Standard deviation

This is homework, but I have come up with two answers and don’t know which is correct. I am working with three numbers: 8.33, 8.37, and 8.41. I did it first in MS Excel by entering the numbers down a column, entering the =STDEV(A2:A4) function, and it came up with .04 as the number. I did it again with my calculator according to the directions in my lab manual and came up with .0016.

Which is correct?

Plugging all three numbers into the TI-83’s STAT menu, I get .04 for the sample standard deviation, and .033 for the population standard deviation.

Thanks. I must have done my calculations wrong.

Perhaps you’re mixing up standard deviation and variance?

.0016 is the variance of the three numbers. .04 is the standard deviation. (The standard deviation is basically the square root of the variance.)

I don’t know Excel so I can’t say why it’s giving you the variance when you ask for the standard deviation. If you do it on paper you’ll come up with .04


Standard deviation is the square root of the variance.
The variance is the difference between the average of the squares and the square of the average.

The average of 8.33, 8.37, and 8.41 is (8.33 + 8.37 + 8.41)/3 = 8.37
8.37[sup]2[/sup] = 70.0569
8.33, 8.37, and 8.41 squared are 69.3889, 70.0569, and 70.7281, respectively
The average of 69.3889, 70.0569, and 70.7281 is (69.3889 + 70.0569 + 70.7281)/3 = 70.05796…
The difference between 70.05796… and 70.0569 is 0.00106…
The square root of 0.00106… is approximately 0.0326598632371090412

The variance is

sum (x[sub]i[/sub]-x[sub]mean[/sub])[sup]2[/sup]/n-1

In this case you’ve got (.16 + 0 + .16)/2 or .16

Larry has the sample standard deviation equation, but not the result. It’s V = (.0016 + 0 + .0016)/2 = .0016. Standard deviation is the square root of variance, so s = sqrt(V) = sqrt(.0016) = .04. Excel is correct. You forgot to take the square root.

Punoqllads is calculating population standard deviation. The formula is the same as sample deviation, except you divide by N instead of (N-1). You would use the population standard deviation if you knew that the three numbers represented the entire population. You would use sample standard deviation if the three numbers were picked at random from a larger population.

Your answer of 0.0016 should have raised some red flags in your head. Standard deviation is the number after “plus or minus”. Thus, for instance, one might characterize the measurement in the OP as “8.37 plus or minus 0.04”. And indeed, looking at the values, this makes sense, since that’s about the range by which the numbers vary (in fact, it’s exactly the range by which they vary, but I suspect that the problem was specifically constructed that way). But 8.37 plus or minus 0.0016 is completely out of line with those numbers.

Another way to notice this it to keep track of the units. Those numbers, if they actually came from something in the lab, would probably have some sort of units attached to them, such as 8.37 meters. If you went through all of the steps in the lab manual, and kept track of the units, you’d find yourself with a result of 0.0016 square meters. Since you want meters, not square meters, you need to take the square root.

Thanks for all the help. The numbers are my actual numbers from a lab experiment, units for those numbers is percent iron in an unknown sample (found via titration with KMnO[sub]4[/sub]). I actually did four titrations, but my solution turned a dark purple instead of a light pink on one; I added too much KMnO[sub]4[/sub], so my professor told me to throw that number out.

I have big problems looking at a group of numbers and making sense of them so it wasn’t obvious to me, but I can see now that I did forget to take the square root of .0016 in my calculations.

oops. :smack:

and I should have finished by taking the square root for the standard deviation.

Also, Boscibo, may I suggest the next time you have a small number of results, doing the variance and standard deviation on paper using the formulas. I realize you are taking chemistry, not statistics, and this may be more work than it’s worth, given your schedule. Nevertheless, I think working it out on paper a couple of times may help you avoid some confusion in the future. Just a thought