Calculating adequate sample sizes

Originally posted by amolv06
The standard deviation of the mean can be calculated based off of a single sample of data. At least this is the way I've learned it. I study physics, so I don't really have a theoretical foundation behind this, but what we were told was that the standard deviation of the mean can be found by taking the standard deviation and dividing by the root of the sample size. My numbers seem to concur with the ones found by SH72 using his macro.

Did you just start your physics major? You'll get your fair share of derivations and theoretical foundations. 😀

Nope, in fact I should be finishing up by May. That said, I've never taken a course in probability, or statistics, nor am I required to. So far, the only exposure I've had to probability or statistics has been in quantum or my experimental lab class. In my lab class, they quoted how to calculate standard deviation, standard deviation of the mean, and propagation of error, and explained how to apply them to error analysis. They never really explained why or how they work -- I just know that they simply do. A mathematical black box, if you will.

This has been sufficient for me as far as classes are concerned. I do want to learn, or at least have an idea of the theory behind it, but haven't gotten around to it.

Originally posted by sh76
I figure that mathematics is a science, so I hope I have the right forum.

I need to determine what level of sample size is necessary to generate statistically reliable data for exam results. In other words, if I give an exam to X random college students, I want to be able to assert that based on those results, I can be confident that Y% of random students wi ...[text shortened]... or; and

2) Someone can give me a layman's tip on how to make that determination.

Thanks!

I think this is what you originally meant, but I don't know if you still want it.

Let's say M is the true mean. If I understand you correctly, you wanted to see how big should your sample size have been if you wanted a confidence interval of [M-Z,M+Z] with 95% confidence.

The formula for a 95% confidence interval for the sample mean is [M - 2*SD\sqrt(N),M + 2*SD/sqrt(N)].

So you want 2*SD\sqrt(N) = Z. Imagine that by good fortune you know the population SD was the one you get in your sample. Then you get the following formula for N as a function of Z:

N = (2*13.01/Z)^2.

For example, if you wanted to see how big the sample would need to be to get a sample in a confidence interval of [M-Z,M+Z] with

Z=1: N = 678
Z=2: N = 339
Z=3: N = 225

Of course the sample SD is not the true sample, so using it isn't usually good practice but I guess it's close enough for what you need. I don't know what Excel uses but you want the sample standard deviation (formula uses N-1 and is unbiased), not the standard deviation of the sample (formula uses N and is biased downward).

Originally posted by Palynka
I think this is what you originally meant, but I don't know if you still want it.

Let's say M is the true mean. If I understand you correctly, you wanted to see how big should your sample size have been if you wanted a confidence interval of [M-Z,M+Z] with 95% confidence.

The formula for a 95% confidence interval for the sample mean is [M - 2*SD\sqrt( ...[text shortened]... iased), not the standard deviation of the sample (formula uses N and is biased downward).

Thank you, Pal. I think I'm going to use the other numbers because they're simpler for a lay person to understand, but this certainly helps my understanding for the future.

Originally posted by amolv06
Nope, in fact I should be finishing up by May. That said, I've never taken a course in probability, or statistics, nor am I required to. So far, the only exposure I've had to probability or statistics has been in quantum or my experimental lab class. In my lab class, they quoted how to calculate standard deviation, standard deviation of the mean, and prop ...[text shortened]... to learn, or at least have an idea of the theory behind it, but haven't gotten around to it.

Surely you've taken a statistical physics course? 🙂

Not yet. That will be part of my last semester.

Originally posted by KazetNagorra
Surely you've taken a statistical physics course? 🙂

That is a whole division of physics on its own. My son-in-law has a Phd in Statistical Physics. He just finished a book called the Physics of foraging. About analyzing animal foraging techniques with statistical physics, by Cambridge press. It has been said with that book, he will be known as the father of foraging!