Expected Value Calculator

Calculate expected value, variance, and standard deviation from five possible outcomes and their probabilities. Expected Value Calculator helps students, analysts, and decision-makers evaluate uncertain results and measure financial or statistical risk.

Highlights

  • Calculate expected value from up to five possible outcomes.
  • Measure risk with variance and standard deviation.
  • Analyze relative risk using the coefficient of variation.
  • Enter custom outcomes and probabilities with flexible values.
  • Useful for probability analysis, financial planning, and decision-making.

Table of Contents

  1. How Expected Value Calculator Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How Expected Value Calculator Works?

The Expected Value Calculator evaluates five possible outcomes and their probabilities. Each outcome is represented by xi, while its entered percentage probability is converted to a decimal probability using pi = Pi/100. The probabilities should describe mutually exclusive possibilities and normally add up to 100%.

p=P100

The expected value, also called the probability-weighted average, gives the long-run average result if the same probability model were repeated many times. The calculator multiplies every outcome by its probability and adds the results.

μ=∑i=15pixi

Variance measures the probability-weighted squared distance of each outcome from the expected value. Squaring the differences prevents positive and negative deviations from canceling each other.

Variance=σ2=∑i=15pi(xi−μ)2

Standard deviation is the square root of variance, so it expresses the typical distance between an outcome and the expected value. The coefficient of variation compares this typical spread with the size of the expected value, providing a relative-risk measure.

σ=σ2;CV=σ|μ|×100%

OutputMeaning
Expected ValueProbability-weighted average outcome.
VarianceAverage squared deviation from the expected value.
Standard DeviationTypical distance of outcomes from the expected value.
Coefficient of VariationStandard deviation expressed relative to the absolute expected value.

For example, using the default outcomes and probabilities, the weighted contributions are calculated as (-10 × 0.10), (0 × 0.20), (10 × 0.30), (25 × 0.25), and (100 × 0.15), which are then added to obtain the expected value. The same five outcome-probability pairs are used for variance, standard deviation, and coefficient of variation.

Practical Use Cases

Students and educators can use the calculator to check probability theory exercises involving discrete random variables. Entering up to five outcomes makes it possible to compare the expected value with variance and standard deviation in one calculation.

Financial analysts can model possible gains and losses for an investment, project, or business decision. A positive expected value may indicate a favorable average result, while the standard deviation shows how widely individual scenarios can differ.

Risk managers can assign probabilities to several loss or incident scenarios and quantify the resulting spread. Variance and standard deviation help distinguish a decision with a stable result from one with substantial uncertainty.

Business decision-makers can compare possible project outcomes such as weak, moderate, and strong demand. The expected value summarizes the average result, while the coefficient of variation helps compare relative risk between alternatives with different expected values.

Game designers and odds analysts can evaluate rewards, penalties, or payouts associated with up to five possible events. The expected value identifies the long-run average payoff, and the dispersion measures show how unpredictable individual results may be.

Accuracy and Limitations

The calculator applies the standard discrete-probability formulas to five entered outcome-probability pairs and reports results based on the precision of the supplied values. Probabilities should be between 0% and 100% and should represent a complete probability distribution, typically totaling 100%; incomplete or overlapping scenarios can make the results misleading. The coefficient of variation is undefined when the expected value is zero because it divides by the absolute expected value, and it can become highly sensitive when that value is close to zero. The calculation also assumes that each outcome is a single numerical result with a known probability and does not estimate probabilities, model correlations, or account for uncertainty in the inputs.

Quick Examples

Example #1: Investment Return Scenarios

Estimate the expected return and risk of an investment across five possible financial outcomes.

Example #2: Product Demand Forecast

Calculate the expected sales volume and demand variability for five possible market conditions.

Example #3: Project Profitability Risk

Evaluate the expected profit and financial risk of a project under different completion outcomes.

Frequently Asked Questions

Expected value is the probability-weighted average of all possible outcomes. Over many repeated trials, the average result tends to approach this value.

Yes. A negative expected value indicates that potential losses outweigh potential gains when considering their probabilities.

Variance measures the average squared distance from the expected value, while standard deviation is its square root and uses the same units as the outcomes.

A high standard deviation indicates that outcomes are widely spread around the expected value, suggesting greater uncertainty or absolute risk.

Probabilities should represent the estimated likelihood of each outcome and collectively add up to 100%. They may be based on historical data, research, or informed assumptions.

Cite This Calculator