Expected Value & Standard Error Calculator

Here’s a comprehensive table summarizing the key aspects of Expected Value and Standard Error:

Concept	Expected Value	Standard Error
Definition	The anticipated average value of a random variable or investment in the long run	A measure of the accuracy of a sample statistic (usually the mean) as an estimate of a population parameter
Formula	E(X) = Σ [x * P(x)]	SE = s / √n
Interpretation	Long-term average outcome of a random variable	Variability of the sample mean around the true population mean
Uses	– Estimating future returns on investments – Analyzing probability distributions – Decision-making under uncertainty	– Assessing the precision of sample estimates – Constructing confidence intervals – Hypothesis testing
Relationship to sample size	Not directly affected by sample size	Decreases as sample size increases
Key characteristics	– Can be positive, negative, or zero – Represents the center of a probability distribution	– Always positive – Smaller values indicate more precise estimates
Calculation example	For a die roll: E(X) = (1/6 * 1) + (1/6 * 2) + … + (1/6 * 6) = 3.5	For a sample mean: SE = s / √n, where s is the sample standard deviation and n is the sample size
Importance in statistics	Fundamental in probability theory and decision analysis	Critical for inferential statistics and understanding sampling variability

This table provides a concise overview of Expected Value and Standard Error, highlighting their definitions, formulas, interpretations, uses, and key characteristics. It’s important to note that while Expected Value is a concept primarily used in probability and decision theory, Standard Error is crucial in inferential statistics for assessing the reliability of sample estimates.