Standard Deviation Calculator

What is standard deviation?

Standard deviation tells you how spread out a set of numbers is around their average. A small standard deviation means the values sit close to the mean, so they're consistent. A large one means they're scattered far from the mean. It's the most common way to describe spread, because it comes back in the same units as the data, unlike variance.

Sample vs population standard deviation

The only difference is the number you divide by. If your numbers are the whole group you care about, that's a population, and you divide by n. If they're a sample drawn from a bigger group, you divide by n minus 1, which nudges the answer up a little to fix the way a sample understates the real spread. Most real-world data is a sample, so when you aren't sure, the sample version is the safer pick. This tool shows both, so you don't have to choose blind.

Variance and the formula

Variance is the step right before the standard deviation: it's the average of the squared differences from the mean, and the standard deviation is its square root. So the standard deviation formula is the square root of Σ(x − mean)² ÷ n for a population, or the same thing over n − 1 for a sample. Because variance is in squared units, like dollars squared, people usually report the standard deviation, which lands back in plain dollars. The result above lists the variance next to the standard deviation, so you've got both at a glance.

Standard deviation and z-scores

Once you have the mean and standard deviation, you can say how unusual a single value is with a z-score, which counts how many standard deviations a value sits from the mean. A z-score of 2 means a value's two standard deviations above average. It's a handy way to compare scores from different scales, and it leans directly on the standard deviation you work out here. The z-score calculator takes it from there, turning a value into a z-score, a percentile, and a probability.

The 68-95-99.7 rule

For data that follows a normal, bell-shaped curve, the standard deviation maps out where the values fall. About 68% of them land within one standard deviation of the mean, roughly 95% within two, and about 99.7% within three. That's the empirical rule, and it's why the standard deviation is so useful: it turns a single number into a picture of how the whole set is spread. If test scores average 70 with a standard deviation of 10, then around 95% of students scored between 50 and 90.

It only holds for roughly normal data, so skewed sets won't follow it as neatly, but a lot of natural measurements come close. The rule also flags outliers: a value more than three standard deviations out is rare, under 0.3% of the time, so it's worth a second look. To pin down exactly how far one value sits from the mean, the z-score calculator turns it into a percentile.

What is a high or low standard deviation?

There's no fixed cutoff, since high and low only make sense next to the mean. A standard deviation of 5 is tiny on values that average 1,000, but huge on values that average 8. A low standard deviation means the numbers cluster tightly around the mean, so they're consistent and predictable. A high one means they're scattered widely, with more ups and downs. That's why two data sets with the same average can behave completely differently.

To compare spread across data sets on different scales, divide the standard deviation by the mean to get the coefficient of variation, a percentage that strips out the units. A class with a 5% coefficient is far more consistent than one at 30%, whatever the raw scores. For the center of your data alongside the spread, the mean median mode calculator covers the averages, and the average calculator handles the mean on its own.