Financial term · Reviewed August 17, 2026
Standard deviation
Standard deviation: A measure of how spread out a set of values is around its mean. A small standard deviation means values cluster tightly; a large one means they scatter widely. The Standard Deviation Calculator returns this figure from population σ = √(Σ(x−mean)²÷n); sample s = √(Σ(x−mean)²÷(n−1)). At that calculator’s documented worked-example inputs the population standard deviation is 4.5; holding every other input steady, moving value 1 from 10 to 14 moves it from 4.3 to 4.9, a spread of 0.585.
Standard deviation gives the typical distance from the average, in the data's own units, and that is what makes a mean interpretable. Two datasets with the same average can be a tight cluster or a wide scatter, and the average alone cannot tell them apart. A single example makes it concrete. Two classes average 70 on a test. In one, almost everybody scored between 65 and 75. In the other, half the class scored 95 and half scored 45. Same average. Completely different classes, and only the spread shows it.
| Term | Standard deviation |
|---|---|
| Category | Statistics & Probability |
| How it is found | Population σ = √(Σ(x−mean)²÷n); sample s = √(Σ(x−mean)²÷(n−1)) |
| Worked example | Population standard deviation: 4.5 |
| Computed by | Standard Deviation Calculator |
| Related terms | 6 defined in this glossary |
Before you rely on it
What to check
Note whether the sample or population formula was used.
The common error
Where people go wrong with standard deviation
Using the population formula on a sample. On small datasets the two differ noticeably, and the sample version is nearly always what was wanted.
Worked through
Standard deviation in plain numbers
Take the data 4, 8, 6, 5 and 7. The mean is 6. Each value's distance from the mean is −2, 2, 0, −1 and 1, and squaring those gives 4, 4, 0, 1 and 1, which sum to 10. Divide by 5 for the population variance, 2, and take the square root: a standard deviation of 1.41. If the numbers are a sample from a larger group, divide by 4 instead and the sample standard deviation is 1.58. Two datasets can share a mean and differ hugely in spread: 5, 6, 7 and 0, 6, 12 both average 6, with standard deviations of 1 and 6; the measure captures that difference.
Easily confused
Standard deviation and its neighbours
Standard deviation is not the variance, the range or the standard error. Variance is the average squared distance and is in squared units; the standard deviation is its square root, back in the original units, and the range is only the gap between highest and lowest values. The standard error describes uncertainty in an estimated mean and shrinks with sample size. Population and sample versions differ in the denominator. Volatility in finance is essentially a standard deviation of returns.
In real decisions
Where standard deviation turns up
Scientists use it to describe variability, statisticians use it in confidence intervals and tests, and quality engineers use it in control charts; investors treat it as a measure of risk; teachers use it to see how spread out scores are. In a roughly bell-shaped distribution, about 68% of values fall within one standard deviation of the mean and about 95% within two. That rule of thumb turns the number into a practical picture.
Know the edges
Where standard deviation stops being reliable
It is sensitive to outliers and to skewed data, where the median and quartiles may describe the spread better; the 68-95 rule holds only for approximately normal data. Small samples give unstable estimates. It says nothing about the shape of the distribution. Plot the data as well, and choose the population or sample formula to match your data.
Quick answers
What people ask about standard deviation
When do I divide by n minus one? For a sample, to correct the tendency to underestimate the spread of the population. Use n when you have the whole population.
What does a large standard deviation mean? The values are spread widely around the mean. A small one means they cluster closely.
Put it to work
How standard deviation is calculated
At the standard deviation calculator’s worked-example inputs, population standard deviation is 4.5, sample standard deviation is 4.9. Moving value 1 from 10 to 14 moves that first figure from 4.3 to 4.9.
Open the Standard Deviation Calculator →
These are standard formulas from introductory statistics. Real-world inference depends on assumptions, normality, independence, a genuinely random sample, that this calculator does not check for you.
Related vocabulary
Terms used alongside this one
Keep reading
Nearby in the glossary
- ← Standard deduction
- Standard drink →
- All 174 defined termsThe complete Free Calculators Online financial glossary.