Test Statistic Calculator
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Test Statistic Calculator
Test Statistic (t or z)
—
Understanding and Calculating the Test Statistic
In statistical hypothesis testing, the test statistic is a crucial value calculated from sample data. It quantifies how far the sample result deviates from the null hypothesis (the default or assumed state of affairs). This deviation is measured in terms of standard error. Essentially, it tells us whether the observed difference between our sample and the hypothesized population is likely due to random chance or represents a genuine effect.
What is a Test Statistic?
The test statistic serves as a bridge between sample data and the population. It's a single number that summarizes the evidence from the sample against the null hypothesis. The value of the test statistic is then compared to a critical value from a probability distribution (like the t-distribution or the standard normal distribution) or used to calculate a p-value to determine if we should reject the null hypothesis.
Common Types of Test Statistics
The specific formula for the test statistic depends on the type of data, the population parameters known (or unknown), and the hypothesis being tested. Two common types are:
- Z-statistic: Used when the population standard deviation (σ) is known, or when the sample size is very large (n > 30) and the sample standard deviation (s) is used as a good approximation of σ.
- T-statistic: Used when the population standard deviation (σ) is unknown and must be estimated using the sample standard deviation (s), especially with smaller sample sizes (n ≤ 30).
How to Calculate the Test Statistic (using the T-statistic formula)
This calculator computes the test statistic using the formula for a one-sample t-test, which is appropriate when the population standard deviation is unknown:
$$ t = \frac{\bar{x} – \mu}{\frac{s}{\sqrt{n}}} $$
Where:
- $ \bar{x} $ (Sample Mean): The average of your sample data.
- $ \mu $ (Population Mean): The hypothesized mean of the population under the null hypothesis.
- $ s $ (Sample Standard Deviation): A measure of the spread or variability of your sample data.
- $ n $ (Sample Size): The number of observations in your sample.
The denominator, $ \frac{s}{\sqrt{n}} $, is known as the standard error of the mean. It represents the standard deviation of the sampling distribution of the mean.
When to Use This Calculator
- When you want to test a hypothesis about a single population mean (μ).
- When the population standard deviation (σ) is unknown.
- When your sample data is approximately normally distributed (or the sample size is large enough for the Central Limit Theorem to apply).
By inputting your sample mean, hypothesized population mean, sample standard deviation, and sample size, this calculator will provide the calculated test statistic. This value is a critical step in determining whether to reject or fail to reject your null hypothesis.
function calculateTestStatistic() {
var sampleMean = parseFloat(document.getElementById("sampleMean").value);
var populationMean = parseFloat(document.getElementById("populationMean").value);
var sampleStdDev = parseFloat(document.getElementById("sampleStdDev").value);
var sampleSize = parseInt(document.getElementById("sampleSize").value);
var resultDiv = document.getElementById("result");
// Clear previous error messages
resultDiv.classList.remove("error");
// Input validation
if (isNaN(sampleMean) || isNaN(populationMean) || isNaN(sampleStdDev) || isNaN(sampleSize)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
resultDiv.classList.add("error");
return;
}
if (sampleSize <= 0) {
resultDiv.innerHTML = "Sample size must be greater than zero.";
resultDiv.classList.add("error");
return;
}
if (sampleStdDev < 0) {
resultDiv.innerHTML = "Sample standard deviation cannot be negative.";
resultDiv.classList.add("error");
return;
}
if (sampleSize === 1 && sampleStdDev === 0) {
resultDiv.innerHTML = "Cannot calculate with sample size 1 and standard deviation 0.";
resultDiv.classList.add("error");
return;
}
if (sampleStdDev === 0 && sampleMean !== populationMean) {
resultDiv.innerHTML = "Standard deviation is zero. The sample mean must equal the population mean for a valid test statistic.";
resultDiv.classList.add("error");
return;
}
if (sampleStdDev === 0 && sampleMean === populationMean) {
resultDiv.innerHTML = "0.00"; // If std dev is 0 and means are equal, t is 0.
return;
}
// Calculation for t-statistic
var standardError = sampleStdDev / Math.sqrt(sampleSize);
var testStatistic = (sampleMean – populationMean) / standardError;
// Display the result, formatted to 2 decimal places
resultDiv.innerHTML = testStatistic.toFixed(2);
}