Calculate Variance of Continuous Variable Integral

Variance is a fundamental measure of how spread out a set of values is. For continuous variables, we calculate variance using probability density functions and integrals. This guide explains how to compute variance for continuous variables, including the formula, step-by-step calculation, and practical interpretation.

What is Variance of a Continuous Variable?

Variance measures the spread of a continuous random variable around its mean. For continuous variables, we use probability density functions (PDFs) to calculate variance. The variance is the expected value of the squared deviation from the mean.

Key points about variance for continuous variables:

Variance is always non-negative
It's measured in the square of the variable's units
A higher variance indicates greater spread of values
Variance is a key parameter in many probability distributions

Variance Formula for Continuous Variables

The general formula for variance of a continuous random variable X with probability density function f(x) is:

Var(X) = E[(X - μ)²] = ∫[from -∞ to ∞] (x - μ)² f(x) dx

Where:

Var(X) = variance of X
μ = mean of X = E[X] = ∫[from -∞ to ∞] x f(x) dx
f(x) = probability density function of X

This formula calculates the expected value of the squared deviation from the mean. For many common probability distributions, there are simplified formulas for variance.

How to Calculate Variance Using Integrals

Step 1: Determine the Probability Density Function

First, identify the PDF of your continuous variable. Common PDFs include:

Normal distribution
Uniform distribution
Exponential distribution
Gamma distribution

Step 2: Calculate the Mean

Compute the expected value (mean) using the integral of x times the PDF:

μ = ∫[from -∞ to ∞] x f(x) dx

Step 3: Calculate the Variance

Compute the variance by integrating (x - μ)² times the PDF:

Var(X) = ∫[from -∞ to ∞] (x - μ)² f(x) dx

For many distributions, this integral can be simplified using known properties of the distribution.

Step 4: Interpret the Result

Once you have the variance, you can:

Compare it to other variances
Calculate the standard deviation (square root of variance)
Assess the spread of your data

Worked Example

Let's calculate the variance of a continuous variable X with the following PDF:

f(x) = 1/2 for x ∈ [0, 2]

f(x) = 0 otherwise

Step 1: Calculate the Mean

μ = ∫[from 0 to 2] x (1/2) dx = (1/2) ∫[from 0 to 2] x dx

= (1/2) [x²/2] from 0 to 2 = (1/2)(4/2 - 0) = (1/2)(2) = 1

Step 2: Calculate the Variance

Var(X) = ∫[from 0 to 2] (x - 1)² (1/2) dx

= (1/2) ∫[from 0 to 2] (x² - 2x + 1) dx

= (1/2) [x³/3 - x² + x] from 0 to 2

= (1/2) [(8/3 - 4 + 2) - (0 - 0 + 0)]

= (1/2)(8/3 - 2) = (1/2)(8/3 - 6/3) = (1/2)(2/3) = 1/3 ≈ 0.333

The variance of this continuous variable is 1/3, which means the values are spread around the mean by approximately 0.577 (the square root of 1/3).

Interpreting the Variance Result

When you calculate the variance of a continuous variable:

A higher variance indicates greater spread of values
The standard deviation (square root of variance) is in the same units as the variable
Variance is useful for comparing distributions
It's a key parameter in many statistical tests and models

Remember that variance is sensitive to outliers. For skewed distributions, the interquartile range (IQR) might be a more robust measure of spread.

FAQ

What is the difference between variance and standard deviation?: Variance is measured in squared units, while standard deviation is in the original units of the variable. Standard deviation is simply the square root of variance.
How do I calculate variance for a sample of continuous data?: For a sample, you use the sample mean and divide by n-1 (Bessel's correction) instead of n. The formula is: s² = Σ(xi - x̄)² / (n-1).
What is the relationship between variance and probability density functions?: For continuous variables, variance is calculated using integrals over the probability density function. The PDF represents the relative likelihood of different values.
Can variance be negative?: No, variance is always non-negative because it's based on squared deviations. The smallest possible variance is zero, which occurs when all values are identical.
How does variance compare to range?: Variance provides a more comprehensive measure of spread than range, as it considers all data points and their distances from the mean, not just the difference between the maximum and minimum values.