Standard Deviation Calculator
Calculate standard deviation, variance, mean, median, mode, and other descriptive statistics for any data set.
Enter Your Data
Example Data Sets:
Sorted Data:
[10, 12, 16, 16, 21, 23, 23, 23]
Sample Standard Deviation (σ)
5.237229
Advanced Statistics:
Mode:
23
Standard Error:
1.851640
Coefficient of Variation:
29.10%
Sum of Squares:
192.0000
What is Standard Deviation?
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.
Standard Deviation Formulas
Sample Standard Deviation
s = √[Σ(xᵢ - x̄)² / (n-1)]
Use when data is a sample of a larger population
Population Standard Deviation
σ = √[Σ(xᵢ - μ)² / n]
Use when data represents the entire population
What Is Standard Deviation?
Standard deviation measures how spread out data is from the mean. A low standard deviation means values cluster tightly around the average, while a high standard deviation indicates values are widely dispersed. It's the most commonly used measure of variability and is expressed in the same units as the original data.
Standard deviation is the square root of variance, making it more interpretable since it's in the original units (not squared units). It's fundamental to understanding data distributions, quality control, and statistical inference.
| Standard Deviation | Interpretation | Example |
|---|---|---|
| Low (small) | Values tightly clustered | Precision manufacturing: SD = 0.01 mm |
| High (large) | Values widely spread | Stock returns: SD = 20% |
| Zero | All values identical | Fixed price: every item costs $10 |
Standard Deviation Formulas
Where:
- σ (sigma)= Population standard deviation
- s= Sample standard deviation
- μ / x̄= Mean (population / sample)
- N / n= Count (population / sample)
Population vs Sample Standard Deviation
The critical distinction between population and sample standard deviation lies in the denominator. Sample SD divides by (n-1) instead of n—this is called Bessel's correction and produces an unbiased estimate of the population SD.
| Aspect | Population SD (σ) | Sample SD (s) |
|---|---|---|
| When to use | You have ALL data | Data is a sample (most cases) |
| Divisor | N (population size) | n-1 (degrees of freedom) |
| Symbol | σ (sigma) | s |
| Bias | Exact (if truly population) | Unbiased estimator of σ |
| Example | All test scores in a class | Survey of some voters |
Why n-1? When we estimate the mean from the sample, we "use up" one piece of information. Dividing by n-1 compensates for this, giving an unbiased estimate. For large n, the difference is negligible; for small n, it matters.
Calculating Standard Deviation Step by Step
Understanding the calculation process helps interpret what standard deviation actually measures. Here's how it works:
| Step | Description | Example: Data = 2, 4, 4, 4, 5, 5, 7, 9 |
|---|---|---|
| 1. Find mean | Sum ÷ count | 40 ÷ 8 = 5 |
| 2. Subtract mean from each | Find deviations | -3, -1, -1, -1, 0, 0, 2, 4 |
| 3. Square each deviation | Remove negatives | 9, 1, 1, 1, 0, 0, 4, 16 |
| 4. Sum squared deviations | Total variation | 32 |
| 5. Divide by (n-1) | Variance (sample) | 32 ÷ 7 = 4.57 |
| 6. Take square root | Standard deviation | √4.57 = 2.14 |
The Empirical Rule (68-95-99.7)
For normally distributed data, the empirical rule (also called the 68-95-99.7 rule) describes what percentage of values fall within 1, 2, and 3 standard deviations of the mean. This is fundamental to understanding probability distributions and detecting outliers.
| Range | % of Data (Normal) | Interpretation |
|---|---|---|
| μ ± 1σ | ~68.3% | About 2/3 of values |
| μ ± 2σ | ~95.4% | Most values |
| μ ± 3σ | ~99.7% | Nearly all values |
| Beyond 3σ | ~0.3% | Rare (potential outliers) |
Quality control application: In manufacturing, a "Six Sigma" process has defects only beyond 6 standard deviations—approximately 3.4 defects per million opportunities.
Interpreting Standard Deviation
Standard deviation is only meaningful in context. What's "high" or "low" depends on the scale and application. Comparing SD to the mean (via coefficient of variation) helps standardize this comparison.
| Measure | Formula | Use Case |
|---|---|---|
| Standard Deviation | s or σ | Absolute spread (same units as data) |
| Variance | s² or σ² | Mathematical properties (squared units) |
| Coefficient of Variation (CV) | (s / x̄) × 100% | Relative spread (dimensionless) |
| Standard Error (SE) | s / √n | Precision of mean estimate |
Example: Investment A has mean return 10% with SD 5%; Investment B has mean 20% with SD 15%. Which is riskier per unit return? CV_A = 50%, CV_B = 75%. Investment B has more relative variability.
