A) Short:
The estimated variance, s^2 (s squared), is based on a concrete, finite sample; it _estimates_ the variance of the related population, sigma^2. [Note: s^2 is NOT the same thing as sigma^2.]
Calculating s^2 using a divisor of n (the sample size) gives a biased estimate of sigma^2. It tends to estimate on the low side. A divisor of (n-1) gives an unbiased estimate -- a lot of estimates together (multiple samples of n each) will approach sigma^2 from both sides, and come closer to the "true" sigma^2 than will the divisor of n.
Side note: yes, the difference between n and n-1 is small when n is greater than 30 or 100. We have enough problems with variation and measurement error now; why add even a small bias, when we have machines to do the calculations?
B) Not so short, but physical.
1) Find a pen with a clip on it. Toss the pen into the air. It can spin along its length, around the center through the clip, and around the center at right angles to the clip. It can rise above your head, it can move left or right, and toward or away from you. A total of 6 different directions of motion. Mechanical engineers call this 6 degrees of freedom. Written df.
2) If I hide four numbers on four slips of paper in a bowl, you would have no idea what those numbers might be. They could be anything (since you don't know how big the paper slips are :) They are "loose." Those numbers have four degrees of freedom. Now I tell you that three of those numbers are 10, 24, and 9. I tell you that the _average_ of the four numbers is 11.75. From this information you can _calculate_ the fourth number. Once you know the average, the fourth number is not "loose." When you calculate the average you 'give up' one degree of freedom from your sample.
If you stand next to a wall and toss the pen in the air so that it is next to the wall, it has lost one degree of freedom - it can't go into the wall. Same as the average - the range of potential values of your numbers has been reduced.
The divisor in the variance estimate, s squared (s^2), _is_ the degrees of freedom. Degrees of freedom of a sample equals sample size - n. To calculate s^2, we need to use the average. Hence, one less degree of freedom, and n-1.
In Addition: In calculating the residual error for a linear regression analysis (functionally equivalent to the standard deviation of a sample), we find that the divisor is (n-2) because we computed the slope and the intercept from the sample. We lost two degrees of freedom in this calculation.
Does this help at all?
Jay
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On Nov 6, 2008, at 8:31:04 AM, walmes zeviani wrote:
Hello moccalatte2003,
You can see the following link:
http://en.wikipedia.org/wiki/Variance#Population_variance_and_sample_variance
Have a good weekend.
Walmes,
Statistic and Design of Experimets - UFLA
Lavras, Minas Gerais, Brazil
De: moccalatte2003 <amymoore20@...>
Para: Statisticians_group@...
Enviadas: Quinta-feira, 6 de Novembro de 2008 11:11:07
Assunto: [Statisticians_group] standard deviationHey people,
I have a question does anyone remember the logic behind the denominator
(N-1) for standard deviation? I need a simpliest terms explanation.
Thanks.
-Amy
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