📐 Math

Standard Deviation

Definition

A statistical measure of the amount of variation or dispersion in a dataset, showing how spread out values are from the mean.

Why is Standard Deviation Important?

Standard Deviation is a foundational mathematical concept used across science, engineering, finance, and everyday problem-solving. From analyzing data sets to optimizing business decisions, this concept provides the analytical framework needed to interpret quantitative information accurately.

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What is Standard Deviation?

Standard deviation (σ or s) measures how spread out numbers are from the mean (average) in a dataset. A low standard deviation means data points cluster tightly around the mean; a high standard deviation means they are spread out over a wide range.

How to Calculate

Population SD: σ = √(Σ(xᵢ − μ)² / N)

Sample SD: s = √(Σ(xᵢ − x̄)² / (n − 1))

Step	Process	Example: {4, 8, 6, 2, 10}
1. Find the mean	Sum ÷ count	(4+8+6+2+10)/5 = 6
2. Subtract mean from each	xᵢ − μ	-2, 2, 0, -4, 4
3. Square each difference	(xᵢ − μ)²	4, 4, 0, 16, 16
4. Average the squares	Σ/N (population) or Σ/(n-1) (sample)	40/5 = 8 (population)
5. Take square root	√variance	√8 ≈ 2.83

The 68-95-99.7 Rule (Empirical Rule)

For normally distributed data:

68% of values fall within ±1 SD of the mean
95% fall within ±2 SD
99.7% fall within ±3 SD