Z-Test (for large samples or known population variance)
T-Test (for small samples, unknown population variance)
Chi-Squared Test
F-Test
Understanding Test Statistics
In inferential statistics, a test statistic is a value derived from sample data that is used to decide whether to reject or fail to reject a null hypothesis. It quantifies how far a sample statistic (like the sample mean) deviates from the hypothesized population parameter (like the population mean), considering the variability within the sample. The calculation of the test statistic depends heavily on the type of statistical test being performed, the nature of the data, and the assumptions made about the population.
Common Types of Test Statistics:
Z-Test Statistic: Used when the sample size is large (typically n > 30) or when the population standard deviation is known. It measures the difference between the sample mean and the population mean in terms of standard errors. The formula is:
Z = (X̄ - μ₀) / (s / √n) Where:
X̄ is the sample mean
μ₀ is the hypothesized population mean
s is the sample standard deviation
n is the sample size
T-Test Statistic: Used when the sample size is small (typically n ≤ 30) and the population standard deviation is unknown. It is similar to the Z-test but uses the sample standard deviation and accounts for the degrees of freedom (n-1). The formula is:
t = (X̄ - μ₀) / (s / √n) Where:
X̄ is the sample mean
μ₀ is the hypothesized population mean
s is the sample standard deviation
n is the sample size
The interpretation of the t-statistic requires comparing it to a t-distribution with n-1 degrees of freedom.
Chi-Squared (χ²) Test Statistic: Used to test hypotheses about categorical data, such as whether observed frequencies differ significantly from expected frequencies in one or more categories. The formula is:
χ² = Σ [ (Oᵢ - Eᵢ)² / Eᵢ ] Where:
Oᵢ is the observed frequency for category i
Eᵢ is the expected frequency for category i
Σ denotes the sum over all categories
This test is used for goodness-of-fit tests and tests of independence.
F-Test Statistic: Often used in Analysis of Variance (ANOVA) to compare the variances of two or more groups. For comparing two variances, the formula is the ratio of the larger variance to the smaller variance:
F = s₁² / s₂² (where s₁² is the larger sample variance)
The interpretation involves comparing the calculated F-value to an F-distribution with specific degrees of freedom (numerator and denominator).
How to Use This Calculator:
1. Select Test Type: Choose the appropriate statistical test from the dropdown menu based on your data and hypothesis.
2. Enter Values: Input the required sample statistics (means, standard deviations, sample sizes, observed/expected frequencies, variances) corresponding to the selected test type. Ensure you use consistent units and accurate measurements. For observed and expected frequencies, enter values separated by commas.
3. Calculate: Click the "Calculate Test Statistic" button.
4. Interpret: The calculator will display the computed test statistic. This value is then used in conjunction with critical values from the relevant statistical distribution (Z, t, Chi-Squared, or F) and the chosen significance level (alpha) to make a decision about your hypothesis.
Note: This calculator provides the test statistic value. Further steps involving comparing this value to critical values or calculating p-values are necessary for complete hypothesis testing.
