Standard Deviation Calculator – Variance & SD
Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6 (Measures of Scale) · Source verified August 6, 2026
A standard deviation calculator quantifies how far the values in a dataset typically sit from their mean, in the same units as the data. It computes the population standard deviation, which applies when the values are the entire group, and the sample standard deviation, which estimates variability from a subset and divides by n - 1 under Bessel's correction. Statisticians, researchers, and quality control analysts use it to judge consistency and spread.
Paste or type a list of numbers separated by commas, spaces, or line breaks. This sd calculator computes the mean, standard deviation, and variance with step-by-step math.
Quick Answer
Compute the sample or population standard deviation for a set of numbers. Choose which one you need, then read the standard deviation alongside the variance, mean, and count.
Is this a sample or a population?
A subset drawn from a larger group. Divides by n - 1.
Separate with commas, spaces, semicolons or new lines. Decimals and negative numbers are fine. Nothing is calculated until every entry reads as a number.
Sample standard deviation (s)
13.490738
s = √(910 / 5) = 13.490738
On average, values sit about 13.490738 away from the mean of 18, in the same units as the numbers you entered. Whether that is a lot or a little depends entirely on what you are measuring and what a typical spread looks like for it. A standard deviation is not high or low on its own.
Where the values sit
Inspect a value
Choose a value to see its distance from the mean and that distance squared.
Describe this chart in words
Dot plot of 6 values on a horizontal number line, from 4 to 42. The sample mean x̄ is 18, marked with a dashed line. A hatched band spans one standard deviation either side of the mean, from 4.5092624 to 31.490738, a width of 13.490738 in each direction. Each value is drawn as a dot; repeated values are stacked so the count is visible. This is a picture of the entered data. It is not a distribution curve and assumes nothing about the shape of the data.
How this standard deviation was reached
- Formula for the selected mode
- s = √(Σ(xᵢ − x̄)² / (n − 1))
- Substitution
- s = √(910 / 5) = 13.490738
- Why this denominator
- Dividing by n − 1 rather than n is Bessel's correction. A sample's own mean sits closer to its values than the true population mean does, so squared deviations measured from it are too small on average. The smaller denominator compensates.
- Units
- Standard deviation is in the same units as your data, which is why it is usually the figure people quote. Variance is in those units squared, so it is harder to interpret directly but easier to work with algebraically.
| Value (x) | x − x̄ | (x − x̄)² |
|---|---|---|
| 4 | -14 | 196 |
| 8 | -10 | 100 |
| 15 | -3 | 9 |
| 16 | -2 | 4 |
| 23 | 5 | 25 |
| 42 | 24 | 576 |
| Sum of squared deviations | 910 | |
Examples
2, 4, 4, 4, 5, 5, 7, 9 (sample)
x̄ = 5 · s ≈ 2.1381 · σ = 2 if this were the whole population
4, 8, 15, 16, 23, 42 (sample)
x̄ = 18 · s ≈ 13.4907 · s² = 182
10, 12, 14, 16, 18 (sample)
x̄ = 14 · s ≈ 3.1623 · s² = 10
5, 5, 5, 5, 5 (identical values)
x̄ = 5 · s = 0 · no spread at all
How it works
Formula · s = √(Σ(xᵢ − x̄)² / (n − 1)) for a sample; σ = √(Σ(xᵢ − μ)² / N) for a population
Standard deviation measures how spread out the values in a data set are around the mean. A small standard deviation means most values are close to the mean; a large one means they are scattered.
Sample s · √(Σ(xᵢ − x̄)² / (n − 1))
Population σ · √(Σ(xᵢ − μ)² / N)
We report both because the right one depends on whether your data is the entire population or just a sample of it. For most real-world analysis, you want the sample standard deviation.
What is standard deviation?
This standard deviation calculator (or sd calculator) computes a single number that summarizes how spread out the values in a dataset are around the mean. The standard deviation, or std dev, is reported in the same units as the data. A standard deviation of zero means every value is identical; larger values mean wider spread.
For a longer overview written for first-time readers, see What Is Standard Deviation?.
Sample vs population standard deviation
There are two versions of standard deviation. The difference is the denominator inside the formula.
- Sample standard deviation (s) divides the sum of squared deviations by n − 1. Use it when your data is a sample drawn from a larger population.
