In statistics there are 2 different standard deviation formulas that look different but are algebraically the same.
It is hard to type them here but I will giveit a shot.
1. sqroot - SUM (x - bar x or mean)2 / n-1
2. sqroot - n(SUMx2) - (SUMx)2 / n(n-1)
My problem is what is the process to algebraically turn the first formula into the second to prove they are algebraically equal.
Can someone solve that?
Proving the 2 Standard Deviation formulas
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orlandomike
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barretttomlinson
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Re: Proving the 2 Standard Deviation formulas
Hi,
You can rewrite your first equation by recognizing that x squared is x*x. Expand the expression by writing out the expression as the product of sums. Then notice you can regroup the expression as summations of individual terms. You will have a sum of sum of x bar2, which since x bar is a constant, reduces to n sum (xbar). You also have to notice that xbar is just (sum (x))/n, make that substitution in your expansion, and collect terms, and you should have the second form. You may also need to use the theorem that the sum of a difference is the difference of the sums of the individual terms. It is not hard to work this out except that you have to deal with a lot of terms in the expressions in the middle of the process, and it is easy to lose one.
My text editor is no better than yours so I cannot easily record the detailed steps. I hope you can decipher the hints well enough to work out the proof to your own satisfaction.
You may find these two sites helpful:
http://richardbowles.tripod.com/maths/n ... lc_sd1.htm
http://www.thestudentroom.co.uk/showthread.php?t=676968
Best regards,
Barrett L Tomlinson
You can rewrite your first equation by recognizing that x squared is x*x. Expand the expression by writing out the expression as the product of sums. Then notice you can regroup the expression as summations of individual terms. You will have a sum of sum of x bar2, which since x bar is a constant, reduces to n sum (xbar). You also have to notice that xbar is just (sum (x))/n, make that substitution in your expansion, and collect terms, and you should have the second form. You may also need to use the theorem that the sum of a difference is the difference of the sums of the individual terms. It is not hard to work this out except that you have to deal with a lot of terms in the expressions in the middle of the process, and it is easy to lose one.
My text editor is no better than yours so I cannot easily record the detailed steps. I hope you can decipher the hints well enough to work out the proof to your own satisfaction.
You may find these two sites helpful:
http://richardbowles.tripod.com/maths/n ... lc_sd1.htm
http://www.thestudentroom.co.uk/showthread.php?t=676968
Best regards,
Barrett L Tomlinson