Standard error, uncertainty?

[Bozeman]

Suppose you have n measurements of x (with random but no systematic error). What is your uncertainty in the value of x (such that correct value is within one uncertainty of your average 68% of the time)? What if you know the exact value of x? Does it change for n very small?

Thanks,
Craig

[MrBlini]

If the random error is normal, this is Student's t-Distribution.

The distribution of t, the variable defined in Equation (1), is approximately normal for large sample sizes. Thus the difference between the sample mean and the population mean--the numerator of Equation (1)--is approximately normally distributed with standard deviation s/sqrt(N). One standard deviation is thus easy to estimate--just divide the sample standard deviation s by the square root of N, the number of samples.

For example, several members of the Zoo decide to measure the average length of a circulated 1996-series $100 bill. They gather their shoeboxes and find N=44 bills from that series, enough samples that Student's t-Distribution is almost identical to a normal distribution. With a plastic engineering ruler (a poor choice IMO), they find that the sample mean is 6.1374 inches and the sample standard deviation s is .0192. They divide this by the square root of N=44 to estimate the standard deviation of the difference between the sample mean and the population mean, finding that s/sqrt(N)=.0029 inches. They conclude that, to within one standard deviation of the t-Distribution, the mean length of a 1996 series $100 bill is 6.1374 +- .0029 inches, or between 6.1345 inches and 6.1402 inches.

As you can see, for large N, the uncertainty in the difference between the sample mean and the population mean decreases as the square root of N.

Your intuition that this all changes when N is small is spot on. For small values of N, you cannot use the normal approximation to the t-Distribution for anything but a very rough guess. You need to use actual t values, which means you'll need mathematical software or statistical tables to solve such problems. The differences between the t-Distribution and the normal distribution are illustrated nicely on this page at Stanford.