Standard Deviation Calculator

The standard deviation is a measure of the spread of a set of numbers. It gives an indication of how far away from the mean each value is on average. A small standard deviation indicates that the data are clustered closely around the mean, while a large standard deviation indicates that the data are more widely dispersed.

This calculator will find the standard deviation of a set of numbers, as well as the variance, mean, count and sum. Select whether your data is a sample or population, enter all observations separated by commas, spaces or new lines, and see all results on one page.

What standard deviation tells us.

Each value in a data set is a certain distance away from the mean. The standard deviation takes those distances and summarizes them into one number. So it tells you how far each value is on average from its mean.

Even if two data sets have the same mean, they can be very different in other respects. For example, a group with values 70, 70, and 70 has the exact same mean as a group with values 40, 70, and 100 - but the spread of the second group is much greater. The mean alone cannot show this difference, but the standard deviation can.

A key advantage over the variance is that standard deviation can never be negative and uses the same units as the original data. If the standard deviation is zero, it means all values are equal.

Sample standard deviation and population standard deviation.

The only choice that changes the result is whether the numbers are for a sample or population. A population is the entire group being analyzed while a sample is just part of that group.

For a population, the sum of squared deviations is divided by n, the number of values. For a sample it is divided by the number n minus 1. This smaller divisor is called Bessel's correction. It makes the sample standard deviation slightly larger which corrects for the problem that the sample tends to underestimate the spread in the population from which it was drawn.

The formula for the population is only used when all members of the population are actually known. For example, if you were analyzing test scores from a class of students. If the data represents a larger population, then the sample formula should be used, and this tool defaults to using the sample formula.

Here, σ is the population standard deviation, μ is the population mean, s is the sample standard deviation and x̄ is the sample mean.

This is how standard deviation is calculated.

The formula may seem complicated but it actually consists of five simple steps.

In the first step all numbers are added and then divided by the number of numbers to calculate the average. In the second step each number is subtracted from the average to determine the deviation. In the third step the individual deviations are squared to ensure that negative numbers do not cancel out positive ones.

In the fourth step, the sum of squared deviations is calculated. This sum is the quadratic sum. In the fifth step, the variance is divided by the number of elements in the population (n) or by n minus 1 (in case of a sample), and then the square root of this division is taken to obtain the standard deviation.

A complete example:

We'll use these eight numbers: 6, 6, 8, 10, 10, 12, 14, 14. This is a data set that's already loaded into the calculator tool. The sum of it is 80, so the mean is 10.

The deviations of each value from the mean are in order: negative 4, negative 4, negative 2, 0, 0, 2, 4, and 4. Squaring them gives us: 16, 16, 4, 0, 0, 4, 16, and 16, which add up to 72.

When the sum of squares (72) is divided by n (equal to 8), we get the population variance (9), and the square root of that is the population standard deviation (3).

If a group of identical numbers is considered as the sample, then dividing by 7 instead of 8 will result in a sample variance of approximately 10.286 and a sample standard deviation of approximately 3.207. Although the data are all equal, the results can be somewhat larger since the formula for the sample helps to avoid underestimating the variability.

Variance and standard deviation

Variance is the standard deviation before taking the square root and is equal to the average of squared deviations. These two quantities are always related: The standard deviation is the square root of the variance, and the variance is the square of the standard deviation.

Since the variance squares the data, it also changes the unit to a squared unit which makes interpretation difficult. If the units of the numbers are centimeters, then the unit of the variance is square centimeters while the standard deviation still has the unit of centimeters. This is why usually the standard deviation is reported.

Rules of thumb

When the data follows a normal distribution with bell-shaped curve, standard deviation can predict how spread out the values are around the mean. About 68% of the data falls within one standard deviation from the mean, about 95% within two standard deviations and about 99.7% within three standard deviations. This is an empirical rule also known as the 68-95-99.7 rule.

Thus, values that are more than two or three standard deviations from the mean appear to be outliers. This forms the basis for quality control limits and the concept of outliers in statistics. By opening the "rule of thumb" level you can check these areas for your own data.

Coefficient of variation and standard error

There are two related measures based on the standard deviation: The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage. This allows comparison of the spread of data sets that have wildly different means.

The standard error of the mean is the sample standard deviation divided by the square root of the sample size. It estimates how much the sample mean would change if a new sample were to be drawn, and decreases as the sample size increases. These two measures are displayed in the Related Measures level.

Applications of standard deviation.

In education, standard deviation is used to measure the spread of test scores and also in grading systems where points are assigned based on how far a score deviates from the average.

In manufacturing and quality control, a small standard deviation indicates that the process is stable and consistent while an increasing standard deviation indicates that product measurements are deviating from the specification. Control limits are set based on multiples of the standard deviation.

In finance, standard deviation is a standard measure of volatility and risk. Portfolios with large fluctuations have high standard deviations while stable portfolios have low standard deviations.

Used symbols:

Each symbol used above is spelled out below.