Lang
Math & Number Tools Calculator
Standard Deviation Calculator
Use this free Standard Deviation Calculator to calculate standard deviation with a cleaner layout, instant results, formulas, examples, and helpful interpretation notes.
See how spread out your data is
Compare population and sample standard deviation, follow every calculation step, and explore the distribution visually. Results update as you type.
Separate values with commas, spaces, semicolons or new lines. For decimal commas, put one value on each line or use semicolons. Up to 300 values.
Spread at a glance
Standard deviation
Variance
Mean
Data count
Minimum
Maximum
Range
Sum of squared deviations
Coefficient of variation
The coefficient of variation is omitted when the mean is zero, negative or too close to zero. It is most meaningful for positive ratio-scale data.
How the values are distributed
Dots show up to 40 individual values; larger data sets use a histogram. Vertical lines mark the mean and one or two standard deviations. The shaded bands do not imply a normal distribution.
Mean±1 SD±2 SDData valuesLarge deviation
Observed within 1 SD
Observed within 2 SD
Values with |z| ≥ 2
An orange value has a large standardized deviation; it is not automatically an error or a statistical outlier. Check its context.
How the result was calculated
Calculation for every value
The table uses unrounded values internally. Display rounding does not affect the final result.
| # | Value x | x − mean | (x − mean)² | z-score |
|---|
Standard deviation FAQ
Should I choose population or sample?
Choose population when the data includes every member of the group of interest. Choose sample when the data is a subset used to estimate a larger group; sample variance divides by n − 1.
Why does sample standard deviation need two values?
With one value, n − 1 is zero, so sample variance and sample standard deviation are undefined.
Does a value outside two standard deviations mean an error?
No. It is only a large standardized deviation in this data set. Investigate the value and the distribution before interpreting it.
Does the chart assume a normal distribution?
No. The bands show measured distances from the mean. Normal-distribution coverage rules do not automatically apply.
How the result was calculated
Entire population (divide by n)
Choose population when the data includes every member of the group of interest. Choose sample when the data is a subset used to estimate a larger group; sample variance divides by n − 1.
σ² = Σ(x − μ)² ÷ n
σ = √σ²
Sample (divide by n − 1)
With one value, n − 1 is zero, so sample variance and sample standard deviation are undefined.
s² = Σ(x − x̄)² ÷ (n − 1)
s = √s²
Try the basic example
Data values: 10 · 12 · 23 · 23
n = 4; Σx = 68; μ = 68 ÷ 4 = 17
Σ(x − μ)² = 49 + 25 + 36 + 36 = 146
σ² = 146 ÷ 4 = 36.5; σ = √36.5 ≈ 6.0415
s² = 146 ÷ 3 ≈ 48.6667; s ≈ 6.9761
Σ(x − μ)² = 49 + 25 + 36 + 36 = 146
σ² = 146 ÷ 4 = 36.5; σ = √36.5 ≈ 6.0415
s² = 146 ÷ 3 ≈ 48.6667; s ≈ 6.9761
Try daily returns
Data values: 0.01 · -0.02 · 0.03 · 0.02 · -0.01 · 0 · 0.02
n = 7; Σx = 0.05; μ = 0.00714286
Σ(x − μ)² = 0.00194286
σ² = 0.00027755; σ ≈ 0.01665986
Σ(x − μ)² = 0.00194286
σ² = 0.00027755; σ ≈ 0.01665986
No. The bands show measured distances from the mean. Normal-distribution coverage rules do not automatically apply.
An orange value has a large standardized deviation; it is not automatically an error or a statistical outlier. Check its context.
How to use this calculator
- Enter the values requested by the Standard Deviation Calculator.
- Use the optional fields when they match your real situation.
- Read the result, then compare it with the formula notes and examples below.
Accuracy tips
- Keep intermediate values visible when possible so you can spot typing mistakes.
- Use the examples to confirm whether the calculator expects percentages, decimals, or whole numbers.
- If the answer is used for school or work, round only after the final calculation.
Why this helps
- Designed for quick math & number tools checks with a focused input area.
- Helpful explanations are kept on the same page so the result is easier to understand.
- The page can be edited directly from the synced WordPress HTML file.
Compare two variables
Standard deviation describes spread in one data set. For paired x-y values, use the Correlation Coefficient Calculator.
Math & Statistics directory
Need another math or statistics tool?
Browse the full math and statistics calculator collection for percentages, algebra, geometry, probability, z-scores, confidence intervals, regression, correlation, percentiles, matrices, and number conversions.