Standard Deviation & Variance Calculator

Compute sample vs. population standard deviation, variance, and coefficient of variation.

Sample Standard Deviation (s)
9.487
Arithmetic Mean (x̄) 24
Population Std Dev (σ) 8.485
Sample Variance (s²) 90
Standard Error (s/√n) 4.243
Inputs
First observation x₁.
Second observation x₂.
Third observation x₃.
Fourth observation x₄.
Fifth observation x₅.
Results & Graph
Sample Standard Deviation (s)
9.487
Arithmetic Mean (x̄)
24
Population Std Dev (σ)
8.485
Sample Variance (s²)
90
Standard Error (s/√n)
4.243

Sample Standard Deviation (s)

Graph 1 • Value1 Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 6.2 12.5 50%: 11.541 11.5 50% 75%: 10.479 10.5 75% 100%: 9.487 9.5 100% 125%: 8.591 8.6 125% 150%: 7.823 7.8 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Sample Standard Deviation (s) 9.487 Arithmetic Mean (x̄) 24 Population Std Dev (σ) 8.485 Sample Variance (s²) 90
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Sample Standard Deviation (s)9.487Arithmetic Mean (x̄)24Population Std Dev (σ)8.485Sample Variance (s²)90Standard Error (s/√n)4.243
Summary: Sample standard deviation s = 9.4868, population σ = 8.4853, and arithmetic mean x̄ = 24.
Summary: Sample standard deviation s = 9.4868, population σ = 8.4853, and arithmetic mean x̄ = 24.

Formula & Mathematical Methodology: Standard Deviation Mathematical Identity

s = Σ(xi − x̄)² / (n − 1)
σ = Σ(xi − μ)² / n
s = √[ Σ(x_i − x̄)² / (n − 1) ]; σ = √[ Σ(x_i − μ)² / n ]
  • value1: Value1 (numeric parameter)
  • value2: Value2 (numeric parameter)
  • value3: Value3 (numeric parameter)
  • value4: Value4 (numeric parameter)
  • value5: Value5 (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Standard Deviation

Compute sample vs. population standard deviation, variance, and coefficient of variation.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: s = √[ Σ(x_i − x̄)² / (n − 1) ]; σ = √[ Σ(x_i − μ)² / n ].
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Standard Deviation via s = √[ Σ(x_i − x̄)² / (n − 1) ]; σ = √[ Σ(x_i − μ)² / n ].

Frequently Asked Questions

What mathematical formula does the Standard Deviation & Variance Calculator use?

It directly evaluates s = √[ Σ(x_i − x̄)² / (n − 1) ]; σ = √[ Σ(x_i − μ)² / n ] in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Descriptive Statistics Guide

How to Use the Standard Deviation & Variance Calculator: Practical Guide & Visual Analysis

Calculate sample standard deviation (s), population standard deviation (σ), variance, mean, and standard error of the mean. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching standard deviation & variance calculator, standard deviation, standard deviation variance, descriptive statistics calculator, this interactive tool helps you calculate sample standard deviation (s), population standard deviation (σ), variance, mean, and standard error of the mean.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Value1, Value2, Value3, Value4, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Standard Deviation), Evaluates Standard Deviation via s = √[ Σ(x_i − x̄)² / (n − 1) ]; σ = √[ Σ(x_i − μ)² / n ]. Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).