This average and standard deviation calculator finds the mean and standard deviation of your data in one step, and shows the full working. Paste a list of numbers or a frequency table, then choose sample or population. You get the mean, standard deviation, variance, coefficient of variation, standard error, median, mode and range. A deviations table walks through the arithmetic line by line, and a frequency plot shows the spread around the mean.
The formulas match the NIST/SEMATECH e-Handbook and OpenStax Introductory Statistics, so the numbers agree with a statistics textbook, a TI calculator or a spreadsheet using the same mode.
How to use the calculator
- Choose List of values and paste your numbers, or choose Frequency table and type one “value frequency” pair per line.
- Pick Sample (n − 1) if the numbers are drawn from a bigger group, or Population (N) if they are the whole group.
- Read the mean and standard deviation at the top. The other mode’s result is shown too, so you can see the difference.
- Open the step-by-step working to copy the deviations table into homework or a lab report. Each row also shows the value’s z-score.
- Copy the link to share the exact data set and settings.
Average and standard deviation formulas
The average (arithmetic mean) is the sum of the values divided by how many there are: x̄ = Σx ÷ n. Standard deviation measures how far the values typically fall from that mean. OpenStax describes it as “a numerical measure of the overall amount of variation in a data set”. It is small when the values cluster near the mean and larger when they spread out.
| Quantity | Sample | Population |
|---|---|---|
| Mean | x̄ = Σx ÷ n | μ = Σx ÷ N |
| Variance | s² = Σ(x − x̄)² ÷ (n − 1) | σ² = Σ(x − μ)² ÷ N |
| Standard deviation | s = √s² | σ = √σ² |
| From a frequency table | s = √[Σf(x − x̄)² ÷ (n − 1)] | σ = √[Σf(x − μ)² ÷ N] |
| Coefficient of variation | CV = s ÷ x̄ | σ ÷ μ |
| Standard error of the mean | s ÷ √n (estimate) | σ ÷ √n |
Variance is the quantity inside the square root, so it is in squared units. Standard deviation is back in the units of the data, which is why it is the number usually reported next to an average, as in “mean 72 kg, SD 9 kg”.
Step-by-step example with a deviations table
Take the eight values 2, 4, 4, 4, 5, 5, 7, 9. Their sum is 40 and n = 8, so the mean is 40 ÷ 8 = 5. Subtract 5 from each value and square the result:
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
| Sum | 0 | 32 |
The deviations always add to zero, which is a handy check on the mean. The squared deviations add to 32. For a population, divide by N = 8: σ² = 4 and σ = 2. For a sample, divide by n − 1 = 7: s² ≈ 4.5714 and s ≈ 2.138. The same data give two different standard deviations, so always say which one you used.
Sample vs population standard deviation
The only difference between the two formulas is the denominator. Divide by N when your list is the entire population you care about. Divide by n − 1 when the list is a sample and you want to estimate the spread of the larger population. OpenStax says the sample variance is an estimate of the population variance, and dividing by n − 1 gives a better estimate. The NIST e-Handbook defines the standard deviation with N − 1 for the same reason.
The n − 1 is also where the degrees of freedom of a one-sample t procedure come from. The deviations must sum to zero, so only n − 1 of them are free to vary. See the degrees of freedom calculator for how that carries into t-tests and chi-square tests. In practice, choose sample for survey responses, lab measurements, quality-control samples and most homework. Choose population for things like the scores of every student in your own class, when the class itself is the subject.
| Your data | Denominator | Symbol | Spreadsheet function |
|---|---|---|---|
| Sample from a larger group | n − 1 | s | STDEV.S |
| Every member of the group | N | σ | STDEV.P |
Standard deviation from a frequency table
When values repeat, typing each one is slow. Enter each distinct value once with its frequency f instead. The sum of squared deviations becomes Σf(x − x̄)², and n is the total of the frequencies. OpenStax Example 2.32 uses the ages of 20 fifth-graders: 9 (×1), 9.5 (×2), 10 (×4), 10.5 (×4), 11 (×6), 11.5 (×3). The frequency form gives a mean of 10.525 years and a sample variance of 0.5125, so s ≈ 0.72 years. That matches the textbook’s s = 0.72, and the calculator loads this example when you switch to frequency mode.
A common mistake is to divide by the number of rows (6 here) instead of the total frequency (20). Another is to forget to multiply each squared deviation by its frequency. The f(x − x̄)² column in the working makes both errors easy to spot.
How to interpret the standard deviation
A standard deviation on its own is just a distance. It becomes meaningful next to the mean and the shape of the data. OpenStax gives two rules of thumb. For any data set, at least 75% of the values lie within two standard deviations of the mean. For bell-shaped, symmetric data, about 68% lie within one standard deviation and about 95% within two. The calculator prints the mean ± 1 SD and ± 2 SD intervals so you can check these against your own data.
To place one value, turn it into a z-score, z = (x − x̄) ÷ s: the number of standard deviations it sits above or below the mean. The deviations table shows z for every row. To convert a z-score into a percentile under a normal model, use the z-score to percentile calculator. If your data are skewed, rank the value against the data itself with the percentile rank calculator instead.
