Discrete Variance Calculator

Unravel the spread of your discrete data with ease. Enter your outcomes and probabilities to calculate variance and visualize distributions.

Input Data

Outcomes (x_i):

Enter comma-separated numerical outcomes.

Probabilities (p_i):

Enter comma-separated probabilities for each outcome (sum must be 1).

Variance (Var[X]):

Formula used: $$Var(X) = \sum_{i=1}^{n} (x_i - \mu)^2 P(x_i)$$, where $$\mu = E[X] = \sum_{i=1}^{n} x_i P(x_i)$$ is the expected value.

Probability Distribution Visualization

Understanding Discrete Variance

Variance measures how spread out a discrete random variable's possible values are. A higher variance indicates that the values are more spread out from the expected value (mean), while a lower variance suggests they are clustered closer to the mean.

For a discrete random variable X, the variance, denoted as Var(X) or σ², is calculated using the formula: $$Var(X) = \sum_{i=1}^{n} (x_i - \mu)^2 P(x_i)$$, where:

x_i are the possible outcomes of X
P(x_i) is the probability of each outcome x_i
μ is the expected value (mean) of X, calculated as $$\mu = E[X] = \sum_{i=1}^{n} x_i P(x_i)$$

This calculator helps you quickly compute the variance by inputting the outcomes and their corresponding probabilities, providing a clear measure of data dispersion.

Learn more about variance and discrete random variables on resources like Wikipedia and introductory statistics textbooks.