What does standard deviation actually mean?

A typical distance from the mean

The AP course describes standard deviation as a typical deviation of the data values from their mean. That is the sentence to hold onto. If a data set has a mean of 17 minutes and a standard deviation of 3.69 minutes, then a value about 3.69 minutes away from 17 is unremarkable, while one sitting 15 minutes away is not.

The word "typical" is doing real work there, and it is meant literally rather than exactly. Standard deviation is not the plain average of the distances. It is built from squared distances, so values far from the mean count for more, and it usually comes out a little larger than the average distance.

The symbols: $s$ is the sample standard deviation, $\bar{x}$ ("x-bar") is the sample mean, $\sigma$ ("sigma") is the population standard deviation, and $\mu$ ("mu") is the population mean. This page uses $s$, since nearly all data you meet is a sample.

Where the number comes from

Each part of the formula answers a problem created by the part before it.

1. Deviations. Subtract the mean from each value, $x_i - \bar{x}$. That is the distance-from-center idea in its rawest form, with a sign attached.
2. Squares. The raw deviations always add to exactly zero, so their average tells you nothing. Squaring makes every one of them positive, and it also weights the far-out values more heavily.
3. Average. Add the squares and divide by $n - 1$. This average squared distance is the variance, $s^2$. The divisor is $n - 1$ rather than $n$ because the deviations were measured from the sample's own mean, which dividing by n - 1 explains in full.
4. Square root. Undo the squaring from step 2.

Skip any step and the result stops meaning anything. Skip the squares and you get zero every time; skip the root and you are left in the wrong units. The five-step calculation by hand runs the whole thing on real numbers.

Why the square root matters

Variance is measured in squared units. If your data is in minutes, the variance is in minutes squared, which is not a quantity anyone can picture. The square root returns the answer to minutes. That is the whole reason standard deviation is reported more often than variance, even though the two carry identical information.

Because $s$ is in the original units, you can set it beside the data and it means something. You can say a value is one standard deviation above the mean, which is exactly what a z-score counts. The AP course defines a standardized score as the number of standard deviations a value falls above or below the mean, and that definition works only because $s$, $x$, and $\bar{x}$ share units.

Units follow the data through conversions, too. Convert your minutes to seconds and $s$ is multiplied by 60 along with everything else, since changing the units of measurement changes the values of the calculated statistics.

What small and large look like on a dotplot

Here are two sets of five delivery times, in minutes, both with a mean of 20 minutes.

• Set A: 18, 19, 20, 21, 22 minutes. The dots sit in a tight clump. The squared deviations 4, 1, 0, 1, 4 total 10, so $s = \sqrt{10/4} = \sqrt{2.5} \approx 1.58$ minutes.
• Set B: 10, 15, 20, 25, 30 minutes. The same five dots stretch across a range five times as wide. The squared deviations 100, 25, 0, 25, 100 total 250, so $s = \sqrt{250/4} = \sqrt{62.5} \approx 7.91$ minutes.

Set B is Set A pulled away from the mean by a factor of 5, and its standard deviation is exactly 5 times as large: $1.5811 \times 5 = 7.9057$. On a dotplot, a small standard deviation looks like a narrow pile and a large one looks like dots scattered thinly across a wide axis.

Two data sets can share a mean and look nothing alike, which is why a center is never a description on its own. A description names shape, center, and variability plus any unusual features, and how to describe a distribution walks through the full checklist.

Interpretation sentences that earn credit

An interpretation names the variable, the individuals, and the units. Restating the number is not interpreting it.

The template for one distribution: the [variable] of the [individuals] typically vary about [s] [units] from the mean of [mean] [units].

Filled in: "The wait times of these six patients typically vary about 3.69 minutes from the mean wait time of 17 minutes."

For a comparison, name both numbers and say which is larger: "The Set B delivery times are more variable than the Set A times, because Set B's standard deviation of 7.91 minutes is larger than Set A's 1.58 minutes."

Three phrasings that do not earn the point:

• "The standard deviation is 3.69." That repeats the calculation with no context and no units.
• "The data is 3.69 away from the mean." No variable, no individuals, and it claims every value is exactly 3.69 away.
• "3.69 is a small standard deviation." Small next to what? Spread is only small or large relative to the mean or to another group.

Never negative, and zero only when every value is identical

A standard deviation cannot come out negative. Every squared deviation is zero or positive, so their sum is zero or positive, $n - 1$ is positive, and the square root of a number that is not negative is not negative either. A negative $s$ is an arithmetic error, with no exceptions.

It reaches zero in exactly one situation: every value in the set is the same number. Take 7, 7, 7, 7. The mean is 7, all four deviations are 0, the sum of squares is 0, and $s = 0$. There is nothing for a value to be typically distant from.

Turn that around. If any two values differ, at least one squared deviation is positive, so $s > 0$. A standard deviation of zero is a claim that the data has no variability at all, which for real measurements usually means a column got copied wrong.

One outlier can move it a lot

Standard deviation is not resistant. The AP course lists the range and standard deviation as nonresistant measures of variability, and the median and IQR as resistant ones, because outliers barely touch that second pair.

The squaring is the reason. A value 40 units from the mean contributes $40^2 = 1600$ to the sum of squares, while a value 4 units away contributes 16. One far-out point can dominate the entire sum, as the second worked example below shows: changing a single value leaves the median exactly where it was and multiplies $s$ by more than five.

There is a second consequence worth knowing. One of the two outlier rules in the AP course flags any value more than 2 standard deviations from the mean, and that rule uses the very $s$ the outlier has already inflated, which can hide the point it was meant to catch. The 1.5 IQR rule does not have that weakness, because quartiles ignore how extreme the extremes are.

When a distribution is strongly skewed or holds outliers, report the median and IQR instead. The comparison of standard deviation and IQR covers which one is the honest summary in each case.