Varianformeln

The Varianformeln, also known as the variance formulas, are a set of mathematical equations used to calculate the variance of a random variable. Variance is a measure of the dispersion of a set of data points from their mean value. The Varianformeln are fundamental in statistics and probability theory, providing a way to quantify the variability of a dataset.

The basic formula for the variance of a random variable X is:

Var(X) = E[(X - μ)^2]

where E denotes the expected value (mean) of the random variable, and μ is the mean of X. This formula can be expanded into two commonly used forms:

1. Population Variance: For a finite population with n data points, the population variance is given by:

Var(X) = Σ[(X_i - μ)^2] / n

where X_i represents each individual data point, and Σ denotes the sum over all data points.

2. Sample Variance: For a sample of data, the sample variance is calculated using:

s^2 = Σ[(X_i - x̄)^2] / (n - 1)

where x̄ is the sample mean, and the division by (n - 1) is known as Bessel's correction, which provides an unbiased estimate of the population variance.

These formulas are essential in various fields, including finance, engineering, and social sciences, where understanding the variability of data is crucial for making informed decisions and predictions.