Statistics
Standard Deviation Calculator
Quantify how widely values fan out around the mean with both sample and population σ plus supporting spread metrics.
Quantify spread with precise σ, variance, and coefficient of variation pulled straight from your dataset.
Standard deviation formulas
Sample s = √[Σ(x − x̄)² / (n − 1)] Population σ = √[Σ(x − μ)² / n]
Sample mode divides by n − 1 to remove bias when the data is a subset. Population mode divides by n when you have every observation.
How to use
- Paste numbers separated by commas, spaces, or new lines.
- Select sample or population mode to match your assignment or compliance rule.
- Optional: adjust decimal places to tighten or loosen the reported precision.
Example
Input: Data = 72, 74, 75, 80, 81, 84 (sample mode)
Output: Mean ≈ 77.67, s ≈ 4.68, Range = 12, CV ≈ 6.0%
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Standard Deviation Calculator: Standard Deviation Calculator. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Standard Deviation Calculator: Computes σ, variance, range, and coefficient of variation from any dataset. It's built around standard deviation, sigma, volatility, dispersion, so you can go from a raw question to a checked answer without switching tools.
The math behind it: Sample mode divides by n − 1 to remove bias when the data is a subset. Population mode divides by n when you have every observation. The core relationship is Sample s = √[Σ(x − x̄)² / (n − 1)] Population σ = √[Σ(x − μ)² / n], shown above the calculator so you can see exactly how your inputs turn into the result.
To use it well: (1) Paste numbers separated by commas, spaces, or new lines. (2) Select sample or population mode to match your assignment or compliance rule. (3) Optional: adjust decimal places to tighten or loosen the reported precision. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering Data = 72, 74, 75, 80, 81, 84 (sample mode) returns Mean ≈ 77.67, s ≈ 4.68, Range = 12, CV ≈ 6.0%. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
When should I choose sample mode?
Use sample mode whenever you are estimating σ from a subset of a larger population—stats classes, surveys, or experiments. Population mode only applies when you have every possible observation.
Why is the coefficient of variation blank?
CV requires both a non-zero mean and a finite standard deviation. If your data is centered at zero or too short for sample σ, the metric is suppressed to avoid misleading percentages.
What formula does the Standard Deviation Calculator use?
Sample mode divides by n − 1 to remove bias when the data is a subset. Population mode divides by n when you have every observation.
How do I use the Standard Deviation Calculator?
Paste numbers separated by commas, spaces, or new lines. Select sample or population mode to match your assignment or compliance rule. Optional: adjust decimal places to tighten or loosen the reported precision.