// Function to show/hide relevant input fields based on selected statistic type
function updateInputFields() {
var type = document.getElementById("statisticType").value;
document.getElementById("z_test_inputs").style.display = (type === "z-test") ? "block" : "none";
document.getElementById("t_test_inputs").style.display = (type === "t-test") ? "block" : "none";
document.getElementById("chi_squared_inputs").style.display = (type === "chi-squared") ? "block" : "none";
document.getElementById("f_test_inputs").style.display = (type === "f-test") ? "block" : "none";
}
// Initial call to set up fields on page load
document.addEventListener("DOMContentLoaded", function() {
updateInputFields();
});
document.getElementById("statisticType").addEventListener("change", updateInputFields);
function calculateTestStatistic() {
var statisticType = document.getElementById("statisticType").value;
var resultDiv = document.getElementById("result").querySelector("span");
var testStatistic = NaN; // Default to NaN
try {
if (statisticType === "z-test") {
var sampleMean_z = parseFloat(document.getElementById("sampleMean_z").value);
var populationMean_z = parseFloat(document.getElementById("populationMean_z").value);
var sampleStdDev_z = parseFloat(document.getElementById("sampleStdDev_z").value);
var sampleSize_z = parseFloat(document.getElementById("sampleSize_z").value);
if (!isNaN(sampleMean_z) && !isNaN(populationMean_z) && !isNaN(sampleStdDev_z) && !isNaN(sampleSize_z) && sampleStdDev_z > 0 && sampleSize_z > 0) {
var standardError_z = sampleStdDev_z / Math.sqrt(sampleSize_z);
testStatistic = (sampleMean_z – populationMean_z) / standardError_z;
} else {
resultDiv.textContent = "Please enter valid numbers for all Z-Test fields.";
return;
}
} else if (statisticType === "t-test") {
var sampleMean_t = parseFloat(document.getElementById("sampleMean_t").value);
var populationMean_t = parseFloat(document.getElementById("populationMean_t").value);
var sampleStdDev_t = parseFloat(document.getElementById("sampleStdDev_t").value);
var sampleSize_t = parseFloat(document.getElementById("sampleSize_t").value);
if (!isNaN(sampleMean_t) && !isNaN(populationMean_t) && !isNaN(sampleStdDev_t) && !isNaN(sampleSize_t) && sampleStdDev_t > 0 && sampleSize_t > 1) { // n must be > 1 for t-test degrees of freedom
var standardError_t = sampleStdDev_t / Math.sqrt(sampleSize_t);
testStatistic = (sampleMean_t – populationMean_t) / standardError_t;
} else {
resultDiv.textContent = "Please enter valid numbers for all T-Test fields (sample size > 1).";
return;
}
} else if (statisticType === "chi-squared") {
var observedFreqStr = document.getElementById("observedFreq").value;
var expectedFreqStr = document.getElementById("expectedFreq").value;
var observedFreq = observedFreqStr.split(',').map(function(item) { return parseFloat(item.trim()); });
var expectedFreq = expectedFreqStr.split(',').map(function(item) { return parseFloat(item.trim()); });
if (observedFreq.length === expectedFreq.length && observedFreq.length > 0 && observedFreq.every(function(val) { return !isNaN(val); }) && expectedFreq.every(function(val) { return !isNaN(val) && val > 0; })) {
var chiSquaredSum = 0;
for (var i = 0; i 0 && variance2 > 0 && sampleSize1 > 1 && sampleSize2 > 1) {
// F-test for equality of variances typically places the larger variance in the numerator
if (variance1 >= variance2) {
testStatistic = variance1 / variance2;
} else {
testStatistic = variance2 / variance1;
}
} else {
resultDiv.textContent = "Please enter valid numbers for variances and sample sizes (variances > 0, sample sizes > 1).";
return;
}
}
if (!isNaN(testStatistic)) {
resultDiv.textContent = "Test Statistic: " + testStatistic.toFixed(4);
} else {
resultDiv.textContent = "Calculation Error. Please check inputs.";
}
} catch (error) {
resultDiv.textContent = "An error occurred: " + error.message;
console.error("Calculation error:", error);
}
}
// Add CSS class to hide elements initially
document.querySelectorAll('div[id$="_inputs"]').forEach(function(el) {
el.classList.add('hidden');
});
// Modify updateInputFields to toggle class
function updateInputFields() {
var type = document.getElementById("statisticType").value;
document.querySelectorAll('div[id$="_inputs"]').forEach(function(el) {
el.classList.add('hidden');
});
var currentInputsId = type.replace('-', '_') + "_inputs";
var currentInputs = document.getElementById(currentInputsId);
if (currentInputs) {
currentInputs.classList.remove('hidden');
}
}
// Initial call and event listener setup
document.addEventListener("DOMContentLoaded", function() {
updateInputFields();
});
document.getElementById("statisticType").addEventListener("change", updateInputFields);