Test Statistic Calculator
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Test Statistic Calculator
Calculate common test statistics for hypothesis testing.
Understanding Test Statistics
In inferential statistics, a test statistic is a value calculated from sample data that summarizes the evidence against a null hypothesis. It's a crucial component in hypothesis testing, as it allows us to determine whether to reject or fail to reject the null hypothesis. The specific formula for a test statistic depends on the type of data, the sample size, and the assumptions made about the population(s).
Common Test Statistics and Their Calculations
1. Z-test (Population Variance Known)
The Z-test is used to test hypotheses about a population mean when the population standard deviation is known, or when the sample size is large (typically n > 30) and the sample standard deviation is used as an estimate.
Formula:
Z = (x̄ – μ₀) / (σ / √n)
- x̄: Sample mean
- μ₀: Hypothesized population mean (from the null hypothesis)
- σ: Population standard deviation
- n: Sample size
If comparing two samples with known population variances (σ₁² and σ₂²), the formula is:
Formula (Two-Sample):
Z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)
2. T-test (Independent Samples)
The T-test is used when the population standard deviation is unknown and the sample size is small. It assumes the data follows a normal distribution.
- Equal Variances Assumed (Pooled T-test): Used when the variances of the two populations are assumed to be equal.
- Unequal Variances (Welch's T-test): Used when the variances are not assumed to be equal. This is generally preferred as it's more robust.
Formula (One-Sample):
t = (x̄ – μ₀) / (s / √n)
(Where 's' is the sample standard deviation)
Formula (Two-Sample, Equal Variances):
First, calculate the pooled standard deviation (sₚ):
sₚ² = [ (n₁ – 1)s₁² + (n₂ – 1)s₂² ] / (n₁ + n₂ – 2)
Then, the t-statistic:
t = (x̄₁ – x̄₂) / √(sₚ²/n₁ + sₚ²/n₂)
Degrees of freedom (df) = n₁ + n₂ – 2
Formula (Two-Sample, Unequal Variances – Welch's):
t = (x̄₁ – x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Degrees of freedom (df) are calculated using the Welch–Satterthwaite equation (complex, often approximated or calculated by software).
3. Paired T-test
Used when the samples are dependent (e.g., before-and-after measurements on the same subjects). It tests the mean of the differences between paired observations.
Formula:
t = d̄ / (s / √n)
- d̄: Mean of the differences between paired observations
- s: Standard deviation of the differences
- n: Number of pairs
4. Chi-Squared (χ²) Goodness-of-Fit Test
Used to determine if a sample distribution matches a hypothesized population distribution. It compares observed frequencies to expected frequencies.
Formula:
χ² = Σ [ (Oᵢ – Eᵢ)² / Eᵢ ]
- Oᵢ: Observed frequency for category i
- Eᵢ: Expected frequency for category i
- Σ: Summation over all categories
Degrees of freedom (df) = number of categories – 1 (or – number of parameters estimated from the data).
How to Use This Calculator
- Select the Test Type from the dropdown.
- Depending on the selected test, relevant input fields will appear or remain visible.
- Enter the required sample data (means, standard deviations, sizes, etc.) accurately.
- Click "Calculate Test Statistic".
- The calculated test statistic value will be displayed below.
Remember that the test statistic is only one part of hypothesis testing. You would typically compare this value to a critical value from a distribution table or use it to calculate a p-value to make a decision about your null hypothesis.