- Population standard deviation (σ) divides by N. Use it only when your data is the entire population you care about.
The n − 1 in the sample formula is called Bessel's correction. It makes the sample variance an unbiased estimator of the population variance. The full comparison is in the Sample vs Population Standard Deviation guide.
When to use sample standard deviation
Use sample standard deviation when your data is a random sample drawn from a larger group, and you want to use the sample to say something about that larger group. Most real-world data fits this case: a class of 30 test scores standing in for all students, a survey of 500 voters standing in for the electorate, 12 measurements of a part standing in for the manufacturing run. Statistical software defaults to the sample version for the same reason.
When to use population standard deviation
Use population standard deviation only when your numbers describe the entire population you are interested in, with no inference to a larger group. Examples: every test score in a single class when the class is the population, the closing prices of a stock for every day this year, or every student in a school for a one-time school-wide report. If there is any larger group your data is meant to represent, switch to the sample version.
The standard deviation formula
The two formulas share the same shape. The numerator is the sum of squared deviations from the mean. The denominator differs.
Sample standard deviation
s = √(Σ(xᵢ − x̄)² / (n − 1))
Population standard deviation
σ = √(Σ(xᵢ − μ)² / N)
Variance relationship
standard deviation = √variance
For a longer breakdown of each piece of these formulas, see the Standard Deviation Formula guide. For the symbol cheat sheet (σ, s, μ, x̄, n, n − 1), see Standard Deviation Symbol.
How to calculate standard deviation step by step
- 1. Compute the mean of the data set.
- 2. Subtract the mean from each value to get the deviation of that value.
- 3. Square each deviation.
- 4. Sum the squared deviations.
- 5. Divide by N for the population variance, or by n − 1 for the sample variance.
- 6. Take the square root of the variance to get the standard deviation.
The calculator above runs all six steps as you type. To run them in Excel or Google Sheets, see How to Calculate Standard Deviation in Excel.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
A classic textbook example. Eight values whose mean is a clean integer, so the arithmetic stays easy to follow. Paste this list into the calculator above to verify each line.
- n = 8
- Sum: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
- Mean (x̄ = μ): 40 / 8 = 5
- Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
- Sum of squared deviations: 32
From the sum of squared deviations:
- Population variance: 32 / 8 = 4
- Population standard deviation: √4 = 2
- Sample variance: 32 / 7 ≈ 4.5714
- Sample standard deviation: √(32 / 7) ≈ 2.1381
Example summary
2, 4, 4, 4, 5, 5, 7, 9
μ = 5 · σ = 2 · s ≈ 2.1381
Variance vs standard deviation
Variance and standard deviation describe the same idea on two different scales. Variance is the average squared deviation from the mean; standard deviation is its square root. Variance is convenient for the math (it adds nicely when you combine independent variables); standard deviation is convenient for reading (same units as the data).
The full comparison, with worked numbers, is in Variance vs Standard Deviation. If you only want variance, the variance calculator runs the same math and reports just the squared figure.
Empirical rule (68-95-99.7)
For a normal distribution, the empirical rule (also known as the 68-95-99.7 rule) describes the percentage of data that falls within a specific number of standard deviations from the mean.
| Distance from the Mean | Percentage of Data |
|---|---|
| Within 1σ of the mean | 68% |
| Within 2σ of the mean | 95% |
| Within 3σ of the mean | 99.7% |
For example, if test scores average 75 with a standard deviation (σ) of 5, about 95% of scores will fall between 65 and 85 (within 2σ of the mean).
Common mistakes
- Dividing by N when you should divide by n − 1. If your data is a sample (almost always), use n − 1. Dividing by N gives you a slightly biased underestimate of the population standard deviation.
- Skipping the square step. The deviations from the mean must be squared before you add them. Without squaring, the positive and negative deviations cancel out and the sum is always zero.
- Reporting variance and calling it standard deviation. Variance is in squared units. Standard deviation is the square root of variance and is in the original units of the data.
- Treating standard deviation as a sign of accuracy. Standard deviation describes spread, not correctness. Precise measurements can be inaccurate, and accurate measurements can be imprecise. For comparing a value to an accepted reference, see the percent error calculator.
