Standard Deviation Calculator

What is Standard Deviation?

Standard deviation (σ or s) measures how spread out numbers are from their average (mean). A low standard deviation means values are close to the mean; a high standard deviation means values are spread out.

Population vs. Sample

Variance

Variance is the square of standard deviation:

• Population variance: σ² = Σ(x - μ)² / N
• Sample variance: s² = Σ(x - x̄)² / (n-1)

Standard deviation is more intuitive because it's in the same units as the original data.

Z-Score (Standard Score)

Z-score tells you how many standard deviations a value is from the mean:

z = (x - μ) / σ

• z = 0: Value equals the mean
• z = 1: Value is 1 standard deviation above mean
• z = -2: Value is 2 standard deviations below mean
• |z| > 3: Unusual value (outlier)

Coefficient of Variation (CV)

CV expresses standard deviation as a percentage of the mean:

CV = (σ / μ) × 100%

• Low CV (<15%): Low variability, consistent data
• Medium CV (15-30%): Moderate variability
• High CV (>30%): High variability, inconsistent data

Real-World Applications

• Quality Control: Manufacturing processes with low SD are more consistent
• Finance: Stocks with high SD are more volatile/risky
• Education: Test score SD shows how varied student performance is
• Medicine: Clinical trials measure treatment effect variability
• Weather: Temperature SD shows climate stability

Example

Test scores: [70, 75, 80, 85, 90]

1. Mean = (70+75+80+85+90) / 5 = 80
2. Deviations: [-10, -5, 0, 5, 10]
3. Squared deviations: [100, 25, 0, 25, 100]
4. Sum of squared deviations = 250
5. Variance (population) = 250 / 5 = 50
6. Standard deviation = √50 ≈ 7.07

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