function calculateTestStatistic() {
var testType = document.getElementById("testType").value;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous result
var sample1Mean = parseFloat(document.getElementById("sample1Mean").value);
var sample1StdDev = parseFloat(document.getElementById("sample1StdDev").value);
var sample1Size = parseFloat(document.getElementById("sample1Size").value);
var sample2Mean = parseFloat(document.getElementById("sample2Mean").value);
var sample2StdDev = parseFloat(document.getElementById("sample2StdDev").value);
var sample2Size = parseFloat(document.getElementById("sample2Size").value);
var testStatistic = NaN;
var errorMessage = "";
// Input validation helper function
function isValidNumber(value) {
return !isNaN(value) && isFinite(value);
}
try {
switch (testType) {
case "z-test-known-variance":
var populationVariance = parseFloat(document.getElementById("populationVariance").value);
var populationMean = parseFloat(document.getElementById("populationMean").value);
var populationStdDev = Math.sqrt(populationVariance); // Calculate sigma from sigma^2
if (!isValidNumber(sample1Mean) || !isValidNumber(populationMean) || !isValidNumber(populationStdDev) || !isValidNumber(sample1Size) || populationStdDev <= 0 || sample1Size <= 0) {
errorMessage = "Please enter valid numbers for Sample Mean, Population Mean, Population Standard Deviation, and Sample Size.";
} else {
testStatistic = (sample1Mean – populationMean) / (populationStdDev / Math.sqrt(sample1Size));
resultDiv.innerHTML = "Z = " + testStatistic.toFixed(4);
}
break;
case "t-test-independent": // Assuming equal variances for now
if (!isValidNumber(sample1Mean) || !isValidNumber(sample1StdDev) || !isValidNumber(sample1Size) ||
!isValidNumber(sample2Mean) || !isValidNumber(sample2StdDev) || !isValidNumber(sample2Size) ||
sample1StdDev <= 0 || sample2StdDev <= 0 || sample1Size <= 1 || sample2Size <= 1) {
errorMessage = "Please enter valid positive numbers for Sample Means, Standard Deviations, and Sample Sizes (greater than 1).";
} else {
var pooledVariance = ((sample1Size – 1) * Math.pow(sample1StdDev, 2) + (sample2Size – 1) * Math.pow(sample2StdDev, 2)) / (sample1Size + sample2Size – 2);
if (pooledVariance < 0) { // Should not happen with valid variances, but a safeguard
errorMessage = "Calculated pooled variance is negative. Check input standard deviations.";
} else {
testStatistic = (sample1Mean – sample2Mean) / Math.sqrt(pooledVariance / sample1Size + pooledVariance / sample2Size);
var degreesOfFreedom = sample1Size + sample2Size – 2;
resultDiv.innerHTML = "t = " + testStatistic.toFixed(4) + " (df = " + degreesOfFreedom + ")";
}
}
break;
case "t-test-independent-unequal": // Welch's t-test
if (!isValidNumber(sample1Mean) || !isValidNumber(sample1StdDev) || !isValidNumber(sample1Size) ||
!isValidNumber(sample2Mean) || !isValidNumber(sample2StdDev) || !isValidNumber(sample2Size) ||
sample1StdDev <= 0 || sample2StdDev <= 0 || sample1Size <= 1 || sample2Size <= 1) {
errorMessage = "Please enter valid positive numbers for Sample Means, Standard Deviations, and Sample Sizes (greater than 1).";
} else {
testStatistic = (sample1Mean – sample2Mean) / Math.sqrt(Math.pow(sample1StdDev, 2) / sample1Size + Math.pow(sample2StdDev, 2) / sample2Size);
// Welch-Satterthwaite approximation for df (complex, simplified here for display)
var df_num = Math.pow(Math.pow(sample1StdDev, 2) / sample1Size + Math.pow(sample2StdDev, 2) / sample2Size, 2);
var df_den = (Math.pow(Math.pow(sample1StdDev, 2) / sample1Size, 2) / (sample1Size – 1)) + (Math.pow(Math.pow(sample2StdDev, 2) / sample2Size, 2) / (sample2Size – 1));
var degreesOfFreedomWelch = df_num / df_den;
resultDiv.innerHTML = "t = " + testStatistic.toFixed(4) + " (df ≈ " + degreesOfFreedomWelch.toFixed(2) + ")";
}
break;
case "paired-t-test":
// For paired t-test, we typically need the means and std devs of the *differences*.