- Ignoring outliers. A single very large or very small value can pull standard deviation up sharply. Investigate outliers before reporting; do not silently drop them, but make sure they are not data-entry errors.
Related guides
- What Is Standard Deviation? plain-English overview for first-time readers.
- Standard Deviation Formula the formula explained piece by piece.
- Variance vs Standard Deviation how the two measures relate.
- Sample vs Population Standard Deviation when to divide by n − 1 versus N.
- How to Calculate Standard Deviation in Excel STDEV.S and STDEV.P in spreadsheets.
- Standard Deviation Symbol σ, s, μ, x̄, and the rest of the notation.
- Z Score Formula how standard deviation drives z scores.
Related calculators
- Variance Calculator for variance alone, in squared units.
- Average Calculator for the mean, median, mode, and range of the same list.
- Standard Error Calculator for standard error of the mean (SEM), reflecting sample mean precision.
- Expected value calculator
- Z Score Calculator to standardize a value against a mean and standard deviation.
- Percent Error Calculator when comparing a measurement to a fixed accepted value rather than a mean.
- Confidence Interval Calculator for the interval that uses standard deviation in the standard error.
- Margin of Error Calculator for the half-width that goes into a confidence interval.
- T-Test Calculator for one- or two-sample mean comparisons that use standard deviation.
Source: NIST/SEMATECH e-Handbook of Statistical Methods. Last reviewed: June 2026.
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Frequently asked questions
Standard deviation is a measure of how spread out the values in a data set are around the mean. A small standard deviation means most values cluster close to the mean; a large one means they are scattered. It is reported in the same units as the data, so it is easier to interpret than variance.
Compute the mean of the data set. Subtract the mean from each value and square the result to get the squared deviations. Add up the squared deviations. Divide that sum by N for the population variance, or by n − 1 for the sample variance. Take the square root of the variance to get the standard deviation.
Sample standard deviation: s = √(Σ(xᵢ − x̄)² / (n − 1)). Population standard deviation: σ = √(Σ(xᵢ − μ)² / N). The numerator is the sum of squared deviations from the mean. The denominator is n − 1 for samples (Bessel's correction) and N for populations. Take the square root to bring the result back into the original units.
Population standard deviation (σ) divides by N and treats your data as the entire population you care about. Sample standard deviation (s) divides by n − 1 and treats your data as a random sample drawn from a larger population. The n − 1 denominator (Bessel's correction) makes s an unbiased estimator of σ when your data is a sample.
Use sample standard deviation when your numbers are a sample drawn from a larger group you want to learn about. Use population standard deviation only when your numbers are the entire population. For most everyday data analysis, the sample version is the right pick, and most statistical software defaults to it.
Variance is the average squared deviation from the mean. Sample variance divides the sum of squared deviations by n − 1; population variance divides by N. Variance is in the squared units of the data (square dollars, square inches, etc.), which is hard to interpret directly. Standard deviation is the square root of variance and is reported in the original units.
Standard deviation is the square root of variance. The relationship works in both directions: if you have one, you can square or square-root it to get the other. They describe the same idea (spread around the mean) on two different scales.
No. Standard deviation is the square root of variance, and variance is itself a sum of squared values divided by a positive count, so both quantities are zero or positive. A standard deviation of zero means every value in the data set is identical; any other case is positive.
A high standard deviation means the values in the data set are spread out widely from the mean. Some values are much higher and some much lower. That spread is sometimes useful information (a wide range of incomes, test scores, or measurement noise) and sometimes a sign that the data needs to be cleaned, segmented, or summarized differently.
A low standard deviation means most values cluster close to the mean. The data set is tight and consistent. In a manufacturing context that often means a stable process; in a measurement context it often means precise instruments. A standard deviation of zero means every value is identical.
The empirical rule (also known as the 68-95-99.7 rule) states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This rule is a quick way to estimate the spread of a dataset and identify potential outliers.
A 'good' standard deviation depends entirely on the context and the scale of your data, as there is no single ideal number. A low standard deviation indicates that data points are clustered tightly around the mean (high consistency), while a high standard deviation indicates that they are spread out widely. To evaluate standard deviation relative to the size of the mean, you can calculate the coefficient of variation (standard deviation divided by the mean).
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