Variance

Statistical measure of the dispersion of data points

Variance

File:Comparison standard deviations.svg

Comparison of standard deviations and variance

File:Variance visualisation.svg

Visualisation of variance

Variance is a statistical measure that represents the degree of spread in a set of data points. It is calculated as the average of the squared differences from the mean, providing a measure of how much the data points differ from the mean value of the dataset.

Definition

Variance is denoted by \( \sigma^2 \) for a population and \( s^2 \) for a sample. The formula for the variance of a population is:

\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \]

where \( N \) is the number of observations, \( x_i \) is each individual observation, and \( \mu \) is the mean of the population.

For a sample, the variance is calculated as:

\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

where \( n \) is the number of observations in the sample, \( x_i \) is each individual observation, and \( \bar{x} \) is the sample mean.

Properties

• Variance is always non-negative because it is the average of squared deviations.
• A variance of zero indicates that all the values are identical.
• Variance is sensitive to outliers, as it involves squaring the deviations.

Applications

Variance is widely used in statistics and probability theory. It is a fundamental concept in fields such as finance, where it is used to measure the risk of an investment. In quality control, variance is used to assess the consistency of a process.

Relationship with Standard Deviation

The standard deviation is the square root of the variance. It provides a measure of dispersion in the same units as the data, making it more interpretable than variance.

Related Concepts

• Standard deviation
• Covariance
• Chi-squared distribution

Related pages

• Standard deviation
• Probability theory
• Statistics

References

• Hogg, R. V., & Craig, A. T. (1995). Introduction to Mathematical Statistics. Prentice Hall.
• Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Duxbury Press.