Standard Deviation

Statistics is the math of data. You collect numbers, describe what they show, and then decide how much of the pattern could just be chance.

Standard Deviation: the central idea

Standard deviation measures typical distance from the mean in the original units. Variance is the corresponding average squared distance.

Words you need

Standard deviation

The standard deviation is the typical distance between a value and the mean, reported in the same units as the original data.

Variance

The variance is the average squared distance from the mean, so it is the standard deviation before you take the square root and its units are squared.

Deviation

A deviation is one value minus the mean, and the deviations of a data set always add up to zero, which is why they get squared.

Population standard deviation

The population standard deviation divides the sum of squared deviations by $N$, the size of the entire group you care about, and is written with the Greek letter sigma.

Sample standard deviation

The sample standard deviation divides the sum of squared deviations by $n-1$ and is written $s$, because it estimates the spread of a big group from a small piece of it.

Bessel's correction

Bessel's correction is the choice to divide by $n-1$ instead of $n$, and it exists because the sample mean is pulled toward the sample, making the raw squared distances a little too small.

What to know before this lesson

Know the mean, subtraction with signed values, squares, square roots, and the difference between a sample and population.

If one of those prerequisites is uncertain, use the Statistics subject guide to locate the earlier concept before memorizing a procedure.

Standard deviation: a worked example

Every step, with the arithmetic

1. Step 1 - Write the five quiz scores7, 7, 10, 13, 13
2. Step 2 - Add them and divide to get the mean7 + 7 + 10 + 13 + 13 = 50, so the mean is 50 / 5 = 10
3. Step 3 - Subtract the mean from each score7 - 10 = -3, 7 - 10 = -3, 10 - 10 = 0, 13 - 10 = 3, 13 - 10 = 3
4. Step 4 - Square each deviation so nothing cancels(-3)^2 = 9, (-3)^2 = 9, 0^2 = 0, 3^2 = 9, 3^2 = 9
5. Step 5 - Add the squares9 + 9 + 0 + 9 + 9 = 36
6. Step 6 - Divide by n - 1 for a samplesample variance = 36 / 4 = 9, so the sample standard deviation is the square root of 9 = 3 points
7. Step 7 - Divide by N instead for a populationpopulation variance = 36 / 5 = 7.2, so the population standard deviation is about 2.68 points
8. Step 8 - Sanity-check the sizethe scores run from 7 to 13, a spread of 6 points, and a typical distance from 10 of about 3 points fits that range

If every observation is identical, every deviation from the mean is zero, so both variance and standard deviation equal zero.

The full deviation table for the five quiz scores 7, 7, 10, 13, 13 (mean = 10)

The step-by-step method for standard deviation

1. Find the mean and compute every deviation $x_i-\bar x$ or $x_i-\mu$.
2. Square and add the deviations, then divide by $n-1$ for a sample or $N$ for a population.
3. Take the square root and interpret the size relative to the data scale and distribution.

How to check your answer

Confirm the result is nonnegative, uses the original measurement unit, and becomes larger when observations spread farther from the mean.

Verify that probabilities stay between 0 and 1, measures of spread are nonnegative, and numerical conclusions match the shape, units, and direction visible in the data.

A mistake that changes the mathematics

Do not mix sample and population denominators. Dividing by $n-1$ accounts for estimating the sample mean from the same data.

Where you will actually use this

Comparing two climates

San Diego and Kansas City can share nearly the same yearly mean temperature, but Kansas City has a much larger standard deviation, which is exactly why one needs a winter coat and the other does not.

Factory quality control

A machine cutting bolts to 50 mm is judged by the standard deviation of its output; if it grows from 0.1 mm to 0.4 mm the operators stop the line even though the mean is still 50 mm.

Test scores and z-scores

Dividing your distance from the mean by the standard deviation turns any raw score into a z-score, which is how a 1300 SAT and a 29 ACT can be compared on one scale.

How standard deviation connects to the rest of statistics

• Mean, median, and mode — Every standard deviation calculation starts by finding the mean, because deviations are measured from it.
• Linear regression — The spread of the residuals around a regression line is measured with a standard deviation, and it decides how much to trust a prediction.
• Probability foundations — The 68-95-99.7 rule turns a standard deviation into a probability by saying how much of a bell-shaped data set falls within one, two, or three deviations of the mean.

Try a transfer problem

Compute and compare the standard deviations of $4,4,4,4$ and $1,3,5,7$ without changing their common mean.

Show the worked answer

For 4, 4, 4, 4 the mean is 16 / 4 = 4. Every deviation is 4 - 4 = 0, so every squared deviation is 0 and the sum of squares is 0. Dividing 0 by 3 (sample) or by 4 (population) still gives 0, and the square root of 0 is 0. The standard deviation is exactly 0 points. For 1, 3, 5, 7 the mean is (1 + 3 + 5 + 7) / 4 = 16 / 4 = 4 as well. The deviations are -3, -1, 1, 3, the squares are 9, 1, 1, 9, and the sum of squares is 20. As a sample, 20 / 3 = 6.67 and the standard deviation is about 2.58; as a population, 20 / 4 = 5 and it is about 2.24. Both sets balance at 4, so no measure of center can tell them apart. Only the standard deviation reveals that the first set has no spread at all while the second set typically sits two and a half units away from its mean.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to statistics practice questions.

Questions about standard deviation

Why divide by n - 1 instead of n for a sample?

The sample mean is computed from the very same numbers you are measuring against, so it sits a little closer to them than the true population mean would. That makes the sum of squared deviations slightly too small. Dividing by $n-1$ instead of $n$ makes the answer a bit bigger and corrects the bias, which is why 36 / 4 = 9 beats 36 / 5 = 7.2 when the five scores are only a sample.

Can a standard deviation ever be negative?

No. Every deviation gets squared, and squares are never negative, so the sum of squares is at least 0 and its square root is at least 0. A standard deviation of exactly 0 means every single value equals the mean; anything larger means there is spread. If you compute a negative answer you dropped a sign or forgot to square.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean and standard deviation is its square root. For the scores 7, 7, 10, 13, 13 the sample variance is 9 squared points and the standard deviation is 3 points. Variance is easier to add up in formulas, but standard deviation is the one you can read off a graph because it uses the original units.

What is the 68-95-99.7 rule?

For data shaped like a bell curve, about 68% of the values land within one standard deviation of the mean, about 95% within two, and about 99.7% within three. If adult heights have a mean of 170 cm and a standard deviation of 8 cm, roughly 68% of adults fall between 162 cm and 178 cm, and someone at 194 cm is three deviations out, which is rare.