Standard Deviation Calculator
Calculate standard deviation, mean, median, mode, and variance for your dataset. Analyze data distribution and spread with statistical measures.
How to Use
1. Enter your numbers (separated by commas or spaces)
2. Get instant statistical calculations
3. View comprehensive data analysis
Results Include
Mean & Median: Central tendency measures
Standard Deviation: Data spread measure
Variance & Mode: Additional statistics
Enter numbers separated by commas or spaces
Statistical Theory
Standard deviation represents a fundamental measure of statistical dispersion, quantifying the typical deviation of observations from their mean value. This concept, developed through the work of Karl Pearson, provides a standardized measure of variability that maintains the original units of measurement. The theoretical foundation combines principles of probability theory with geometric interpretations of data spread.
The relationship between standard deviation and the normal distribution emerges from the central limit theorem, establishing connections between sample statistics and population parameters. This theoretical framework enables precise quantification of probability distributions and forms the basis for statistical inference.
Mathematical Framework
The standard deviation is defined through precise mathematical formulas:
Population Standard Deviation:
σ = √(Σ(x - μ)²/N)
Sample Standard Deviation:
s = √(Σ(x - x̄)²/(n-1))
Where:
- σ = Population standard deviation
- s = Sample standard deviation
- μ = Population mean
- x̄ = Sample mean
- N = Population size
- n = Sample size
Probability Distribution
In the context of the normal distribution, standard deviation defines probability intervals:
Empirical Rule:
- • μ ± 1σ contains ~68% of data
- • μ ± 2σ contains ~95% of data
- • μ ± 3σ contains ~99.7% of data
Normal Density Function:
f(x) = (1/(σ√(2π)))e^(-(x-μ)²/(2σ²))
Statistical Properties
Standard deviation possesses important mathematical properties:
Key Properties:
- • Scale dependence: σ(ax) = |a|σ(x)
- • Translation invariance: σ(x + b) = σ(x)
- • Non-negativity: σ ≥ 0
- • Pythagorean theorem for variances
These properties enable reliable comparison of variability across different scales and measurements.
Advanced Applications
Standard deviation extends to multivariate analysis through covariance and correlation:
Correlation Coefficient:
ρ = Cov(X,Y)/(σₓσᵧ)
Standardized Score:
z = (x - μ)/σ
Standard Error:
SE = σ/√n
These relationships form the basis for advanced statistical analysis and inference methods.