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σ

Standard Deviation: Spread of Data

σ (population) or s (sample) measures spread. σ = √(variance). Population: divide by N; sample: divide by n−1 (Bessel). Same units as data.

Concept Fundamentals
Population
σ
Sample
s
Variance
σ²
σ/μ × 100
CV

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σ = √(variance). Population: divide by N; sample: n−1. CV = σ/μ×100. Relative spread, unitless. 68–95–99.7 rule: ~68% within 1σ of mean.

Key quantities
Population
σ
Key relation
Sample
s
Key relation
Variance
σ²
Key relation
σ/μ × 100
CV
Key relation

Ready to run the numbers?

Why: Standard deviation quantifies spread. σ = √(variance). Sample uses n−1 for unbiased estimate. CV = σ/μ×100 for relative spread. Used in quality control, finance.

How: Population: σ = √[(1/N)Σ(x−μ)²]. Sample: s = √[(1/(n−1))Σ(x−x̄)²]. Same units as data. CV = σ/μ×100 (or s/x̄×100).

σ = √(variance). Population: divide by N; sample: n−1.CV = σ/μ×100. Relative spread, unitless.
Sources:NCTM StandardsKhan Academy

Run the calculator when you are ready.

Calculate Standard DeviationEnter data values
stddev.sh
CALCULATED
$ stddev --values "2, 4, 4, 4, 5, 5, 7, 9"
Mean (μ)
5
s
2.13809
Variance
4.571429
CV %
42.761799%
Standard Deviation Calculator
s = 2.13809
Mean: 5 | Variance: 4.571429 | n = 8
numbervibe.com
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Summary Metrics

Values Within σ Bands

📐 Step-by-Step Breakdown

INPUTS
n
8
MEAN
Mean (μ)
5
\text{Sigma} x_i / n
DEVIATIONS
Squared deviations
(2 − 5)², (4 − 5)², (4 − 5)²...
Sum of squared deviations
32
\text{Sigma} (x_i - \text{mu} )^{2}
Divisor
7
n − 1 (Bessel)
RESULT
s
2.13809
√( ext{variance})
Variance
4.571429
\text{sigma} ^{2} ext{or} s^{2}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

σ

σ = √(variance)

— Standard deviation

📊

Sample: divide by n−1 (Bessel)

— Unbiased

📋 Key Takeaways

  • • Population σ: divide by N. Sample s: divide by (n−1) — Bessel correction for unbiased estimate
  • • σ = √(variance) — same units as data; variance has squared units
  • • 68-95-99.7 rule: ~68% within μ±σ, ~95% within μ±2σ, ~99.7% within μ±3σ (normal distribution)
  • • CV = (σ/μ)×100% — coefficient of variation for comparing spread across different means
  • • σ = 0 iff all values are identical

💡 Did You Know?

📜Karl Pearson introduced the term "standard deviation" in 1893.Source: History of Statistics
⚖️Sample s with (n−1) is unbiased for population σ.Source: Bessel Correction
📐Adding a constant does not change σ; multiplying by c multiplies σ by |c|.Source: Linear Transform
📈In finance, σ measures volatility and risk.Source: Quantitative Finance
🏭Quality control uses σ to monitor process consistency (Six Sigma).Source: Manufacturing
🔢Z-score = (x − μ) / σ standardizes any value.Source: Standardization

📖 How Standard Deviation Works

Compute the mean μ, then for each value find (x_i − μ)², sum them, divide by N (population) or n−1 (sample), and take the square root. The result measures typical distance from the mean. Outliers inflate σ because squaring amplifies large deviations.

σ² = (1/N) Σ(x_i − μ)² → σ = √σ²

📝 Worked Example: 2, 4, 4, 4, 5, 5, 7, 9

Step 1: Mean μ = (2+4+4+4+5+5+7+9)/8 = 5

Step 2: Squared deviations: (2−5)²=9, (4−5)²=1, … Sum = 24

Step 3: Sample variance = 24/(8−1) = 3.43

Step 4: s = √3.43 ≈ 1.85

🚀 Real-World Applications

📊 Finance & Investing

Volatility of stock returns, risk assessment, portfolio analysis.

🔬 Scientific Research

Reporting uncertainty in measurements, error bars.

🏭 Quality Control

Process capability, Six Sigma, manufacturing consistency.

📈 Data Science

Feature scaling, outlier detection, model evaluation.

🎓 Education

Test score distributions, grading curves.

🌡️ Meteorology

Temperature variability, climate analysis.

⚠️ Common Mistakes to Avoid

  • Using population when you have a sample: Use n−1 for sample to get unbiased estimate.
  • Confusing variance and std dev: Variance = σ²; std dev = √variance. Report in same units as data.
  • Ignoring outliers: σ is sensitive to outliers; consider robust measures like MAD.
  • Comparing σ across different scales: Use CV when means differ greatly.

🎯 Expert Tips

💡 Population vs Sample

Use population when you have all data; sample when it is a subset. Bessel correction (n−1) corrects bias.

💡 Coefficient of Variation

CV = (σ/μ)×100%. Compare spread across datasets with different units or scales.

💡 Z-Score

z = (x − μ) / σ. Tells how many standard deviations x is from the mean.

💡 Outliers

Squaring amplifies outliers. Consider MAD or IQR for robust spread.

📊 Reference Table

TypeDivisorSymbol
PopulationNσ
Samplen − 1s

❓ FAQ

Population vs sample?

Population: entire group (divide by N). Sample: subset (divide by n−1 for unbiased estimate of population σ).

What is coefficient of variation?

CV = (σ/μ)×100%. Relative spread; compare across different means or units.

Why n−1 for sample?

Bessel correction: corrects bias when estimating population σ from a sample. Degrees of freedom = n−1.

Relationship to variance?

σ = √(variance). Variance has squared units; σ has same units as data.

When σ = 0?

All values are identical. No spread.

What is the 68-95-99.7 rule?

For normal distributions: ~68% within μ±σ, ~95% within μ±2σ, ~99.7% within μ±3σ.

📌 Summary

Standard deviation measures the typical spread of data around the mean. Population σ divides by N; sample s divides by n−1 (Bessel correction). The 68-95-99.7 rule describes normal distributions. Use CV to compare spread across different scales. Outliers inflate σ; consider robust alternatives for skewed data.

⚠️ Disclaimer: For very large datasets, consider computational formulas. This calculator uses the definitional formula. Results are for educational purposes.

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