Variance

In probability theory and statistics, the variance is a way to measure how far a set of numbers is spread out. Variance describes how much a random variable differs from its expected value. The variance is defined as the average of the squares of the differences between the individual (observed) and the expected value. This means that it is always positive. A variance is often represented by the symbol $\sigma ^{2}$ , if the data is the entire population, and $s^{2}$ , if the data is from a sample.^[1]^[2]^[3]

In practice, variance is a measure of how much something changes. For example, temperature has more variance in Moscow than in Hawaii.

The variance is not simply the average difference from the expected value. The standard deviation, which is the square root of the variance and comes closer to the average difference, is also not simply the average difference. Variance and standard deviation are used because it makes the mathematics easier—when adding two random variables together.

In accountancy, a variance refers to the difference between the budget for a cost, and the actual cost.

History

Karl Pearson, the father of biometry, first used the term variance as follows:

"It is here attempted to (show) the biometrical properties of a population of a more general type that has (..) been examined, inheritance in which follows this scheme. It is hoped that in this way it will be possible to make a more exact analysis of the causes of human variability. The great body of available statistics shows us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and that therefore, the variablility may be uniformly measured by the standard deviation, corresponding to the square root of the mean square error."^[4]

Normal distribution, with variance and mean as parameters

References

↑ "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-09-11.
↑ Weisstein, Eric W. "Variance". mathworld.wolfram.com. Retrieved 2020-09-11.
↑ "Standard Deviation and Variance". www.mathsisfun.com. Retrieved 2020-09-11.
↑ Ronald Aylmer Fisher: The correlation between relatives on the supposition of Mendelian inheritance., Trans. Roy. Soc. Edinb. 52: 399-433, 1918.

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic cubic generalized/power Median Mode
Dispersion	Variance Standard deviation Average absolute deviation Coefficient of variation Percentile Range Interquartile range
Shape	Central limit theorem Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Optimal design Sample size determination Replication Missing data
Survey methodology	Sampling stratified cluster Standard error Opinion poll Questionnaire
Controlled experiments	Scientific control Randomized experiment Randomized controlled trial Random assignment Blocking Interaction Factorial experiment
Adaptive Designs	Adaptive clinical trial Up-and-Down Designs Stochastic approximation
Observational Studies	Cross-sectional study Cohort study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Specific tests

Z-test (normal) Student's t-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity
Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)
Frequency domain	Spectral density estimation Fourier analysis Wavelet Whittle likelihood

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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Variance

History

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