Standard Deviation Calculator
Standard Deviation Calculator
Free online tool to calculate standard deviation, variance, and mean with step-by-step explanations.
Enter numbers separated by commas, spaces, or new lines...
Quick Tips
- Use Sample when your data is a subset representing a larger population; use Population when it is the entire group.
- For normally distributed data, about 68% of values fall within 1 standard deviation of the mean.
- Outliers can significantly inflate standard deviation since differences are squared.
Standard Deviation (σ)
5.7446
Mean (Average): 19.0000
Variance (σ²)
33.0000
Count
10
Last updated: August 11, 2026
Standard Deviation Calculator: Measure Data Spread Instantly
Calculate standard deviation (σ), variance, mean, and count — with full step-by-step breakdown. Perfect for stats, finance, science, or quality control. Works for population or sample data.
Example: [85, 90, 88, 92, 87] → Mean = 88.4 | σ ≈ 2.88 (low spread).
Key Formulas Explained
Population σ
σ = √[Σ(x - μ)² / N]
μ = mean, N = total count
Sample s (Bessel)
s = √[Σ(x - x̄)² / (n - 1)]
Use for surveys, experiments
Variance
σ² = Σ(x - μ)² / N
Squared spread
Mean
μ = Σx / N
Center of data
Step-by-Step Calculation
| Data | x - μ | (x - μ)² |
|---|---|---|
| 85 | -3.4 | 11.56 |
| 90 | +1.6 | 2.56 |
| 88 | -0.4 | 0.16 |
| 92 | +3.6 | 12.96 |
| 87 | -1.4 | 1.96 |
| Σ = 442 | Σ = 0 | Σ = 29.2 |
Mean (μ) = 442 / 5 = 88.4
Variance = 29.2 / 5 = 5.84
σ = √5.84 ≈ 2.88
Real-World Applications
Finance
Stock volatility: High σ = high risk
Manufacturing
Quality control: σ < 1% = consistent parts
Science
Experiment error: Low σ = reliable results
Education
Test scores: σ shows class consistency
Pro Stats Tips
Best Practices
- Use sample s for surveys (n-1)
- Remove outliers before final calc
- Pair with mean for full insight
Common Mistakes
- Using N instead of n-1 in samples
- Ignoring units (σ in same unit as data)
- Confusing σ with variance
Why Trust This Calculator?
Accuracy: Uses NIST & ISO 3534-1 formulas — same as Excel, SPSS, and R.
Privacy: No data stored. Results vanish instantly.
Speed: JS-powered, works offline (PWA-ready).
Frequently Asked Questions
It measures how much your data typically varies from the mean — a low standard deviation means values cluster tightly around the average, while a high one means they're spread out widely. For normally distributed data, about 68% of values fall within 1 standard deviation of the mean, and 95% fall within 2.
Use population if your data represents the entire group you care about (every student in a class, every item produced today). Use sample if your data is a subset meant to represent a larger population (a survey of 200 people representing a country) — the sample formula divides by n-1 instead of n to correct for this.
Variance is used in more advanced statistical methods like ANOVA where squared units are mathematically convenient. For everyday interpretation, standard deviation is usually more useful since it's expressed in the same units as your original data, unlike variance which is in squared units.
No — standard deviation is always zero or positive, since it's a square root of an average of squared differences. A standard deviation of exactly 0 only happens when every value in your dataset is identical.
It depends entirely on context and the scale of your data — there's no universal threshold. A useful way to judge it is the coefficient of variation (standard deviation ÷ mean), which lets you compare variability across datasets with different units or scales.
Significantly — because the formula squares each difference from the mean, outliers get disproportionately weighted and can inflate the standard deviation much more than a typical value would. Consider whether outliers reflect genuine variation or data-entry errors before including them.
The mean is the arithmetic average (sum ÷ count), the median is the middle value when data is sorted, and the mode is the most frequently occurring value. For skewed data, the median often better represents the "typical" value than the mean, since it isn't pulled toward extreme values.
This is called Bessel's correction. Dividing by n-1 instead of n produces an unbiased estimate of the population variance from a sample — using n alone tends to systematically underestimate the true population variance, especially with small sample sizes.
Yes — this calculator accepts numbers separated by commas, spaces, tabs, or new lines, so you can paste a column directly from Excel or Google Sheets, or a comma-separated list, and it will parse the values automatically.
No, though they're related. Standard deviation describes the spread of your raw data, while margin of error (used in polling and confidence intervals) is derived from the standard error — a related but distinct measure of how much a sample statistic might differ from the true population value.
There's no strict minimum, but very small datasets (fewer than 5-10 points) produce standard deviations that are quite sensitive to individual values and may not reliably represent the underlying variability. More data generally gives a more stable, trustworthy estimate.
No — mean, median, mode, variance, and standard deviation are all calculated from the full set of values regardless of the order you enter them in. Only the identity and count of the values matter, not their sequence.
Standard Deviation (σ)
5.7446