Standard Deviation Calculator
Last updated: July 2026 · Free · No sign-up required
Enter values above and click Calculate to see your result instantly.
Quick reference
| Metric | Population | Sample |
|---|---|---|
| Standard deviation | σ | s |
| Mean symbol | μ | x̄ |
| Denominator | N | n − 1 |
| Use case | All values | Subset estimating a larger group |
functions Shows the working, not just the answer
For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.
calculate Accurate to the spec
Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.
school Free for classroom use
Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.
tips_and_updates Pair with the spoke articles
Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.
Interpretation guide
| Rule of thumb | Normal distribution meaning |
|---|---|
| ±1 SD | About 68% of values |
| ±2 SD | About 95% of values |
| ±3 SD | About 99.7% of values |
lightbulb A real-world example — analyzing test scores
Result: Mean: 82.25. Sample Standard Deviation (s): 9.45. Population Standard Deviation (σ): 8.84. Variance (sample): 89.36.
What this means: A standard deviation of ~9.5 around a mean of 82 means most students scored between 73 and 92 (one σ below to above). Sample vs population SD: use sample (n-1 denominator) when your data is a subset of a larger group (these 8 students out of all the class's possible scores); use population (n denominator) when the data is the entire group of interest.
Formula & methodology
Formula: Population σ = √(Σ(x−μ)^2/N); sample s = √(Σ(x−x̄)^2/(n−1))
Standard deviation is the square root of variance. Use population SD when your dataset is complete; use sample SD with n−1 (Bessel's correction) when estimating a larger population from a sample.
Frequently asked questions
Explore the full statistics toolkit
Every CalcaTools statistics calculator — descriptive measures, probability distributions, hypothesis tests, confidence intervals, and Six Sigma process metrics — with step-by-step workings on each result.