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What is standard deviation?

Standard deviation (σ) measures how spread out data is around the mean. A low value means data points are tightly clustered (consistent); a high value indicates high variability. It's the most widely used measure of dispersion in statistics, quality control, finance, and science, because it's expressed in the same units as the original data.

Standard deviation formula

Full population (σ):
σ = √( Σ(xᵢ − μ)² / N )

Sample (s) — more common in practice:
s = √( Σ(xᵢ − x̄)² / (n−1) )

The difference: when analyzing a sample rather than the full population, divide by (n−1) instead of n (Bessel's correction) for an unbiased estimate.

Step-by-step example

The maximum temperatures over 5 days in May were: 22, 25, 21, 28, 24 °C.

1. Mean (μ) = (22+25+21+28+24) / 5 = 24°C
2. Squared differences: (22−24)²=4, (25−24)²=1, (21−24)²=9, (28−24)²=16, (24−24)²=0
3. Sum = 30; divided by 5 = 6 (variance)
4. σ = √6 ≈ 2.45°C

The 68-95-99.7 rule

In a normal distribution, approximately:

• 68% of data falls within ±1 standard deviation of the mean.
• 95% within ±2 standard deviations.
• 99.7% within ±3 standard deviations.

In the example above, 68% of temperatures fall between 24 − 2.45 = 21.55°C and 24 + 2.45 = 26.45°C.

Practical applications

• Quality control: In manufacturing, detects out-of-tolerance parts; the goal is σ < specification limit / 3.
• Finance (volatility): Standard deviation of daily stock returns measures risk. Higher σ = greater uncertainty.
• Education: Normalize exam scores for comparison across different tests (Z-score = (x − μ) / σ).
• Meteorology: Quantify climate variability between regions or decades.