How to use the standard deviation calculator
This free standard deviation calculator helps you understand how spread out your data is. Follow these steps to calculate standard deviation and variance:
- Enter your numbers separated by commas, spaces, or line breaks in the input box at the top of this page.
- The calculator instantly computes the mean, standard deviation, and variance as soon as you type or paste.
- Additional statistics like median, minimum, maximum, and range appear automatically.
- Use the buttons to copy the full results or just the standard deviation value with one click, then paste them into your spreadsheet, report, or homework.
What is standard deviation?
Standard deviation is a measure of how spread out values are around the mean (average) of a dataset. A small standard deviation means the values are tightly clustered around the mean; a large standard deviation means they are more spread out.
In statistics, standard deviation is used to understand variability in test scores, experiment results, financial returns, and many other types of data.
Standard deviation formulas: population vs sample
This calculator can show both population standard deviation and sample standard deviation. For a set of numbers x₁, x₂, …, xₙ with mean μ, the population variance σ² and standard deviation σ are:
Variance (σ²) = ( (x₁ − μ)² + (x₂ − μ)² + ... + (xₙ − μ)² ) / n
Standard deviation (σ) = √σ²
Where μ is the mean (average) of all values and n is the number of values. This is the standard formula used when your data represents an entire population rather than a sample.
For sample standard deviation, you divide by n − 1 instead of n. This gives an unbiased estimate of the population variance when your data is just a sample:
Sample variance (s²) = ( (x₁ − x̄)² + ... + (xₙ − x̄)² ) / (n − 1)
Sample standard deviation (s) = √s²
Example: calculating population standard deviation by hand
Consider the numbers 4, 7, 9, 10, and 12.
- Mean = (4 + 7 + 9 + 10 + 12) ÷ 5 = 8.4
- Squared differences from the mean: (4 − 8.4)², (7 − 8.4)², …
- Variance = average of squared differences ≈ 8.24
- Standard deviation = √8.24 ≈ 2.87
Enter these values into the calculator above to confirm the same standard deviation and variance instantly.
Standard deviation vs variance vs range
This calculator shows several measures of spread side by side:
- Variance is the average of squared distances from the mean.
- Standard deviation is the square root of variance and has the same units as the original data.
- Range is simply max − min and gives a quick sense of the overall spread.
Population vs sample standard deviation
Use population standard deviation when your dataset includes every value you care about (for example, all 30 students in a specific class). In this case, you divide by n.
Use sample standard deviation when your dataset is a subset of a larger group (for example, 30 customers sampled from thousands). In this case, you divide by n − 1 to correct for the fact that you are estimating the population spread from limited data.
The toggle in the results panel lets you switch between population and sample standard deviation instantly, so you can see both views of your data.
Mean, variance, and standard deviation: how they relate
Mean is the central value of your dataset, variance measures the average squared distance from that center, and standard deviation is simply the square root of variance.
In simple terms: you start with the mean, look at how far each value is from the mean, square those distances, average them (variance), and then take a square root to bring the units back to the same scale as your original data (standard deviation).
The calculator shows all three together so you can quickly see both the central tendency (mean) and the amount of variability (variance and standard deviation).
Standard deviation formula explained simply
To compute standard deviation step by step:
- Find the mean (average) of your numbers.
- Subtract the mean from each value and square the result.
- Add up all of those squared differences.
- Divide by n (population) or n − 1 (sample) to get variance.
- Take the square root of the variance to get standard deviation.
Our calculator does this instantly in your browser using a numerically stable method (Welford's algorithm) so it stays accurate even for large numbers and long lists.
Common use cases
People typically use a standard deviation calculator for:
- Checking variability in exam or quiz scores.
- Measuring volatility in finance and investment returns.
- Analyzing experimental data in science or engineering.
- Comparing consistency across different datasets.
For quick averages without spread, try our Average Calculator.