Build intuition for risk with a clean calculator that handles probabilities or raw data and flags errors in real time. See expected value variance and standard deviation with stepwise explanations. Toggle sample or population mode and export results. Designed for learners teachers analysts and everyday decisions Use it for economics finance quality control experiments
Expected value and variance are the twin pillars of quantitative reasoning. Expected value summarizes the long run average outcome of a random variable, while variance measures typical spread around that center. Together they explain central tendency and risk, which makes them essential for probability homework, actuarial work, and data driven decisions in business.
The expected value EV of a discrete distribution is the probability weighted mean. You multiply each possible value by its probability and add the results. In symbols EV equals the sum of x times p. This calculator accepts a probability mass function table, checks whether the probabilities add to one, and can automatically normalize small rounding differences for cleaner reports.
Variance captures average squared distance from the mean. For a population distribution it is the sum of each probability times the squared deviation x minus mean squared. For a sample it uses the unbiased version with n minus one in the denominator. The tool also reports standard deviation as the square root of variance for easier interpretation in original measurement units.
Expected value and variance power everyday choices. An investor compares projects using expected return and risk. A manufacturer monitors quality using mean and variability of part dimensions. A student learns how to compute variance step by step and understands why extreme outcomes raise spread. Clear understanding of population variance versus sample variance prevents common mistakes in reports and dashboards.
You can switch between two input modes. Enter a probability table with values and probabilities or paste raw observations as a dataset. The calculator supports weights or frequencies, detects duplicates, and removes blanks. You may choose population or sample mode, and the engine derives EV variance and standard deviation with exact formulas. Error messages highlight invalid probabilities or non numeric entries so your final numbers are trustworthy.
Use the expected value formula with probabilities to price lotteries and warranty decisions. Estimate the variance of demand to size safety stock. Analyze a probability mass function example for grading policies. Run sensitivity analysis by tweaking probabilities and see how risk changes with heavier tails. With clear steps and labeled formulas, this expected value and variance calculator becomes a reliable learning companion.
Include phrases such as expected value calculator for discrete distribution, how to calculate variance step by step, expected value formula with probabilities, population variance vs sample variance, risk analysis in decision making, and probability mass function example. Bookmark this page, experiment with datasets, export results for documentation, and share downloadable report with teammates or instructors for clarity and reproducibility.
Enter each possible outcome in the value column and its probability in the probability column. Add rows as needed. Select normalize if probabilities do not sum exactly to one.
Population variance divides by the total probability or total weight. Sample variance uses n minus one to correct bias when your numbers represent a finite sample.
Yes. Paste numbers separated by spaces commas or newlines into the Values box. Optionally paste matching frequencies or weights into the Weights box.
If you check normalize the tool rescales probabilities so they add to one while preserving relative weights. This is helpful when rounding has been applied.
Weights act like counts or frequencies for raw data. When normalized they behave like probabilities. The tool supports both interpretations with clear labeling.
Match the precision of the input data or your field standards. The results show many decimal places for transparency but you should round to meaningful digits.
Use your browser print to save as PDF. Copy tables to spreadsheets for further analysis.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.