Calculate standard deviation, variance, mean, sum, and margin of error for a set of numbers.
Use "Population" when your data represents every member of the group you care about. Use "Sample" when your data is a subset drawn from a larger population you're trying to estimate — the sample formula divides by N-1 instead of N, which corrects for the fact that a sample tends to slightly underestimate the true population variance.
The standard error of the mean (SEM) describes how much the sample mean would likely vary if you repeated the sampling process. Multiplying SEM by a z-score gives a margin of error at a chosen confidence level — for example, "95% confidence" means that if you repeated this sampling many times, about 95% of the resulting intervals would contain the true population mean.
This is Bessel's correction. Using a sample's own mean to compute variance slightly underestimates the true population variance, since the sample mean is, by construction, the value that minimizes the sum of squared deviations for that exact sample. Dividing by N-1 instead of N corrects for that bias.
Sample standard deviation is undefined for a single value, since there's no way to divide by N-1=0. Population standard deviation of a single value is always 0, since that one value is exactly equal to the mean.
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