Standard Deviation — Measure of Spread
σ or s: square root of variance. The #1 measure of spread. Population (σ) vs sample (s) with Bessel correction. Mean, median, quartiles, IQR, CV, SE.
Why This Statistical Analysis Matters
Why: SD quantifies spread. Six Sigma QC, risk analysis, research — all rely on SD. 68-95-99.7 rule for normal data.
How: Enter data (comma/space separated). Choose population or sample. Get mean, variance, SD, median, Q1, Q3, IQR, CV, SE.
- ●σ² = Σ(x-μ)²/N
- ●s² uses n−1
- ●68-95-99.7 rule
Mean, Variance, SD, Median, Quartiles, IQR — Histogram & Box Plot
The #1 measure of spread. From Six Sigma QC to Wall Street risk analysis. Step-by-step breakdown with interactive charts.
Real-World Scenarios — Click to Load
Data Input
Distribution Histogram
Box Plot (5-Number Summary)
Empirical Rule (Normal Distribution)
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
📈 Statistical Insights
Population SD divides by N
— Definition
Sample SD divides by n−1
— Bessel
Within ±1σ for normal
— Empirical
Key Takeaways
- • Standard deviation measures how spread out data is from the mean — the most widely used measure of dispersion
- • Use sample SD (s, divide by n−1) when your data is a subset; population SD (σ, divide by N) when you have full data
- • The Empirical Rule: for normal distributions, ~68% within ±1 SD, ~95% within ±2 SD, ~99.7% within ±3 SD
- • Coefficient of Variation (CV%) lets you compare variability across datasets with different units
Did You Know?
How Standard Deviation Works
Standard deviation quantifies variation or dispersion. A low SD indicates data clusters near the mean; a high SD indicates wide spread.
Step 1: Calculate the Mean (x̄)
Add all values and divide by count.
Step 2: Find Each Deviation (x − x̄)
Subtract the mean from each point. Sum of deviations = 0.
Step 3: Square the Deviations
Squaring ensures positivity and amplifies large deviations.
Step 4: Average Squared Deviations (Variance)
Divide by N (population) or n−1 (sample). Bessel's correction for sample.
Step 5: Take the Square Root (SD)
Converts variance back to original units.
Expert Tips
Population vs Sample
Use sample (n−1) when surveying a subset. Use population (N) when you have full data. When in doubt, use sample.
Use CV% to Compare
SD of $5 is meaningless without context. CV% = (SD/mean)×100 lets you compare across different scales.
Outlier Detection
Values beyond ±2 SD are unusual; beyond ±3 SD are potential outliers. IQR method: below Q1−1.5×IQR or above Q3+1.5×IQR.
SD for Symmetric Data
SD works best for symmetric data. For skewed data (income, prices), use IQR or MAD.
Why Use This Calculator vs. Other Tools?
| Feature | This Calculator | Excel | Python |
|---|---|---|---|
| Step-by-step breakdown | ✅ | ❌ | ❌ |
| Histogram + box plot | ✅ | ⚠️ Manual | ⚠️ matplotlib |
| All descriptive stats | ✅ | ⚠️ Multiple | ⚠️ Multiple |
| Population AND sample | ✅ | ⚠️ STDEV vs STDEVP | ⚠️ ddof |
| Copy & share results | ✅ | ❌ | ❌ |
| AI-powered interpretation | ✅ | ❌ | ❌ |
Frequently Asked Questions
Population vs sample standard deviation?
Population (σ) divides by N — use when you have every member. Sample (s) divides by n−1 — use when data is a subset. Research almost always uses sample.
Why n−1 for sample SD?
Bessel's correction. The sample mean is closer to sample data than the true population mean, so sample variance underestimates. Dividing by n−1 corrects this bias.
What is a "good" standard deviation?
Depends on context. Use CV% to compare: CV < 15% low variability, 15–30% moderate, > 30% high.
How does SD relate to the normal distribution?
Empirical Rule: 68% within ±1 SD, 95% within ±2 SD, 99.7% within ±3 SD. Chebyshev: any distribution, ≥75% within ±2 SD.
What units does SD have?
Same as your data. Dollars → SD in dollars. Seconds → SD in seconds. Variance has squared units.
How do outliers affect SD?
SD is sensitive to outliers (squared deviations). For robust measure, use IQR or MAD.
Standard error vs standard deviation?
SD = spread of data. SE = SD/√n = precision of the sample mean. SE decreases with n; SD stays roughly the same.
Can SD be zero or negative?
SD = 0 when all values are identical. SD is never negative (square root of variance).
Standard Deviation by the Numbers
Official Data Sources
Disclaimer: This calculator uses well-established formulas. Verify results for critical applications (clinical research, finance, QC). For weighted or grouped data, specialized methods may be required. Educational and professional reference purposes.