Standard Deviation Calculator
Computes sample and population standard deviation, sum of squares, and data dispersion.
Standard Deviation, Variance & Z-Score Calculator
Compute sample and population standard deviation ($s$ / $\sigma$), variance ($s^2$ / $\sigma^2$), standard error of the mean (SEM), normal distribution Z-scores, and step-by-step variance tables.
Dataset Values
Z-Score & Normal Curve Evaluator
Step-by-Step Deviations
| Value (x) | Deviation (x - x̄) | Squared (x - x̄)² |
|---|---|---|
| 12 | -13.700 | 187.6900 |
| 15 | -10.700 | 114.4900 |
| 18 | -7.700 | 59.2900 |
| 22 | -3.700 | 13.6900 |
| 25 | -0.700 | 0.4900 |
| 28 | +2.300 | 5.2900 |
| 30 | +4.300 | 18.4900 |
| 32 | +6.300 | 39.6900 |
| 35 | +9.300 | 86.4900 |
| 40 | +14.300 | 204.4900 |
Understanding Standard Deviation and Statistical Variance
What is Standard Deviation?
Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates values cluster close to the mean, while a high standard deviation indicates values spread out over a wider range.
Sample ($s$) vs Population ($\sigma$): Which should I choose?
Use Sample ($s$) with Bessel's correction ($n - 1$) when your data is a subset or sample representing a larger population. Use Population ($\sigma$) ($N$) when your data includes every member of the entire group.
What does a Z-Score represent?
A Z-score (z = (x - μ) / σ) indicates how many standard deviations an observation lies above or below the mean. A Z-score of +2.0 means the value is 2 standard deviations higher than average, placing it in the top 2.28% of a normal distribution.
Are calculations private?
Yes. All statistical formulas and dataset parsing run 100% inside your browser's local JavaScript engine via NEOKYRU. No data ever leaves your computer or phone.
What is Standard Deviation?
Standard deviation measures the average distance of data points from their mean.
Low standard deviation indicates data clusters tightly around the mean; high indicates wide dispersion.
Formula & Step-by-Step Calculation
Square root of population variance.
Worked Step-by-Step Examples
Std dev of [10, 12, 23, 23, 16, 23, 21, 16]
Common Real-World & Academic Use Cases
- ✓ Investment risk metrics
- ✓ Manufacturing tolerance control
- ✓ Psychological psychometric grading
How to Use the Standard Deviation Calculator
Enter Data
Type or paste dataset.
View Std Dev
Inspect both sample and population figures.
Frequently Asked Questions
Q: What is the 68-95-99.7 empirical rule?
In a normal distribution, ~68% of data falls within 1σ, ~95% within 2σ, and ~99.7% within 3σ.