Standard deviation is sensitive to outliers because deviations are squared. A single extreme value can inflate it a lot. If the frequency plot shows a long tail or an isolated bar, draw a box plot to flag outliers with the 1.5 × IQR rule. For skewed data, the median and interquartile range are often better summaries than the mean and standard deviation.
Coefficient of variation and standard error
Coefficient of variation (CV)
NIST defines the coefficient of variation as the ratio of the standard deviation to the mean, cv = s/x̄, often written as a percentage. Because units cancel, it compares variability across data sets with different scales. The values 150, 160, 170 and 10, 20, 30 have the same standard deviation (10 and 10), but CVs of 6.3% and 50%. Relative to their means, the second set varies far more. NIST warns that the CV is only meaningful for ratio-scale data with a true zero, such as mass or length, and not for temperatures in °C or °F. It also becomes unstable when the mean is near zero. The calculator therefore prints “not meaningful” when the data contain negatives or the mean is not positive.
Standard error of the mean (SEM)
The standard error describes how precisely the average is known, not how spread out the individual values are. OpenStax §7.1 calls σ/√n the standard error of the mean. When σ is unknown, confidence intervals use s/√n in its place (OpenStax §8.2). For the eight-value example, SEM = 2.138 ÷ √8 ≈ 0.756. Quadrupling the sample size halves the standard error, but it does not shrink the standard deviation. Report SD to describe the data and SEM to describe the precision of the mean.
Common mistakes
- Mixing up modes. Spreadsheets, calculators and textbooks each default differently. Check whether you need s (n − 1) or σ (N).
- Squaring after summing. Square each deviation first, then add. Σ(x − x̄) is always zero, so squaring that sum gives nothing useful.
- Forgetting the square root. Stopping at variance leaves the answer in squared units.
- Rounding the mean too early. Carrying a rounded mean through every row adds error. This calculator uses full precision and a numerically stable update, so data like 1,000,000,001 and 1,000,000,002 still give the correct spread.
- Reporting SEM as SD. The standard error is always smaller once n > 1, which makes data look less variable than they are.
Frequently asked questions
How do I calculate average and standard deviation together?
Add the values and divide by the count to get the average. Then subtract the average from every value, square each difference, and add the squares. Divide that sum by n − 1 for a sample or N for a population, and take the square root. The calculator shows every step in a deviations table.
What is the calculator for standard deviation of a sample?
Choose “Sample (n − 1)” above. The sample standard deviation is s = √[Σ(x − x̄)² ÷ (n − 1)], the formula in the NIST e-Handbook and OpenStax. It is the right choice whenever your numbers are a subset of a larger group you want to describe.
Should I use sample or population standard deviation?
Use sample (n − 1) when the data are observations drawn from a larger group, which covers most lab, survey and classroom data. Use population (N) only when the list contains every member of the group you care about, such as all 30 students in one class when the class is the whole topic.
Why is the sample standard deviation larger than the population one?
It divides the same sum of squared deviations by n − 1 instead of n. OpenStax explains that the sample variance estimates the population variance, and dividing by n − 1 gives a better estimate. The gap shrinks as n grows.
How do I find standard deviation from a frequency table?
Switch the data format to “Frequency table” and enter one “value frequency” pair per line. The calculator uses the OpenStax frequency formula s = √[Σf(x − x̄)² ÷ (n − 1)], where n is the total of the frequencies, not the number of rows.
What is the difference between variance and standard deviation?
Variance is the average squared deviation. Standard deviation is its square root, so it is back in the original units. If heights are in centimetres, the standard deviation is in centimetres and the variance is in square centimetres.
What does a standard deviation of 0 mean?
Every value is identical, so there is no spread at all. A population of one value always has σ = 0. The sample standard deviation of a single value is undefined, because n − 1 = 0.
What is a good standard deviation?
There is no universal good value. Judge it relative to the mean and to what you are measuring. The coefficient of variation (s ÷ mean) helps compare spread across data sets with different units or very different means.
What is the standard error of the mean?
It is the standard deviation divided by the square root of the sample size, σ/√n, and estimated in practice by s/√n. It measures how much the average itself would vary from sample to sample, not how spread out the individual values are.
Can standard deviation be negative?
No. It is the square root of a sum of squares divided by a positive count, so it is always zero or positive. A negative result means a sign error in a hand calculation.
Sources & method
- NIST/SEMATECH e-Handbook of Statistical Methods §1.3.5.6 — Measures of scale (variance, standard deviation, AAD, MAD)
- OpenStax Introductory Statistics 2e §2.7 — Measures of the spread of the data (sample, population and frequency-table formulas)
- OpenStax Introductory Statistics 2e §7.1 — Central limit theorem for sample means (standard error of the mean σ/√n)
- OpenStax Introductory Statistics 2e §8.2 — Single population mean using the t distribution (s/√n, degrees of freedom n − 1)
- NIST Dataplot Reference Manual — Coefficient of variation (cv = s/x̄, ratio-scale requirement)
- Microsoft Support — STDEV.S function (sample, n − 1) and STDEV.P (entire population)
Results are estimates for general information. Found an error? It helps everyone — see our methodology.