Use this standard deviation calculator to find how spread out your data points are from the mean. Enter your numbers, choose sample or population, and get instant results with step-by-step workings.
Standard deviation is a measure of dispersion. It tells you whether your data values cluster tightly around the average or scatter widely. A low standard deviation indicates the numbers stay close to the mean. A high standard deviation means they spread far from it.
This calculator handles both sample and population standard deviation. It also returns the variance, mean, and sum of squared differences from the mean so you can verify every step.
Calculate standard deviation step-by-step
Here is how to calculate standard deviation using our calculator:
- Enter your data set. Type or paste numbers separated by commas, spaces, or line breaks. You can paste data directly from Excel or Google Sheets.
- Choose sample or population. Pick "sample" if your data represents part of a larger population. Pick "population" if your data set covers the entire population.
- Click Calculate. The calculator returns the standard deviation, variance, mean, count, and a full breakdown of each step.
The results update instantly. If you change a number or add more data points, recalculate to refresh every output.
Should I use sample or population standard deviation? Use sample standard deviation when your data comes from a subset of a larger group (survey responses, lab measurements, test scores from one class). Use the population standard deviation when you have data for every member of the group you care about (every employee's salary, every sensor reading in a batch). When in doubt, sample is almost always the safer choice.
Standard deviation formula
Two versions of the formula exist. The only difference is what you divide by before taking the square root.
Sample standard deviation formula
The sample standard deviation formula is:
s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
Where:
- xᵢ = each data point
- x̄ = the sample mean
- n = the sample size (number of data points)
- Σ = sum of all squared differences from the mean
You divide by n − 1 instead of n. This correction (called Bessel's correction) accounts for the fact that a sample tends to underestimate the true variability of the larger population. Dividing by n − 1 removes that bias.
Why does sample variance divide by n−1 instead of n? When you calculate the sample mean from your own data, you've already used one piece of information. That leaves only n − 1 independent data points to estimate spread. Dividing by n − 1 produces an unbiased estimate of the population variance.
Population standard deviation
The population standard deviation formula is:
σ = √[ Σ(xᵢ − μ)² / N ]
Where:
- σ (sigma) = the population standard deviation
- μ = the population mean
- N = the total number of values in the population
Because you already have every value in the population, no correction is needed. You divide by N.
Population vs sample standard deviations
The formulas look nearly identical. The practical difference matters.
| Detail | Sample | Population |
|---|---|---|
| Data covers | Part of a group | The entire population |
| Symbol | s | σ |
| Denominator | n − 1 | N |
| Use when | You sampled from a larger population | You measured every member |
Choosing wrong inflates or deflates your result. Using population SD on sample data underestimates the true dispersion. Using sample SD on a full population slightly overestimates it.
Find the sample standard deviation
Follow these steps to find the sample standard deviation:
- Find the mean. Add all data values and divide by n (the sample size).
- Subtract the mean from each data point. These are the deviations from the mean.
- Square each difference. This removes negative signs and weights larger deviations more heavily.
- Sum the squared differences.
- Divide by n − 1 to get the sample variance.
- Take the square root of the variance. That result is the sample standard deviation.
Why can't I calculate sample SD from one number? With a single data point, n − 1 equals zero. Dividing by zero is undefined, so you need at least two values.
Find standard deviation for a population
The process is the same except at step 5: divide by N (the total count) instead of n − 1. The result is the population standard deviation (σ).
If you have access to every data point in the group you're studying, this is the correct choice.
How to calculate standard deviation by hand
Working through a small example by hand makes the formula concrete.
Data set: 5, 9, 8, 12, 6
- Find the mean. (5 + 9 + 8 + 12 + 6) / 5 = 40 / 5 = 8
- Subtract the mean from each data point.5 − 8 = −3 9 − 8 = 1 8 − 8 = 0 12 − 8 = 4 6 − 8 = −2
- Square each difference.(−3)² = 9 1² = 1 0² = 0 4² = 16 (−2)² = 4
- Sum the squared differences. 9 + 1 + 0 + 16 + 4 = 30
- Divide. For a sample: 30 / (5 − 1) = 7.5. For a population: 30 / 5 = 6.
- Take the square root. Sample SD = √7.5 ≈ 2.74. Population SD = √6 ≈ 2.45.
That is the entire calculation behind this calculator. The tool simply automates it for any number of data points.
Standard deviation examples
Example 1: Test scores
A teacher records quiz scores for 8 students: 5, 9, 8, 12, 6, 10, 6, 8.
- Mean: (5 + 9 + 8 + 12 + 6 + 10 + 6 + 8) / 8 = 64 / 8 = 8
- Squared differences: 9, 1, 0, 16, 4, 4, 4, 0
- Sum of squared differences: 38
- Sample variance: 38 / 7 ≈ 5.43
- Sample standard deviation: √5.43 ≈ 2.33
These scores stay fairly close to the mean. A standard deviation of about 2.33 on a quiz centered at 8 shows moderate spread.
Example 2: Identical groups
Data: 5, 5, 5, 9, 9, 9, 10, 10, 10.
- Mean: 72 / 9 = 8
- Squared differences: 9, 9, 9, 1, 1, 1, 4, 4, 4
- Sum: 42
- Population SD (entire group): √(42 / 9) = √4.67 ≈ 2.16
- Sample SD: √(42 / 8) = √5.25 ≈ 2.29
Notice how the sample SD is slightly larger. That n − 1 adjustment always pushes the estimate up compared to the population version.
What does standard deviation actually tell me? It tells you the typical distance from the mean. In Example 1, a single quiz score lands about 2.33 points away from 8 on average. Pair it with the mean, and you have a concise summary of both center and spread.
A small standard deviation means the data clusters tightly. A large one signals wide variability, which can affect decisions around confidence intervals, margin of error, and data analysis in general.
Standard deviations and the calculation behind this calculator
This standard deviation calculator follows the exact formulas above. It parses your input, computes the mean, calculates each squared difference, sums them, divides by the correct denominator, and takes the square root of the variance.
Variance and standard deviation are closely related. Variance is the square of the standard deviation. Some fields (finance, ANOVA) work in variance because squared units simplify certain equations. Standard deviation converts back to the original units, making it easier to interpret.
How the calculator handles your data:
- Accepts integers, decimals, and negative numbers.
- Ignores blank entries and extra spaces.
- Reports both sample and population results so you can compare.
- Shows intermediate values (mean, count, sum of squares) for verification.
When standard deviation connects to other tools: Standard deviation feeds into many related statistics. The standard error divides the SD by the square root of n, useful for estimating how precise a sample mean is. Confidence intervals use the SD (or standard error) to build a range around an estimate. Relative standard deviation expresses the SD as a percentage of the mean, which helps compare variability across data sets with different scales.
This calculator provides estimates for learning and planning. It is not a substitute for professional statistical analysis. For coursework, research, or business decisions with significant consequences, verify results with your instructor, analyst, or appropriate professional.