Standard Deviation Calculator

Population (÷ n) only when your data is literally everyone or everything you care about — all 30 students in the one class you're describing. Sample (÷ n−1) whenever the data stands in for something bigger — 30 students representing all students, a week of measurements representing the process. In real statistics that's nearly always, which is why sample is the default here. If an exam doesn't specify, the phrase 'a random sample of…' is your answer.

Because you measured spread around the sample's own mean — the value that, by construction, sits closest to your data — so the raw average of squared deviations systematically understates the spread around the true population mean you don't know. Dividing by n−1 instead of n (Bessel's correction) inflates the result just enough to compensate, on average. The formal phrasing: the sample mean consumes one degree of freedom, leaving n−1.

It's the typical distance of a data point from the mean, in the data's own units — test scores with mean 75 and SD 8 mostly live within 8-ish points of 75. For roughly bell-shaped data, the 68-95-99.7 rule applies: about 68% of values fall within 1 SD of the mean, 95% within 2, 99.7% within 3. That's what makes SD the currency of statistics: 'two standard deviations above average' means the same thing regardless of what's being measured.

Variance (the squared version) is what the math wants: variances of independent variables add cleanly, which makes it the workhorse inside statistical theory. But its units are squared — dollars² means nothing to a human — so the square root gets taken at the end to return to interpretable units. Practical rule: think and communicate in standard deviation, and expect variance in formulas and ANOVA tables. This calculator reports both since homework asks for both.