Standard Deviation of a Probability Distribution Calculator
Probability Distribution Data
Enter the possible outcomes (values) and their corresponding probabilities. Ensure probabilities sum to 1.
Calculation Results
Standard Deviation will appear here
Understanding Standard Deviation of a Probability Distribution
The standard deviation of a probability distribution is a crucial measure of the dispersion or spread of possible outcomes around the expected value (mean). It quantifies how much, on average, the individual outcomes are likely to deviate from the expected value. A low standard deviation indicates that the outcomes tend to be close to the expected value, while a high standard deviation suggests that the outcomes are more spread out.
Key Concepts:
Probability Distribution: A function that describes the likelihood of obtaining different possible values for a random variable. For discrete distributions, this involves listing each possible outcome and its associated probability.
Random Variable (X): A variable whose value is a numerical outcome of a random phenomenon.
Expected Value (E[X] or μ): The weighted average of all possible values of a random variable, where the weights are the probabilities of those values. It represents the long-run average outcome. Calculated as: E[X] = Σ (x * P(x))
Variance (Var(X) or σ²): The expected value of the squared deviation from the mean. It measures the average squared difference between each outcome and the expected value. Calculated as: Var(X) = E[(X - E[X])²] = Σ ((x - E[X])² * P(x))
Standard Deviation (SD or σ): The square root of the variance. It's often preferred because it's in the same units as the original data, making it more interpretable. Calculated as: SD = √Var(X)
How to Calculate Standard Deviation:
To calculate the standard deviation (σ) for a discrete probability distribution, follow these steps:
List Outcomes and Probabilities: Identify all possible outcomes (x) and their corresponding probabilities (P(x)). Ensure the sum of all P(x) equals 1.
Calculate the Expected Value (Mean): Multiply each outcome value (x) by its probability (P(x)) and sum these products.
E[X] = x₁*P(x₁) + x₂*P(x₂) + ... + xn*P(xn)
Calculate the Variance: For each outcome:
Subtract the Expected Value (E[X]) from the outcome value (x).
Square the result: (x - E[X])².
Multiply this squared difference by the probability of the outcome: (x - E[X])² * P(x).
Sum these values for all outcomes to get the Variance.
Var(X) = (x₁ - E[X])²*P(x₁) + (x₂ - E[X])²*P(x₂) + ... + (xn - E[X])²*P(xn)
Calculate the Standard Deviation: Take the square root of the Variance.
σ = √Var(X)
Use Cases:
Understanding and calculating standard deviation is vital in many fields:
Finance: Measuring the risk associated with an investment. Higher standard deviation implies greater volatility and risk.
Statistics: Describing the spread of data in samples and populations.
Quality Control: Monitoring variations in manufacturing processes to ensure consistency.
Natural Sciences: Analyzing variability in experimental results.
Insurance: Estimating the potential range of claims.
This calculator helps you quickly compute the standard deviation for any discrete probability distribution you define.
var outcomeCount = 3; // Start with 3 pairs
function addProbabilityPair() {
var container = document.getElementById('probabilityInputsContainer');
var newPairId = 'pair' + outcomeCount;
var newPairDiv = document.createElement('div');
newPairDiv.className = 'probability-pair';
newPairDiv.id = newPairId;
newPairDiv.innerHTML = `
`;
container.appendChild(newPairDiv);
outcomeCount++;
}
function removeProbabilityPair() {
if (outcomeCount > 1) {
var container = document.getElementById('probabilityInputsContainer');
var lastPairId = 'pair' + (outcomeCount – 1);
var lastPair = document.getElementById(lastPairId);
if (lastPair) {
container.removeChild(lastPair);
outcomeCount–;
}
} else {
alert("You must have at least one outcome.");
}
}
function calculateStandardDeviation() {
var expectedValue = 0;
var variance = 0;
var outcomes = [];
var probabilities = [];
var probabilitySum = 0;
for (var i = 0; i < outcomeCount; i++) {
var valueInput = document.getElementById('value' + i);
var probabilityInput = document.getElementById('probability' + i);
var value = parseFloat(valueInput.value);
var probability = parseFloat(probabilityInput.value);
if (isNaN(value) || isNaN(probability)) {
alert("Please enter valid numbers for all outcome values and probabilities.");
return;
}
if (probability 1) {
alert("Probabilities must be between 0 and 1.");
return;
}
outcomes.push(value);
probabilities.push(probability);
probabilitySum += probability;
}
// Check if probabilities sum to approximately 1
if (Math.abs(probabilitySum – 1) > 0.001) {
alert("The sum of probabilities must be equal to 1. Current sum: " + probabilitySum.toFixed(4));
return;
}
// Calculate Expected Value
for (var i = 0; i < outcomes.length; i++) {
expectedValue += outcomes[i] * probabilities[i];
}
// Calculate Variance
for (var i = 0; i < outcomes.length; i++) {
variance += Math.pow(outcomes[i] – expectedValue, 2) * probabilities[i];
}
// Calculate Standard Deviation
var standardDeviation = Math.sqrt(variance);
document.getElementById('result').textContent = standardDeviation.toFixed(4);
document.getElementById('expected-value-display').textContent = "Expected Value (Mean): " + expectedValue.toFixed(4);
}