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

1. Compute the mean (average).
2. Subtract the mean from each data point (deviation).
3. Square each deviation.
4. Average the squared deviations.
5. 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