Standard Deviation Calculator
Enter a list of numbers to calculate the sample and population standard deviation, variance, mean, and standard error — with every step shown.
- 01Calculate sample (n − 1) and population (n) standard deviation instantly.
- 02Get variance, mean, median, range, and standard error in one click.
- 03See step-by-step calculations using the standard deviation formula.
- 04Paste numbers separated by commas, spaces, or new lines.
- 05100% free and private — every calculation runs in your browser.
Standard Deviation Calculator
Try an example data set
Sample Standard Deviation (s)
5.237229
Divided by n − 1 — use for a sample
Population Standard Deviation (σ)
4.898979
Divided by n — use for a full population
Count (n)
8
Sum (Σx)
144
Mean (x̄)
18
Median
18.5
Sample Variance (s²)
27.428571
Population Variance (σ²)
24
Standard Error (SEx̄)
1.85164
Coefficient of Variation
29.095719%
Minimum
10
Maximum
23
Range
13
Sum of Squares (SS)
192
Step-by-step calculation
- 01Mean x̄ = Σx ÷ n = 144 ÷ 8 = 18
- 02Sum of squared deviations Σ(x − x̄)² = 192
- 03Population variance σ² = Σ(x − x̄)² ÷ n = 192 ÷ 8 = 24
- 04Sample variance s² = Σ(x − x̄)² ÷ (n − 1) = 192 ÷ 7 = 27.428571
- 05Population standard deviation σ = √σ² = 4.898979
- 06Sample standard deviation s = √s² = 5.237229
Show deviation table (x − x̄)
| Value (x) | Deviation (x − x̄) | Squared (x − x̄)² |
|---|---|---|
| 10 | -8 | 64 |
| 12 | -6 | 36 |
| 23 | 5 | 25 |
| 23 | 5 | 25 |
| 16 | -2 | 4 |
| 23 | 5 | 25 |
| 21 | 3 | 9 |
| 16 | -2 | 4 |
Why Use This Standard Deviation Calculator
Sample and Population SD
Get both the sample standard deviation (dividing by n − 1) and the population standard deviation (dividing by n) at the same time, so you always have the right value for your statistics problem — whether you are working with a sample or an entire population.
Step-by-Step Working
See exactly how the standard deviation is calculated: the mean, every deviation from the mean, the squared deviations, the sum of squares, the variance, and the final square root. Perfect for homework, exam revision, and checking your own work.
Full Descriptive Statistics
Beyond standard deviation, the calculator reports variance, mean, median, range, minimum, maximum, sum of squares, standard error of the mean, and the coefficient of variation — a complete summary of your data set.
Flexible Data Input
Paste numbers separated by commas, spaces, tabs, or new lines. Decimals and negative numbers are supported, so you can drop in data straight from a spreadsheet column without reformatting.
Instant and Private
Everything runs entirely in your browser with no server round-trips. Your data never leaves your device, results appear instantly, and there is no signup or installation required.
Free with No Limits
Calculate the standard deviation for as many data sets as you need — no daily limits, no account, and no paywall. The full statistics calculator is completely free.
What Is Standard Deviation?
Standard deviation is a measure of how spread out a set of numbers is around their mean (average). A low standard deviation means the values cluster close to the mean, while a high standard deviation means they are spread over a wider range. It is one of the most widely used measures of variability in statistics, finance, science, and quality control.
Whether you are a student learning statistics, a researcher analysing data, or an analyst measuring risk, this standard deviation calculator gives instant, step-by-step results for any data set.
- Sample vs Population Standard Deviation
- Population standard deviation (σ) divides the sum of squared deviations by n and is used when your data covers the entire population. Sample standard deviation (s) divides by n − 1 (Bessel's correction) and is used when your data is a sample drawn from a larger population. This calculator shows both.
- How It Relates to Variance
- Variance is the average of the squared deviations from the mean, and standard deviation is simply the square root of the variance. Standard deviation is often preferred because it is expressed in the same units as the original data, making it easier to interpret.
- The Standard Deviation Formula
- To calculate standard deviation: find the mean, subtract the mean from each value to get the deviations, square each deviation, add them up to get the sum of squares, divide by n (population) or n − 1 (sample) to get the variance, then take the square root.
- Why Standard Deviation Matters
- Standard deviation tells you how reliable an average is and how much individual values typically differ from it. It underpins confidence intervals, z-scores, the normal distribution, risk in finance, and process control in manufacturing.
How to Use the Standard Deviation Calculator
- 01
Enter your numbers
Type or paste your data set into the input box. Separate values with commas, spaces, or new lines — for example 10, 12, 23, 23, 16, 23, 21, 16. Decimals and negative numbers are supported.
