"Types of standard deviation" is searched by students who have met two different formulas in two different places and want to know which is which, and by people who have seen "relative", "pooled" or "weighted" standard deviation in a report and want to know whether those are new things. The honest answer is that there are two standard deviations, population and sample, and everything else is either one of them applied in a particular way or a related measure with a different name. This page gives the two, with one set of numbers worked both ways so the difference is visible, then the others in a table, then how to read a standard deviation once you have one. The printable sheet has the formulas, the worked example and the practice on two pages.
Download the standard deviation formula sheet (PDF) (both formulas with when to use each, the six steps, the worked example both ways, the other types in a table, the 68-95-99.7 rule, five practice questions with answers; two pages, free).
The two types: population and sample
| Population standard deviation (σ, sigma) | Sample standard deviation (s) | |
|---|---|---|
| Use when | You have every value in the group you care about: every student in the class, every item made today, every score in the league | Your data is a sample and you want to estimate the spread of the larger population it came from |
| Formula | σ = √[ Σ(x − μ)² / N ] | s = √[ Σ(x − x̄)² / (n − 1) ] |
| Divide by | N, the number of values | n − 1 |
| Mean symbol | μ (mu), the population mean | x̄ (x-bar), the sample mean |
| Excel and Google Sheets | =STDEV.P(range) | =STDEV.S(range), or the older =STDEV |
| Calculator | σx or σn | sx or σn−1 |
| Size | Smaller | Always slightly larger; the difference shrinks as n grows |
Why n − 1. A sample's values are, on average, closer to their own mean than to the population mean, because the sample mean is calculated from them. Dividing by n therefore underestimates the true spread, and dividing by n − 1 corrects it; the correction is named after Friedrich Bessel. The practical rule: if you would describe your data as "all of them", divide by N; if you would describe it as "some of them", divide by n − 1.
The six steps, for either type
- Find the mean.
- Subtract the mean from each value (some differences are negative).
- Square each difference.
- Add the squares.
- Divide by N for a population, or by n − 1 for a sample. This is the variance.
- Take the square root. This is the standard deviation.
One worked example, both ways
Six quiz scores: 4, 8, 6, 5, 3, 7. The mean is 33 ÷ 6 = 5.5.
| Value x | x − mean | (x − mean)² |
|---|---|---|
| 4 | −1.5 | 2.25 |
| 8 | 2.5 | 6.25 |
| 6 | 0.5 | 0.25 |
| 5 | −0.5 | 0.25 |
| 3 | −2.5 | 6.25 |
| 7 | 1.5 | 2.25 |
| Sum | 17.5 |
| Population (these six are the whole class) | Sample (these six are a sample of a year group) | |
|---|---|---|
| Variance | 17.5 ÷ 6 = 2.917 | 17.5 ÷ 5 = 3.5 |
| Standard deviation | √2.917 = 1.71 | √3.5 = 1.87 |
Same numbers, same steps, one different divisor, and the sample answer is about ten percent larger. With sixty values instead of six the two answers would differ by less than one percent, which is why the choice matters most in school examples with small n.
Which one does the exam want?
| Course | Default |
|---|---|
| AP Statistics | Sample (n − 1); the calculator's sx |
| US high school Algebra 2 and Pre-Calculus | Usually population (N), with the formula given |
| A-level Mathematics (England) | Both on the formula sheet; the question says "sample" or gives the population; S1 and the large-data-set questions mostly use the population form |
| IB Mathematics (AA and AI) | Population, from the calculator; the IB does not test n − 1 |
| CBSE Class 11 (Statistics chapter) | Population (divide by N or Σf) |
| University introductory statistics, research, Excel reports | Sample, almost always |
When the question is silent, use the formula your course teaches and write "population" or "sample" next to the answer; markers credit a correct, labelled method.
The other "types" people mean
| Name | What it is | Formula or rule |
|---|---|---|
| Variance | The standard deviation squared: the step before the square root | σ² or s² |
| Standard error of the mean | How much a sample mean varies from sample to sample; what confidence intervals are built on | s ÷ √n |
| Relative standard deviation (coefficient of variation) | The spread as a percentage of the mean, so data in different units or of different sizes can be compared | (s ÷ mean) × 100% |
| Pooled standard deviation | One spread for two or more groups that are assumed to have the same variance; used in two-sample t-tests | √[ ((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2) ] |
| Weighted standard deviation | Each value counts with a weight: frequencies in a grouped table, or survey weights | √[ Σw(x − x̄)² ÷ Σw ] |
| Standard deviation of grouped data | The weighted version with class midpoints as x and frequencies as w; CBSE Class 11 and GCSE Statistics | As above |
| Mean absolute deviation (not a standard deviation) | The average distance from the mean without squaring; taught in grades 6 to 8 before standard deviation | Σ |
| Standard deviation of a probability distribution | The spread of a random variable rather than of data; binomial: √(np(1 − p)) | Depends on the distribution |
So a report that gives "RSD 4%" is giving the sample standard deviation divided by the mean; a paper that gives a "pooled SD" has combined two groups; and a textbook chapter on "standard deviation of grouped data" is the weighted formula. None of them is a third kind of standard deviation.
How to read a standard deviation
A small standard deviation means the values cluster near the mean; a large one means they spread out. For data that is roughly bell-shaped (heights, test scores in a large group, measurement errors), the 68-95-99.7 rule applies: about 68 percent of values lie within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three. A test with a mean of 70 and a standard deviation of 8 has most scores between 62 and 78; a score of 90 is two and a half standard deviations above the mean and is in the top one percent or so. Two classes with the same mean of 60 and standard deviations of 4 and 12 are very different classes: in the first almost everyone scored between 52 and 68; in the second a third of the class is outside 48 to 72.
Practice
- Find the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.
- Find the sample standard deviation of 10, 12, 23, 23, 16, 23, 21, 16.
- A sample has a variance of 16. What is its standard deviation?
- A bell-shaped set has mean 50 and standard deviation 5. What range holds about 95 percent of the values?
- Two classes both have a mean of 60; one has a standard deviation of 4 and the other 12. Which is more consistent?
Answers: 1) The mean is 5; the squared deviations sum to 32; 32 ÷ 8 = 4; σ = 2. 2) The mean is 18; the squared deviations sum to 192; 192 ÷ 7 = 27.43; s = 5.24. 3) 4. 4) 40 to 60. 5) The first.
Download the standard deviation formula sheet (PDF)
Where to get help
A 1-on-1 online class with a tutor who holds a master's in mathematics, for AP Statistics, Algebra 2, A-level and IB mathematics, CBSE Class 11 and 12 and university introductory statistics: the formulas taught from a worked example on the shared screen, the calculator and spreadsheet functions alongside, and the exam board's own questions from the first class. The recording and notes after every class, the same tutor all year, US$10 to $13 an hour. The AP Statistics and statistics tutoring pages have the courses; the free 30-minute first class is a real lesson on the topic the student is on this week.
Questions parents ask
1What are the two types of standard deviation?
2Why does sample standard deviation divide by n − 1?
3Which standard deviation do I use in an exam?
4What is the difference between standard deviation and variance?
5What does a standard deviation tell you?
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