Standard Deviation Calculator

Take the mean. Subtract it from every value and square each difference. Add those squares up. That total is the sum of squared differences. Divide by n for the population standard deviation, or by n - 1 for the sample one. Take the square root of the result. The n - 1 is called Bessel correction. A sample tends to sit closer to its own mean than to the true mean of the population, so dividing by n underestimates the real spread. Subtracting one from the denominator corrects for it, and matters most when n is small: with 5 values it raises the answer by about 12 percent, with 500 by about a tenth of a percent.

Should I use the sample or the population formula?

If your numbers are every item you care about - the height of all 30 students in one class, when the class is the whole question - use the population formula. If they are a sample standing in for a larger group, use the sample formula with n - 1. In practice most real data is a sample.

What does standard deviation actually tell me?

How far a typical value sits from the average. A small standard deviation means the numbers are bunched together; a large one means they are spread out. Two sets can share a mean and look nothing alike.

Why square the differences instead of just taking them?

Because the plain differences always add up to exactly zero: the positives and negatives cancel. Squaring removes the sign, and it also weights a value far from the mean more heavily than one nearby, which is usually what you want.

What is the difference between variance and standard deviation?

Variance is the average squared difference; standard deviation is its square root. Variance behaves better in algebra, which is why formulas use it, but it is in squared units. If your data is in dollars, the variance is in dollars squared, which means nothing to anybody.

What is the 68-95-99.7 rule?

For data that follows a normal curve, about 68 percent of values lie within one standard deviation of the mean, 95 percent within two and 99.7 percent within three. It only holds for roughly bell shaped data, so check your numbers before leaning on it.

Can standard deviation be negative?

No. It is a square root of a sum of squares, so it is zero or positive. Zero means every value is identical. If you calculate a negative one, a sign went wrong in the arithmetic.