Standard Deviation: A Quick Definition
The standard deviation quantifies how spread out a set of numbers is around its mean. It’s the square root of the average squared distance of each data point from the mean.
1. Core Formulas
Population Standard Deviation
σ = √(1/N ∑(xᵢ − μ)²)
Sample Standard Deviation
s = √(1/(n − 1) ∑(xᵢ − x̄)²)
- N: total number of observations
- n: sample size
- xᵢ: individual data point
- μ: population mean
- x̄: sample mean
2. Step-by-Step Interpretation
- Compute the mean (average).
- Subtract the mean from each data point (deviation).
- Square each deviation.
- Average the squared deviations.
- Take the square root.
3. Numerical Example
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
| xᵢ | xᵢ − μ | (xᵢ − μ)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
Mean (μ) = 5
Population σ = √(32/8) = 2
Sample s = √(32/7) ≈ 2.14
4. Why It Matters
- Units match the original data scale.
- In a normal distribution:
- ~68% of data lies within ±1σ
- ~95% within ±2σ
- Used in confidence intervals, hypothesis testing, and control charts.
5. Beyond the Basics
- Variance: the squared deviation
- Mean Absolute Deviation (MAD): robust alternative
- Skewness & Kurtosis: higher moments
- Continuous Distributions: use integrals
- Bootstrapping: estimate σ from messy data
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