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Standard Deviation & Variance
Financial Math • Grades 9, 10, 11, 12
Financial Math
Standard Deviation & Variance
Measure statistical spread, variance, and standard deviation with σ = √(Σ(x - μ)² / N).
The Formula
σ = √Σ(x - μ)² / N
Used for: Measuring data dispersion, spread, and statistical volatility around the arithmetic mean
σPopulation standard deviation
sSample standard deviation (divided by n - 1)
μMean (average): Σx / N
xIndividual data point
NTotal number of data points
⚡Live Interactive Quick Solver
Instant CalculationPopulation Std Dev (σ):1.7078
Mean (μ): 5.50 • Variance (σ²): 2.9167 (N = 6)
\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}, \quad s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}
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🎯 Key Concepts & Rules
Essential principles for calculating with this formula:
📉
σ = Standard DeviationAverage distance of data points from the arithmetic mean.
🔢
Variance = σ²Square of the standard deviation.
📊
68-95-99.7 RuleIn a normal distribution, 68% of data lies within 1 standard deviation.
📝 Worked Examples & Step-by-Step Solutions
Example 1: Data Set {2, 4, 4, 4, 5, 5, 7, 9}
Problem:
Find mean (μ) and population standard deviation (σ).1
Mean (μ)
Sum = 40 / 8 = 52
Squared Deviations
(2-5)² + (4-5)²... = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 323
Variance (σ²)
32 / 8 = 44
Standard Deviation (σ)
√4 = 2Answer: Mean μ = 5, σ = 2
💡 Pro Tips & Common Mistakes
Population (N) vs Sample (n - 1)
Use n - 1 (Bessel's correction) when calculating sample standard deviation s.
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