How the Calculator Works
This tool uses the standard statistical formulas verified for 2026/2026 educational standards. It automatically distinguishes between:
- Population Standard Deviation (σ)Used when your data represents the entire group of interest. The formula divides by N.
- Sample Standard Deviation (s)Used when your data is a subset of a larger population. The formula divides by n-1 (Bessel's Correction).
Standard Deviation Examples (2026)
The table below demonstrates how standard deviation varies across different datasets. These examples show real-world scenarios where understanding standard deviation is crucial for data analysis.
| Dataset | Mean | Population SD (σ) | Sample SD (s) | Variance | Interpretation |
|---|---|---|---|---|---|
| 10, 10, 10, 10, 10 | 10 | 0 | 0 | 0 | No variation |
| 8, 9, 10, 11, 12 | 10 | 1.41 | 1.58 | 2.0 | Low variation |
| 5, 7, 10, 13, 15 | 10 | 3.74 | 4.18 | 14.0 | Moderate variation |
| 1, 5, 10, 15, 19 | 10 | 6.32 | 7.07 | 40.0 | High variation |
| Test Scores: 85, 87, 88, 90, 92 | 88.4 | 2.58 | 2.88 | 6.64 | Consistent performance |
| Stock Prices: 100, 105, 95, 110, 90 | 100 | 7.91 | 8.84 | 62.5 | Volatile market |
Population SD (σ) divides by N, Sample SD (s) divides by n-1 (Bessel's correction). Variance is the square of standard deviation. Lower SD indicates more consistent data.
Key Concepts & Formulas
What is Standard Deviation?Definition
Standard deviation is a statistical metric that measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (average), while a high standard deviation indicates that the values are spread out over a wider range. To calculate the mean of your dataset first, use our average calculator.
The Formulas
Population Standard Deviation (σ)
Where σ is standard deviation, x is each value, μ is the population mean, and N is the size of the population.
Sample Standard Deviation (s)
Where s is sample deviation, x̄ is the sample mean, and n is the sample size. The n-1 is known as Bessel's Correction.
Frequently Asked Questions (FAQ)
Why use n-1 for sample standard deviation?▼
Using n-1 (Bessel's Correction) corrects for the bias in the estimation of the population variance. When you calculate the variance from a sample, it tends to slightly underestimate the true population variance because the sample mean is specifically positioned to minimize the squared deviations for that specific sample. Dividing by a smaller number (n-1 instead of n) increases the result slightly, compensating for this underestimation.
How do I interpret the result?▼
In a normal distribution (bell curve):
- 68% of data fall within 1 Standard Deviation of the mean.
- 95% of data fall within 2 Standard Deviations.
- 99.7% of data fall within 3 Standard Deviations.
This rule (68-95-99.7) allows you to determine if a specific data point is an outlier or expected.
What is Variance vs Standard Deviation?▼
Variance is simply the Standard Deviation squared. While Variance is mathematically useful, it is expressed in squared units (e.g., "squared dollars"), which is hard to interpret. Standard Deviation brings the units back to the original scale (e.g., "dollars"), making it much easier to understand in real-world terms.
Need more Analysis?
Our tools are constantly updated for accuracy. Check out our other statistical calculators for more in-depth data analysis.
Last Updated: December 2026 • Verified by VerCalc Statistics Team