The choice depends on what the data represent
Use population standard deviation when the values are the complete group you intend to describe. Use sample standard deviation when the observed values are treated as a random sample and you want to estimate variation in a larger population. The formulas begin identically: find the mean, subtract it from every value, square those deviations, and add them. The denominator changes because the two calculations answer different questions.
One dataset, two results
For 2, 4, 4, 4, 5, 5, 7, 9, the count is 8 and the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, totaling 32. If these eight values are the whole population of interest, population variance is 32 ÷ 8 = 4, and population standard deviation is √4 = 2.
If the same eight observations are a sample used to estimate a wider population, sample variance is 32 ÷ (8 − 1) = 32/7 = 4.5714. Sample standard deviation is √(32/7) = 2.1381, rounded to four decimals. The mean, median and mode do not change when you switch the variance basis; only variance and standard deviation do.
Why sample variance uses n − 1
The sample mean is estimated from the same observations. Once that mean is fixed, the deviations must add to zero, so only n − 1 deviations can vary independently. This is the lost degree of freedom. Under independent, identically distributed random sampling with finite variance, dividing the squared-deviation sum by n − 1 makes the usual sample variance an unbiased estimator of population variance. Dividing by n would tend to underestimate that variance.
Variance unbiasedness does not transfer through the square root
The sample standard deviation is the square root of sample variance. Because square root is nonlinear, sample standard deviation is generally still a slightly downward-biased estimator of population standard deviation. Do not call it exactly unbiased. The n − 1 version remains the conventional estimate reported by many statistical tools.
Keep the units and scope visible
Variance is expressed in squared units: squared centimetres if the observations are centimetres. Standard deviation returns to the original unit, which makes it easier to discuss typical spread around the mean. Neither result proves a distribution is normal, identifies bad data, or describes uncertainty in the mean. Record whether the list is a full population or a sample before interpreting the number.
Check the arithmetic with the Descriptive Statistics Calculator, then use the Ratios and Statistics Practice Worksheet for a printable worked exercise.
References: OpenStax: Measures of the Spread of the Data and NIST/SEMATECH: sample variance and standard-deviation bias.