Sample size to determine confidence interval [Elsmar Cove Index (Ar)]

Tim Folkerts

19th January 2005, 02:10 PM

Benoit,

The standard approach would be to collect several groups of 50 beads and weigh them. From that data calculate the mean and standard deviation. 95% of the data should fall within +/- 1.96 sigma.

So if you found mu = 0.25 and sigma = 0.01, then 95% should fall within
0.25 +/- 0.0196

This is going off on a bit of a statistical tangent so feel free to skip down to the end if you aren't into statistics, but I just realized one other detail that applies here and to many other similar situations like control charts that use standard deviations. This analysis estimates mu and sigma and then makes the predictions based on those estimates. But those estimates are not certain so the predictions are not certain.

Suppose we have a set of 30 data points from weighing 30 sets of 50 beads (a fairly typical number). Suppose from those 30 data points we found that mu = 0.250 and sigma = 0.010, as above. If we did this again with a new set of 30 data points, we might get mu = 0.252 and sigma = 0.009. Every time you take a new set of 30 data points, you will get new numbers and hence new estimates of where the limits would be to include 95% of the groups.

For fun ;) I did a simulation using random data in excel. Basically, if you measure 30 data points from a normal distribution, there is less than a 50% chance that the limits calculated by
mu +/- 1.96 sigma

will truly include 95% of the data from the original normal distribution. Sometimes those limits include more that 95% of the true distribution, sometimes less. If you want to be 95% certain that 95% of the original distribution truly falls within your stated range, then you need to expand the range. It this particular case, you would need to use
something more like
mu +/- 2.5 sigma

to be 95% sure that 95% of all possible sets of beads would fall within that range.

NOTE: This has nothing to do with 50 beads (other than ensuring that the distribution of numbers will be approximately normal). It has everything to do with collecting 30 data points from a normal distribution. The more data points you collect, the less you need to expand the limits to be 95% confident.

That was a bigger tangent than I expected! I hope that didn't scare you : :).

The simple answer is tell the customer you used
mu +/- 1.96 sigma
based on a sample of "X" sets of 50 beads and they will almost certainly be happy.

Tim F

Steve Prevette

19th January 2005, 04:11 PM

An additional consideration is that you may be able to get away with a smaller initial sample (say of 15) and get an initial estimate of the standard deviation. With that, you can then estimate how much bigger of a sample you need in order to meet your confidence interval needs.

A lot depends on if the individual samples are expensive, or if you end up adding expense by getting two different batches of samples (initial estimate and a final).