Science Buddies: "Ask an Expert"

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?

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