// This calculator structure is better suited for independent samples.
// To adapt: one would need paired data input or summary stats of differences.
// For this example, we'll simulate by assuming the user inputs summary stats for differences directly.
// Let's repurpose sample1Mean/StdDev/Size for the differences.
var diffMean = parseFloat(document.getElementById("sample1Mean").value); // Using sample1Mean for difference mean
var diffStdDev = parseFloat(document.getElementById("sample1StdDev").value); // Using sample1StdDev for difference std dev
var nPairs = parseFloat(document.getElementById("sample1Size").value); // Using sample1Size for number of pairs
if (!isValidNumber(diffMean) || !isValidNumber(diffStdDev) || !isValidNumber(nPairs) || diffStdDev <= 0 || nPairs 0; }); // Expected must be positive
if (!validObserved || !validExpected) {
errorMessage = "Please ensure all entered frequencies are valid positive numbers.";
} else {
testStatistic = 0;
for (var i = 0; i < observedFreq.length; i++) {
testStatistic += Math.pow(observedFreq[i] – expectedFreq[i], 2) / expectedFreq[i];
}
var degreesOfFreedomChiSq = observedFreq.length – 1;
resultDiv.innerHTML = "χ² = " + testStatistic.toFixed(4) + " (df = " + degreesOfFreedomChiSq + ")";
}
}
}
break;
default:
errorMessage = "Please select a valid test type.";
break;
}
if (errorMessage) {
resultDiv.innerHTML = '
Error: ' + errorMessage;
}
} catch (e) {
resultDiv.innerHTML = '
Calculation Error: Please check your inputs. Details: ' + e.message;
}
}
// Show/hide additional input groups based on test type selection
document.getElementById("testType").addEventListener("change", function() {
var selectedTest = this.value;
document.getElementById("populationVarianceGroup").style.display = "none";
document.getElementById("populationMeanGroup").style.display = "none";
document.getElementById("nullHypothesisValueGroup").style.display = "none";
document.getElementById("chiSquaredObserved").style.display = "none";
document.getElementById("chiSquaredExpected").style.display = "none";
// Clear values in hidden fields
document.getElementById("populationVariance").value = "";
document.getElementById("populationMean").value = "";
document.getElementById("nullHypothesisValue").value = "";
document.getElementById("chiSquaredObservedFreq").value = "";
document.getElementById("chiSquaredExpectedFreq").value = "";
if (selectedTest === "z-test-known-variance") {
document.getElementById("populationVarianceGroup").style.display = "flex";
document.getElementById("populationMeanGroup").style.display = "flex";
// Assume for z-test comparing sample mean to known population mean, no separate nullHypothesisValue needed unless specified
} else if (selectedTest === "t-test-independent" || selectedTest === "t-test-independent-unequal") {
// These t-tests compare two independent samples, so no population mean/variance needed here
// Could add an option for null hypothesis value (difference = 0) if needed
document.getElementById("nullHypothesisValueGroup").style.display = "flex"; // Optional: For difference of means != 0
document.getElementById("nullHypothesisValue").value = "0"; // Default to testing for difference = 0
} else if (selectedTest === "paired-t-test") {
// Paired t-test needs the mean and std dev of the differences.
// We'll use the first sample's inputs for simplicity here, assuming user knows to input differences.
// A more robust UI would have specific 'Difference Mean', 'Difference Std Dev', 'Number of Pairs' inputs.
document.getElementById("nullHypothesisValueGroup").style.display = "flex"; // Optional: For difference = 0
document.getElementById("nullHypothesisValue").value = "0"; // Default to testing for difference = 0
} else if (selectedTest === "chi-squared") {
document.getElementById("chiSquaredObserved").style.display = "flex";
document.getElementById("chiSquaredExpected").style.display = "flex";
}
});
// Initial setup on page load
document.addEventListener("DOMContentLoaded", function() {
document.getElementById("testType").dispatchEvent(new Event("change"));
});