What is standard deviation?

The definition

Formally, the standard deviation is the root mean square deviation from the mean: take how far each value sits from the mean, square those distances, average the squares, and take the square root of the result. Written out, for a whole population of N values with mean μ:

σ = √[ Σ(x − μ)² / N ]

The formula page takes that expression apart term by term. For understanding what the number means, the plain-English version is enough: it is a measure of how far, on average, the data strays from its centre. Statisticians call this a measure of dispersion or spread, and it is the most widely used one there is.

Try it below. The calculator computes the standard deviation of whatever you enter and lays out each step, so you can watch the definition turn into a number.

What standard deviation measures, and what it does not

Standard deviation measures spread, not location. Two datasets can have the same mean and completely different standard deviations, and two datasets can have identical standard deviations while their means are miles apart. The mean tells you where the data is centred; the standard deviation tells you how tightly it gathers around that centre. Neither one substitutes for the other, which is why results are conventionally reported as a pair, "mean ± SD".

It is also not the range. The range is the gap between the largest and smallest value, and it depends on exactly two observations. The standard deviation uses every value, so a single extreme point stretches it far less than it stretches the range, though it still moves it more than a resistant measure such as the interquartile range would. Nor is it the average distance from the mean in the literal sense. That statistic exists and is called the mean absolute deviation; the standard deviation squares the distances first, which gives large deviations extra weight and is why it is never smaller than the MAD on the same data.

Units: standard deviation is in the same units as the data

If your data is in centimetres, the standard deviation is in centimetres. If it is in seconds, dollars or milligrams, so is the standard deviation. This is the reason the formula ends with a square root. The step before it, the variance, is an average of squared deviations, so its unit is the data's unit squared. A set of adult heights with a mean of 170 cm and a variance of 64 cm² is awkward to interpret, because nobody thinks in square centimetres. Take the square root and you get a standard deviation of 8 cm, which reads naturally: a typical person in the set is about 8 cm from the average height.

Having the same unit as the data is what makes the standard deviation directly comparable to the mean, and what makes statements like "two standard deviations above the mean" translate immediately into a real quantity. It is the practical reason standard deviation, rather than variance, is the number that gets reported.

High vs low standard deviation: two datasets with the same mean

Here are two sets of five test scores. Both have a mean of exactly 70.

• Set A: 68, 69, 70, 71, 72
• Set B: 50, 60, 70, 80, 90

The mean is no help in telling them apart. The standard deviation is. Working set B through the definition, using the sample formula (divide by n − 1) that the calculator above defaults to:

s² = 1000 / (5 − 1) = 250 s = √250 = 15.81

Set A goes through the same steps with deviations of −2, −1, 0, 1 and 2. Their squares sum to 10, so s² = 10 / 4 = 2.5 and s = √2.5 = 1.58. Same mean, but one set has a standard deviation ten times the other. A typical score in set A is about a point and a half from average; a typical score in set B is nearly sixteen points from it. Notice also that the middle column sums to zero in both cases. It always does, which is the reason the deviations have to be squared before they are averaged.

So which is "high" and which is "low"? Only relative to each other, and to the mean. There is no universal cut-off. A standard deviation of 15.81 is large for scores that average 70, but it would be tiny for annual salaries that average 70,000. The way to judge a standard deviation is to set it against the mean of the same data, which is what the coefficient of variation does by dividing one by the other, or to compare it with the standard deviation of something similar.

A standard deviation of zero, and why it can never be negative

A standard deviation of zero means every value is the same. If the data is 7, 7, 7, 7, the mean is 7, every deviation is 0, the sum of squares is 0, and so is its square root. There is no other way to reach zero: the moment any single value differs from the rest, at least one deviation is non-zero, its square is positive, and the standard deviation is positive too.