Standard Deviation vs Standard Error
Don't confuse standard deviation (SD) with standard error (SE). SD measures spread in the data; SE measures precision of the sample mean as an estimate of the population mean.
| Concept | Standard Deviation (SD) | Standard Error (SE) |
|---|---|---|
| Measures | Spread of individual data points | Precision of mean estimate |
| Formula | s = √[Σ(x-x̄)²/(n-1)] | SE = s / √n |
| As n increases | Stays roughly constant | Decreases (more precision) |
| Used for | Describing data variability | Confidence intervals for mean |
| Report when | Describing a distribution | Estimating population mean |
Rule of thumb: 95% confidence interval for the mean ≈ x̄ ± 2×SE. Larger samples have smaller SE, giving narrower confidence intervals.
Applications of Standard Deviation
Standard deviation appears throughout science, finance, quality control, and social sciences. Understanding these applications helps interpret statistical reports and make informed decisions.
| Field | Application | Typical SD Values |
|---|---|---|
| Finance | Investment volatility | Stock: 15-25% annually; Bond: 5-10% |
| Manufacturing | Quality control (6σ) | Target: as low as possible |
| Psychology | IQ scores | Mean=100, SD=15 by design |
| Education | Test score scaling | SAT: Mean=500, SD=100 per section |
| Weather | Temperature variability | Desert: high SD; Coastal: low SD |
| Sports | Performance consistency | Consistent player: low SD |
Coefficient of Variation
Where:
- CV= Coefficient of variation (relative SD)
- s= Standard deviation
- x̄= Mean
Worked Examples
Calculate Sample Standard Deviation
Problem:
Calculate the sample standard deviation of test scores: 72, 88, 76, 84, 80
Solution Steps:
- 1Find the mean: (72 + 88 + 76 + 84 + 80) / 5 = 400/5 = 80
- 2Calculate deviations: 72-80=-8, 88-80=8, 76-80=-4, 84-80=4, 80-80=0
- 3Square deviations: 64, 64, 16, 16, 0
- 4Sum squared deviations: 64 + 64 + 16 + 16 + 0 = 160
- 5Divide by (n-1): 160 / 4 = 40 (this is variance)
- 6Take square root: √40 = 6.32
Result:
Sample SD = 6.32 points. About 68% of scores fall within 80 ± 6.32 (between 73.7 and 86.3).
Interpreting SD with the Empirical Rule
Problem:
Heights of adult men are normally distributed with mean 70 inches and SD 3 inches. What percentage are between 64 and 76 inches?
Solution Steps:
- 1Calculate deviations: 64 = 70 - 6 = μ - 2σ; 76 = 70 + 6 = μ + 2σ
- 2This range is mean ± 2 standard deviations
- 3Apply empirical rule: ~95.4% of values fall within ± 2σ
Result:
Approximately 95% of adult men are between 64 and 76 inches tall. Only ~2.5% are shorter than 64 inches, and ~2.5% are taller than 76 inches.
Comparing Variability Using CV
Problem:
Company A stock: mean return 8%, SD 12%. Company B stock: mean return 15%, SD 20%. Which has more relative risk?
Solution Steps:
- 1Calculate CV for A: (12/8) × 100% = 150%
- 2Calculate CV for B: (20/15) × 100% = 133%
- 3Compare: A has higher CV despite lower absolute SD
Result:
Company A has relatively MORE variability (CV = 150%) per unit of return than Company B (CV = 133%), even though A's raw SD is lower. CV allows comparing variability across different scales.
Tips & Best Practices
- ✓Use sample SD (n-1 denominator) for samples; population SD (n denominator) only when you truly have the entire population.
- ✓The empirical rule (68-95-99.7) only applies to normally distributed data—check normality first.
- ✓Compare variability across different scales using coefficient of variation (CV), not raw SD.
- ✓Standard error decreases with sample size (SE = SD/√n); SD doesn't change much as you add more data.
- ✓For data with outliers, consider robust alternatives like median absolute deviation (MAD) or interquartile range (IQR).
- ✓In reports, always clarify whether you're reporting SD or SE—they mean very different things.
- ✓Values beyond 3 standard deviations from the mean are potential outliers worth investigating.
Frequently Asked Questions
Sources & References
Last updated: 2026-01-22