The one difference that matters — the divisor
Both formulas compute variance the same way up to the final division: take each value's squared deviation from the mean, sum them all up. Population variance divides that sum by n (the total count). Sample variance divides by n − 1 instead — known as Bessel's correction. Standard deviation is simply the square root of variance either way, so this one divisor difference is the entire distinction between the two formulas.
A worked example on the same dataset
For the dataset 2, 4, 4, 4, 5, 5, 7, 9 (mean = 5, 8 values): treating it as the full population gives a variance of 4 and a standard deviation of exactly 2. Treating the identical numbers as a sample instead gives a variance of about 4.571 and a standard deviation of about 2.138 — noticeably larger, purely because the divisor shrank from 8 to 7. The underlying data never changed; only the assumption about whether it represents everything or just part of something larger did.
Why sample variance needs to be larger
When you only have a sample, the sample's own mean is itself an estimate of the true population mean — and a dataset's squared deviations from its own sample mean are, on average, slightly smaller than its squared deviations from the true (unknown) population mean would be. Dividing by n instead of n − 1 would systematically underestimate the true population variance. Dividing by the smaller n − 1 compensates for this bias, making the sample variance an unbiased estimator of the population variance it's trying to infer.
How to know which one to use
Use population standard deviation only when your dataset genuinely IS the entire group you care about — every student in a specific class's exam scores, if you only ever want to describe that exact class. Use sample standard deviation whenever your dataset is a subset drawn from a larger group you're trying to draw conclusions about — a survey of 200 customers used to infer something about your entire customer base, a handful of lab measurements meant to estimate a true physical quantity. In practice, most real-world statistics (surveys, experiments, A/B tests) use the sample formula, since it's rare to actually have data for an entire population.
The gap shrinks as your dataset grows
The difference between dividing by n and n − 1 matters most for small datasets — going from n=8 to n=7 is a meaningful 12.5% change in the divisor. For a dataset of 1,000 points, dividing by 999 instead of 1,000 barely moves the result at all. This is why the sample-vs-population choice is especially important to get right for small samples, and comparatively low-stakes for very large ones.