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Variance (probability and statistics)

Variance quantifies dispersion by averaging squared deviations from the mean. It is nonnegative, measured in squared units, and underpins standard deviation, risk measures and many statistical methods.

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Overview: In probability and statistics, variance measures how widely values in a collection or distribution are spread around their central value. Informally, it is the average of the squared differences between each observation and the expected value. This makes variance inherently nonnegative and sensitive to large deviations. The concept arises in probability theory and statistics and applies to a set of observations or to a random variable. The central reference point is the expected value (mean), and the commonly used notation is σ² for a full population and s² for a sample estimate.

Definition and notation

Formally, variance equals the expectation of the squared deviation from the mean. In words: compute each value's difference from the mean, square those differences, and take the average (or expected value). Because differences are squared, the units of variance are the square of the original measurements. The square root of variance is the standard deviation, which returns dispersion to the original units by applying the square root operation.

Key properties

  • Nonnegativity: variance is always ≥ 0; zero indicates all values equal the mean.
  • Units: expressed in squared units of the data (e.g., meters², dollars²).
  • Scaling: multiplying every observation by a constant c multiplies variance by c².
  • Additivity for independent variables: the variance of the sum of independent random variables is the sum of their variances.
  • Estimation: the sample variance often uses a denominator of n−1 to produce an unbiased estimate of the population variance.

History and development

The mathematical concept of measuring dispersion evolved through the 19th and early 20th centuries as statisticians sought ways to quantify variation in biological, economic and social data. Key developments that led to the modern formulation were made by early statisticians and by those who formalized methods of estimation and hypothesis testing. The modern notation and widespread use in inferential statistics became established as probability theory and statistical inference were developed in the early 1900s.

Applications and examples

Variance is used across many fields. In meteorology one might compare temperature variance between cities — for example, seasonal temperatures typically show greater variance in Moscow than in Hawaii. In finance, variance (and its square root, volatility) quantifies investment risk. In experimental design and quality control, analysis of variance (ANOVA) partitions total variability into components attributed to different sources. In everyday business, the word "variance" also denotes the difference between a planned amount and what actually occurred, such as the difference between a budgeted cost and actual spending.

Practical considerations and alternatives

Variance is mathematically convenient but has limitations: it is sensitive to outliers because squaring amplifies large deviations, and its units can be unintuitive. Alternatives used when robustness or interpretability is important include mean absolute deviation and interquartile range. In many analyses practitioners report both variance (or standard deviation) and robust measures to give a fuller picture of dispersion.

Notable distinctions

  • Population vs sample: σ² denotes the true population variance; s² denotes an estimate from sample data.
  • Bias in estimation: the sample variance formula with denominator n−1 corrects bias present when using n.
  • Decomposition: variance can be decomposed into parts (for example, within- and between-group variance), a foundation of ANOVA.

For further technical details and proofs, readers can consult resources linked from the sections above or standard texts in probability theory and statistics.

Data examples • Random variables • Expected value • City climate examples • Climate contrasts • Standard deviation explanation • Mathematical operations • Budget variance

Density functions of two normally distributed random variables X(red) and Y(green) with equal expected value μ {\displaystyle \mu _{X}=\mu _{Y}=0}, but different variances. The horizontal axis shows the value, the vertical axis the frequency. Since the red curve f(x)narrower around the expected value than the green f(y), it has a lower variance ( {\displaystyle \sigma _{X}^{2}<\sigma _{Y}^{2}}). The square root of the variance, the standard deviation, can be read from the turning points in the normal distribution.

Introduction to the problem

As a starting point for the construction of the variance, one considers an arbitrary quantity that is dependent on chance and can thus assume different values. This quantity, which is Xdenoted by following, follows a certain distribution. The expected value of this variable is denoted by

{\displaystyle \mu :=\mathbb {E} (X)}

abbreviated. The expected value indicates the average value of the random variable X. It can be interpreted as the center of gravity of the distribution (see also section Interpretation) and reflects its location. However, in order to sufficiently characterize a distribution, a quantity is missing which, as a key figure, provides information about the strength of dispersion of a distribution around its centroid. This quantity should always be greater than or equal to zero, since negative dispersion cannot be meaningfully interpreted. A first obvious approach would be to use the mean absolute deviation of the random variable from its expected value:

{\displaystyle \mathbb {E} \left(|X-\mu |\right)}.

Since the magnitude function used in the definition of the mean absolute deviation is not differentiable everywhere, and otherwise sums of squares are usually used in statistics, it makes sense to use the mean squared deviation, i.e. the variance, instead of the mean absolute deviation.

Calculation of the variance

Variance for discrete random variables

A random variable Xwith a finite or countably infinite range of values {\displaystyle {\mathcal {T}}=\{x_{1},x_{2},\dotsc ,x_{k},\dotsc \}}is called discrete. Its variance is then calculated as the weighted sum of the squares of the deviations (from the expected value):

{\displaystyle \sigma ^{2}=(x_{1}-\mu )^{2}p_{1}+(x_{2}-\mu )^{2}p_{2}+\ldots +(x_{k}-\mu )^{2}p_{k}+\ldots =\sum _{i\geq 1}(x_{i}-\mu )^{2}p_{i}}.

Here {\displaystyle p_{i}=P(X=x_{i})}the probability that X x_{i}takes the value Thus, in the above sum, each possible expression {\displaystyle (x_{i}-\mu )^{2}} is p_{i}weighted by the probability of its occurrence Thus, for discrete random variables, the variance is a weighted sum with weights {\displaystyle p_{i}\;(i=1,\ldots ,n)}. The expected value of a discrete random variable Xalso represents a weighted sum given by

{\displaystyle \mu =x_{1}p_{1}+x_{2}p_{2}+\ldots +x_{k}p_{k}+\ldots =\sum _{i\geq 1}x_{i}p_{i}}

is given. The sums extend in each case over all values that this random variable can assume. In the case of a countably infinite range of values, the result is an infinite sum. In words, the variance, in the discrete case, is calculated as the sum of the products of the probabilities of the realizations of the random variable Xwith the respective squared deviation.

Variance for continuous random variables

A random variable X is said to be continuous if its value range is a countable quantity. If the random variable is absolutely continuous, then as a consequence of Radon-Nikodým's theorem there exists a probability density function (density for short) f_{X}(x). In the case of a real-valued random variable, the distribution function {\displaystyle F(t)=P(X\leq t)}, {\displaystyle t\in \mathbb {R} }, as an integral as follows:

{\displaystyle F(t)=\int _{-\infty }^{t}f(x)\,\mathrm {d} x}

For the variance of a real-valued random variable Xwith density f(x)now holds

{\displaystyle \operatorname {Var} (X)=\int _{-\infty }^{\infty }(x-\mathbb {E} [X])^{2}f(x)\,\mathrm {d} x\quad }, where its expected value is given by {\displaystyle \mathbb {E} [X]=\int _{-\infty }^{\infty }xf(x)\,\mathrm {d} x}.

If a density exists, the variance is calculated as the integral over the product of the squared deviation and the density function of the distribution. Thus, it is integrated over the space of all possible expressions (possible value of a statistical characteristic).

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  • digital.library.adelaide.edu.au : The correlation between relatives on the supposition of Mendelian inheritance.