How to calculate standard deviation by hand

What standard deviation tells you

Standard deviation is a single number that summarizes how spread out a data set is. A small standard deviation means most values cluster tightly around the mean. A large one means the values are scattered widely.

Two data sets can share the same mean and still look completely different. Scores of 70, 70, and 70 average to 70 with zero spread, while scores of 40, 70, and 100 also average 70 but vary a lot, so the second set has the larger standard deviation. That spread is what standard deviation captures, which is why it is usually reported alongside the mean rather than the median.

Standard deviation belongs to a family of spread measures that also includes the range and the interquartile range. For roughly bell-shaped data, the empirical rule says about 68% of values fall within one standard deviation of the mean, which gives the number a concrete meaning.

The five steps to calculate standard deviation by hand

The AP course describes standard deviation as a typical deviation of the data values from their mean. Here is the five-step recipe you can run on any small data set, step by step.

1. Find the mean. Add up all the values and divide by how many there are. The sample mean is written $\bar{x}$ (say "x-bar"), and $n$ is the number of values.
2. Find each deviation. Subtract the mean from every value, $x_i - \bar{x}$. Some deviations come out negative and some positive.
3. Square each deviation. Squaring removes the negative signs and gives $(x_i - \bar{x})^2$.
4. Add the squares and divide. For a sample, divide the sum by $n-1$. For a full population, divide by $n$. This result is the variance.
5. Take the square root. The square root of the variance is the standard deviation.

The square root in the last step puts the answer back in the original units, so a standard deviation of quiz points is measured in points, not points squared.

Sample vs population standard deviation

The steps are identical until step 4. The choice of $n-1$ or $n$ depends on whether the data is a sample or a whole population.

A population includes every member you care about, so its mean is the true mean, written $\mu$ ("mu"). The population standard deviation is written $\sigma$ ("sigma") and divides by $n$:

A sample is a subset drawn from a larger population. Its standard deviation is written $s$, called the sample standard deviation, and it divides by $n-1$. This is the version on the AP formula sheet:

Most real data sets are samples, so $s$ is the default you will use on the AP exam and the one a calculator reports first. Because $n-1$ is smaller than $n$, the sample standard deviation $s$ comes out slightly larger than the population value $\sigma$ for the same numbers.

The variance formula

Variance is the value you get in step 4, before taking the square root. In other words, standard deviation is the square root of variance, and variance is standard deviation squared.

For a sample, the variance formula is:

The AP course states this directly: the square of the sample standard deviation, $s^2$, is the sample variance. Variance is reported in squared units (points squared, dollars squared), which is hard to interpret on its own, so taking the square root returns the spread to the original units. That is why standard deviation is reported more often than variance, even though the two carry the same information.

Why you divide by n minus 1

When you work with a sample, you do not know the true mean $\mu$, so you use the sample mean $\bar{x}$ in its place. The catch is that $\bar{x}$ is calculated from the same values whose spread you are measuring, so those values sit a little closer to $\bar{x}$ than they would to the true mean. The squared deviations come out slightly too small.

Dividing by $n-1$ instead of $n$ makes the result a little larger, which corrects that downward bias. One common way to describe it: estimating the mean uses up one degree of freedom, so only $n-1$ independent pieces of information about the spread remain.

With a full population you already know $\mu$, so there is no bias to correct, and you divide by $n$.

Common mistakes to avoid

A few errors show up again and again when people calculate standard deviation by hand.

• Forgetting to square the deviations. The raw deviations always add to zero, so you must square them before adding, or the sum is useless.
• Dividing by the wrong number. Use $n-1$ for a sample and $n$ for a population. Mixing them up is the most common slip.
• Stopping at the variance. The variance is only step 4. You still have to take the square root to reach the standard deviation.
• Rounding too early. Keep extra decimals through the middle steps and round only the final answer, or small rounding errors pile up.

Check your work with a calculator

Working through the steps by hand once or twice builds intuition, but on the AP exam a graphing calculator with statistical capabilities is expected, and the sample standard deviation formula is printed on the provided formula sheet. Calculators usually list the sample value as $s$ or $Sx$ and the population value as $\sigma$ or $\sigma x$, so read the label carefully.

To check a calculation, enter the same data set in the standard deviation calculator and confirm your $s$ and $\sigma$ match. To watch the spread change as you drag points around, try the descriptive statistics sandbox.

Standard deviation is not the same as standard error, which measures how much a sample mean varies from sample to sample. The guide on the difference between standard error and standard deviation explains when to use each.