stdevp( ) appears to be mislabeled

[Mike Elzinga]

The current notational standards in statistics are that σX is the population standard deviation and sX is the sample standard deviation. The Stats 1Var app is consistent with the notational standards; and it also agrees with the STATS app on the HP 50G, the TI 89 and TI 83 and 84, and various other statistics packages.

The reason for dividing by N-1 in the sample standard deviation is because we don't have either the actual population mean or the actual population standard deviation when we take a sample. By taking a sample, we hope we have values that are representative of the population so that we can use the sample mean and standard deviation as <i>estimates</i> of the corresponding population parameters. These estimates of population parameters that we take from our sample are called statistics.

However, we lose a degree of freedom in calculating the mean of the sample because we then turn right around and use that sample mean in calculating the sample standard deviation. This makes the sample standard deviation a biased estimate of the population standard deviation unless we divide by N-1 instead of N.

When you calculate the population mean and standard deviation, you have every indivitual in the population, so there is no conflict in subtracting individual measurements from the population mean. So in calculating the population standard deviation it is proper to divide by N.

However, if you apply the calculation to get the sample standard deviation to the population itself, you get a value that is sqrt(N/(N-1)) times larger than it is supposed to be. Remember that the sample is not the population; it is hopefully a representative sample of the population from which we hope to estimate the population parameters when we can't count every member of the population.

Also, when we look at the behaviors of the sample means and sample standard deviations as a function of sample size, we find that the standard deviations of the sample means decreases with sample size as &sigma;/sqrt(N) whereas the standard deviation of the sample standard deviations decreases as &sigma;/sqrt(2N) when sampling a population that has a normal distribution.

I discovered the inconsistency in the notion of stddevp when I was writing an app to demonstrate the behaviors of the sample mean and sample standard deviation as a function of sample size. :-)

So, to be consistent with standard statistical notation, I would think that stddevp should be the same as &sigma;X and stddev (or, better, stddevs) should be the same as sX. At the moment, they are reversed.