That is also why the standard deviation can never be negative. Squaring makes every term in the sum non-negative, an average of non-negative numbers is non-negative, and the square root is taken as the positive root by definition. The smallest possible standard deviation is zero. If you ever calculate a negative one, the arithmetic went wrong somewhere, most often by squaring the sum of the deviations instead of summing the squares.

Reading it with the 68–95–99.7 rule

For data that is roughly bell-shaped, the standard deviation has a very concrete meaning. About 68% of the values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is the empirical rule, and it is the fastest way to turn a standard deviation into a picture of the data.

Adult IQ scores are the textbook case: the tests are scaled to a mean of 100 and a standard deviation of 15, so roughly 68% of people score between 85 and 115, and roughly 95% between 70 and 130. A score of 145 sits three standard deviations above the mean, which puts it in about the top 0.15% of the distribution. The z-score generalises this: it expresses any value as a number of standard deviations from the mean, which is what makes results from different scales comparable.

The caveat matters. The rule describes the normal distribution and holds only approximately for data that resembles one. Skewed data, such as incomes or reaction times, and data with heavy tails will put far more than 5% of values beyond two standard deviations. The standard deviation is still a valid measure of spread for that data; it just no longer maps onto those particular percentages.

Sample and population standard deviation

There are two versions of the calculation. The population standard deviation, written σ, divides the sum of squared deviations by N and describes a complete set of values. The sample standard deviation, written s, divides by n − 1 and estimates the spread of a larger group from the subset in front of you. Dividing by the smaller number makes s a little larger than σ would be on the same data, and that inflation exactly corrects the tendency of a sample to look less spread out than the population it came from.

Which one you want depends on your question rather than on your data. If the numbers are everything you care about, use the population version; if they stand in for something bigger, use the sample version, which is the more common situation. The sample vs population page works through the decision and shows how much difference it makes at each sample size. The notation page covers the symbols.

Where standard deviation is used

Laboratory precision. When an analyst measures the same sample repeatedly, the standard deviation of the results is the precision of the method. It is usually reported as the relative standard deviation, the SD as a percentage of the mean, so that methods working at different concentrations can be compared. Five replicates averaging 5.00 mg with an SD of 0.05 mg have an RSD of 1%.

Finance. The standard deviation of an asset's returns is its volatility. A fund whose monthly returns have a high standard deviation swings more widely around its average than one with a low standard deviation, even if the two average the same. Risk in portfolio theory is defined by this number.

Psychology and education. Standardised tests are built around a fixed mean and standard deviation, IQ at 100 and 15 being the best-known example, so that a score can be read as a position in the population rather than a raw count of correct answers.

Manufacturing. Quality control tracks the standard deviation of a measured dimension across production. Control limits are conventionally placed three standard deviations either side of the target, and a process whose spread is small relative to its tolerance is called capable. "Six sigma" takes its name from the standard deviation symbol.

Common misconceptions

• "A high standard deviation is bad." It is neither good nor bad; it is a description. High variability in manufacturing tolerances is a problem. High variability in a set of exam scores may be exactly what a well-designed exam is supposed to produce.
• "Standard deviation tells you the shape of the data." It does not. Two datasets can share a mean and a standard deviation while one is symmetric and the other is badly skewed. The 68–95–99.7 percentages assume a shape; the standard deviation itself makes no such assumption.
• "It is the average distance from the mean." Close, but that describes the mean absolute deviation. Because standard deviation squares the distances first, it is pulled more by the far-out values and is always at least as large as the MAD.
• "Standard deviation and standard error are the same thing." The standard deviation describes the spread of the data. The standard error describes the uncertainty in the mean, and shrinks as you collect more data. The standard deviation does not.
• "Outliers barely affect it." They affect it a great deal. Because the deviations are squared, one value far from the mean can dominate the sum. Check for outliers before trusting a standard deviation on a small dataset, and consider the interquartile range if they are genuine.

When you are ready to compute one by hand, the step-by-step guide walks through the arithmetic, and the methodology page explains exactly how the calculators on this site do it.