- 02
Click Calculate
Press the Calculate button. The tool parses your numbers and computes the mean, variance, and both sample and population standard deviation instantly in your browser.
- 03
Read the results
The headline cards show the sample standard deviation (s) and population standard deviation (σ). Below them you get the mean, median, variance, standard error, range, and sum of squares for a full picture of your data.
- 04
Review the steps
Open the step-by-step section and the deviation table to see how each value contributes — the deviation from the mean, the squared deviation, the sum of squares, and the final square root. Great for learning and for checking homework.
Tips for Calculating Standard Deviation
Choose Sample or Population
Use the sample standard deviation (n − 1) when your data is a subset drawn from a larger group, and the population standard deviation (n) when it includes every member of the group. Picking the wrong one is the most common mistake.
Check Your Units
Standard deviation is in the same units as your data, while variance is in squared units. When reporting spread alongside the mean, standard deviation is usually the clearer choice.
Watch for Outliers
Standard deviation is sensitive to extreme values because deviations are squared. A single outlier can inflate it significantly, so inspect your data and consider whether outliers are genuine before drawing conclusions.
Pair With the Mean
Standard deviation only makes sense relative to the mean. Reporting both together (for example, 18 ± 4.5) tells a much clearer story about your data than either value alone.
Use the Coefficient of Variation
To compare variability between data sets with different means or units, use the coefficient of variation (standard deviation ÷ mean), which expresses spread as a percentage and is unit-free.
Keep Enough Precision
Avoid rounding the mean or intermediate deviations too early — rounding before the final step can introduce noticeable errors. Round only the final standard deviation for reporting.
Standard Deviation Formulas and Definitions
Definition of standard deviation
Standard deviation measures the typical distance between each value in a data set and the mean. It is the square root of the variance and is reported in the same units as the data.
What standard deviation tells you
- How tightly values cluster around the mean (lower = more consistent).
- How reliable an average is as a summary of the data.
- The basis for z-scores, confidence intervals, and the normal distribution.
- A measure of risk or volatility in finance and quality control.
Sample vs population — which to use
Use the sample standard deviation (÷ n − 1) when your data is a sample of a larger group, and the population standard deviation (÷ n) when it represents the entire population.
Key Statistics Formulas
Mean
x̄ = Σx ÷ n
Example: (2 + 4 + 6) ÷ 3 = 4.
Population variance
σ² = Σ(x − x̄)² ÷ n
Average of the squared deviations from the mean.
Sample variance
s² = Σ(x − x̄)² ÷ (n − 1)
Uses n − 1 (Bessel's correction) for an unbiased estimate.
Population standard deviation
σ = √(Σ(x − x̄)² ÷ n)
Square root of the population variance.
Sample standard deviation
s = √(Σ(x − x̄)² ÷ (n − 1))
Square root of the sample variance.
Standard error of the mean
SEx̄ = s ÷ √n
How precisely the sample mean estimates the population mean.
Standard Deviation Calculator FAQ
Q01How do I calculate standard deviation?
Find the mean of your data, subtract the mean from each value to get the deviations, square each deviation, add them up to get the sum of squares, then divide by n for a population or n − 1 for a sample to get the variance. The standard deviation is the square root of the variance. This calculator does all of these steps for you and shows the working.
Q02What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by n and is used when your data includes the whole population. Sample standard deviation (s) divides by n − 1 (Bessel's correction) and is used when your data is a sample from a larger population. The calculator reports both so you can use whichever fits your problem.
Q03What is the standard deviation formula?
Population: σ = √(Σ(x − x̄)² ÷ n). Sample: s = √(Σ(x − x̄)² ÷ (n − 1)). In both, x̄ is the mean, x is each value, and n is the number of values. Standard deviation is the square root of the variance.
Q04How is standard deviation related to variance?
Variance is the average of the squared deviations from the mean. Standard deviation is the square root of the variance. Variance is in squared units, while standard deviation is in the same units as your original data, which makes it easier to interpret.
Q05What is standard error of the mean?
The standard error of the mean (SEM) estimates how much the sample mean is likely to vary from the true population mean. It equals the sample standard deviation divided by the square root of n. A smaller standard error means a more precise estimate of the mean.
Q06Can I enter decimals and negative numbers?
Yes. You can enter integers, decimals, and negative numbers separated by commas, spaces, or new lines. The calculator parses all of them and ignores extra spacing, so you can paste a column straight from a spreadsheet.
Q07Is my data sent to a server?
No. The calculator runs entirely in your browser using JavaScript. Your numbers are never uploaded or stored anywhere, so it is safe to use with private or sensitive data.
Q08Is this standard deviation calculator free?
Yes, it is completely free with no limits, no signup, and no premium tier. Calculate the standard deviation and variance for as many data sets as you like.