About this calculator
How to measure the average level and volatility of a random variable? Expectation and variance are two of the most important numerical characteristics in probability and statistics. The expectation (mean) E(X) represents the average value of the random variable and reflects the central tendency of the data. The variance Var(X) represents the degree to which the random variable deviates from the expectation and reflects the degree of dispersion of the data. The standard deviation σ is the square root of the variance, which has the same unit as the original data and is more intuitive.
For discrete random variables, the expectation is E(X) = Σ xᵢpᵢ (the sum of each value multiplied by its probability). Variance Var(X) = E[(X-E(X))²] = E(X²) - [E(X)]². For continuous random variables, the expectation and variance are calculated using integrals. Expectation and variance have many important properties, such as E(aX+b) = aE(X)+b, Var(aX+b) = a²Var(X).
In practical applications, expectations and variances are everywhere. In investment decisions, the expected rate of return represents the average return, and the variance represents the risk. In quality control, the expectation of product dimensions is the target value, and the variance represents stability. In test score analysis, the expectation is the average score, and the variance reflects the dispersion of the scores. In actuarial science, expected claims are used for pricing and variances are used for risk assessment.
Our expected variance calculator supports calculations for both discrete and continuous random variables. You can enter a probability distribution table and automatically calculate statistics such as expectation, variance, and standard deviation. Detailed calculation procedures and explanations of statistical significance are also provided to help you understand these concepts. Whether students are learning probability statistics or data analysts are conducting risk assessments, this tool can provide accurate and efficient calculation services.
What it calculates
The expectation and variance calculator finds the expected value, variance, and standard deviation of a discrete random variable.
Formula
- E(X) = sum(x_i * p_i)
- Var(X) = sum((x_i - E(X))^2 * p_i)
- SD(X) = sqrt(Var(X))
Inputs
- Possible values x_i.
- Probability p_i for each value.
- The probabilities should usually sum to 1.
Example
| Value | Probability | Contribution |
|---|---|---|
| 0 | 0.5 | 0 * 0.5 |
| 10 | 0.5 | 10 * 0.5 |
| Expected value | - | 5 |
How to interpret the result
Expected value is the long-run average. Variance measures spread around the expected value, and standard deviation uses the same unit as the original variable.
Common mistakes
- Probabilities should not drift away from a total of 1.
- The expected value does not have to be an actually possible value.
- Variance is measured in squared units.
How to use
Using the expected variance calculator is very simple. Just enter the value of the random variable and the corresponding probability.
**Basic steps:** 1. Select the random variable type (discrete or continuous) 2. Enter the value xᵢ of the random variable 3. Enter the corresponding probability pᵢ (discrete type) or probability density (continuous type) 4. Click the "Calculate" button to view the results
**Example 1:** Expectation and variance of a die roll. X takes values 1,2,3,4,5,6, and the probability is 1/6. Expect E(X) = (1+2+3+4+5+6)/6 = 3.5. E(X²) = (1+4+9+16+25+36)/6 = 15.167. VarianceVar(X) = 15.167 - 3.5² = 2.917. Standard deviation σ ≈ 1.708.
**Example 2:** Expectation and variance of investment returns. Investment A: The probability of a 10% return is 0.5, and the probability of a -5% return is 0.5. Expected E(X) = 10%×0.5 + (-5%)×0.5 = 2.5%. Variance Var(X) = [10%²×0.5 + (-5%)²×0.5] - 2.5%² = 0.005625, standard deviation σ = 7.5%.
**Example 3:** Exam score analysis. The results of a certain class: 10 students scored 60 points, 20 students scored 70 points, 30 students scored 80 points, 20 students scored 90 points, and 20 students scored 100 points. Total number of people: 100. Expected E(X) = (60×10 + 70×20 + 80×30 + 90×20 + 100×20)/100 = 81 points. Calculate variance and standard deviation to assess the dispersion of grades.
The calculator will display statistics such as expectation, variance, standard deviation, coefficient of variation, etc., and provide detailed calculation steps.
Main features
• Discrete random variables: Calculate the expectation and variance of a discrete distribution • Continuous random variables: Calculate the expectation and variance of a continuous distribution • Various statistics: expectation, variance, standard deviation, coefficient of variation • Calculation steps: show detailed calculation process • Probability verification: automatically checks whether the sum of probabilities is 1 • Common distributions: Provides fast calculations of binomial distribution, Poisson distribution, etc. • Data import: Supports importing data from Excel and CSV • Chart display: plot probability distribution and expected position • Statistical significance: Explain what expectations and variances actually mean • Totally free: no registration required, use anytime
Use cases
• Investment decisions: Calculate the expected return and risk of an investment portfolio • Quality Control: Analyze the stability of product quality • Test analysis: Assessing the mean and dispersion of test scores • Actuarial: Calculation of expected claims and risk reserves • Project management: Assessing project duration and cost uncertainties • Data analysis: describe the central tendency and dispersion of data • Probability and statistics learning: students learn the concepts of expectation and variance • Risk assessment: quantifying the magnitude of the risk • Decision analysis: comparing the expected utility of different options • Scientific research: analyzing the statistical characteristics of experimental data