ(X - M)²/N or Σ(X - μ)²/N?

Since M is the mean of the N values of X and since the sum of squared deviations of a set of numbers from their own mean is smaller than the sum of squared deviations from any other number, the quantity Σ(X - M)²/N will always be smaller than Σ(X - μ)²/N.
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