Calculate the Expected Value (E) for Chi-Square test components.
Chi-Square Component (O – E)^2 / E
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Understanding the Expected Value in Chi-Square Tests
The Chi-Square (χ²) test is a powerful statistical tool used to analyze categorical data. It helps determine if there's a statistically significant difference between observed frequencies and expected frequencies. A crucial component in calculating the Chi-Square statistic itself is the expected value (E) for each category.
What is the Expected Value (E)?
The expected value represents the frequency that would be observed in a category if there were no association between the variables being studied (i.e., if the null hypothesis were true). It's a theoretical value calculated based on the marginal totals of the contingency table and the assumption of independence.
The Chi-Square Statistic Formula
The overall Chi-Square statistic is calculated by summing the contributions from each category. The formula for a single category's contribution is:
χ²_component = (O - E)² / E
Where:
O = Observed Frequency (the actual count in a category)
E = Expected Frequency (the theoretical count for that category)
How the Calculator Works
This calculator focuses on computing that individual component (O - E)² / E. You input the observed frequency (O) and the calculated expected frequency (E) for a specific category. The calculator then performs the following steps:
Calculates the difference between the observed and expected values: (O - E).
Squares this difference: (O - E)².
Divides the squared difference by the expected value: (O - E)² / E.
The result is the contribution of that specific category to the overall Chi-Square statistic.
Use Cases for Chi-Square Tests and Expected Values
Goodness-of-Fit Test: To determine if a sample distribution matches a hypothesized population distribution (e.g., Does the observed distribution of car colors sold match the expected distribution based on national trends?).
Test of Independence: To determine if there is a statistically significant association between two categorical variables (e.g., Is there an association between smoking status and lung disease diagnosis?).
Test of Homogeneity: To determine if the distribution of a categorical variable is the same across different populations (e.g., Is the proportion of voters favoring a candidate the same in different age groups?).
Important Considerations
Calculating Expected Values: This calculator assumes you have already calculated the expected value (E) for your category. The method for calculating E depends on the specific Chi-Square test (Goodness-of-Fit, Independence, Homogeneity). For a test of independence, E is typically calculated as: (Row Total * Column Total) / Grand Total.
Assumptions: Chi-Square tests have assumptions, such as sufficient sample size (often requiring expected frequencies to be at least 5 in most cells) and independence of observations.
Interpreting the Result: A larger component value indicates a greater deviation between observed and expected frequencies for that category, thus contributing more significantly to the overall Chi-Square statistic. The final Chi-Square statistic is compared against a critical value (based on degrees of freedom and significance level) to make a decision about the null hypothesis.
function calculateChiSquareExpectedValue() {
var observedValue = parseFloat(document.getElementById("observedValue").value);
var expectedValue = parseFloat(document.getElementById("expectedValue").value);
var resultSection = document.getElementById("result-section");
var chiSquareComponentDisplay = document.getElementById("chiSquareComponent");
var errorMessage = document.getElementById("errorMessage"); // Add an element for error messages if needed
// Clear previous error messages if any
if (errorMessage) {
errorMessage.textContent = "";
}
// Validate inputs
if (isNaN(observedValue) || isNaN(expectedValue)) {
if (errorMessage) errorMessage.textContent = "Please enter valid numbers for both Observed and Expected Values.";
resultSection.style.display = 'none';
return;
}
if (expectedValue === 0) {
if (errorMessage) errorMessage.textContent = "Expected Value cannot be zero to avoid division by zero.";
resultSection.style.display = 'none';
return;
}
if (expectedValue < 0) {
if (errorMessage) errorMessage.textContent = "Expected Value cannot be negative.";
resultSection.style.display = 'none';
return;
}
// Calculate the Chi-Square component
var difference = observedValue – expectedValue;
var squaredDifference = difference * difference;
var chiSquareComponent = squaredDifference / expectedValue;
// Display the result
chiSquareComponentDisplay.textContent = chiSquareComponent.toFixed(4); // Display with 4 decimal places
resultSection.style.display = 'block';
}