function calculateTestStatistic() {
var sampleMean = parseFloat(document.getElementById('sampleMean').value);
var populationMean = parseFloat(document.getElementById('populationMean').value);
var populationStdDev = parseFloat(document.getElementById('populationStdDev').value);
var sampleSize = parseInt(document.getElementById('sampleSize').value);
var resultDiv = document.getElementById('testStatisticResult');
resultDiv.innerHTML = "; // Clear previous results
if (isNaN(sampleMean) || isNaN(populationMean) || isNaN(populationStdDev) || isNaN(sampleSize)) {
resultDiv.innerHTML = 'Please enter valid numbers for all fields.';
return;
}
if (populationStdDev <= 0) {
resultDiv.innerHTML = 'Population Standard Deviation must be greater than zero.';
return;
}
if (sampleSize <= 1) {
resultDiv.innerHTML = 'Sample Size must be greater than 1.';
return;
}
var standardError = populationStdDev / Math.sqrt(sampleSize);
var zScore = (sampleMean – populationMean) / standardError;
resultDiv.innerHTML = 'The calculated Z-Score is: ' + zScore.toFixed(4) + '';
resultDiv.innerHTML += 'This Z-score indicates how many standard errors the sample mean is away from the population mean.';
}
/* Basic styling for the calculator */
.calculator-container {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
}
.calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
}
.calculator-input-grid {
display: grid;
grid-template-columns: 1fr;
gap: 15px;
margin-bottom: 20px;
}
.calculator-input-row {
display: flex;
flex-direction: column;
}
.calculator-input-row label {
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.calculator-input-row input[type="number"] {
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
width: 100%;
box-sizing: border-box;
}
.calculator-button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease;
}
.calculator-button:hover {
background-color: #0056b3;
}
.calculator-result {
margin-top: 20px;
padding: 15px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 4px;
color: #155724;
font-size: 17px;
text-align: center;
}
.calculator-result p {
margin: 0 0 5px 0;
}
.calculator-result p:last-child {
margin-bottom: 0;
}
.calculator-result .error {
color: #721c24;
background-color: #f8d7da;
border-color: #f5c6cb;
padding: 10px;
border-radius: 4px;
}
Understanding the Test Statistic (Z-Score)
In statistics, a test statistic is a numerical summary of a sample data set that is used to perform a hypothesis test. It quantifies how much the sample data deviates from what is expected under the null hypothesis. The value of the test statistic is then compared to a critical value or used to calculate a p-value to determine whether to reject or fail to reject the null hypothesis.
What is a Z-Score?
The Z-score, also known as the standard score, is a specific type of test statistic used when comparing a sample mean to a known population mean, especially when the population standard deviation is known and the sample size is sufficiently large (typically n > 30), or the population is normally distributed. It measures how many standard deviations an element is from the mean.
The Z-Score Formula
The formula for a one-sample Z-test statistic is:
Z = (x̄ – μ) / (σ / √n)
x̄ (Sample Mean): The mean of your sample data.
μ (Population Mean): The hypothesized mean of the population under the null hypothesis.
σ (Population Standard Deviation): The known standard deviation of the population.
n (Sample Size): The number of observations in your sample.
σ / √n (Standard Error of the Mean): This represents the standard deviation of the sampling distribution of the sample mean.
How to Interpret the Z-Score
A larger absolute Z-score indicates that the sample mean is further away from the population mean, making it less likely that the observed difference occurred by chance. To make a decision about your hypothesis, you would compare the calculated Z-score to critical Z-values from a standard normal distribution table, or use it to find a p-value. For example:
If your calculated Z-score falls outside the critical region (e.g., beyond ±1.96 for a 95% confidence level in a two-tailed test), you would typically reject the null hypothesis.
A Z-score close to zero suggests that the sample mean is very close to the population mean, supporting the null hypothesis.
Example Calculation
Let's say a school claims that the average IQ of its students is 100 (μ = 100). A researcher takes a random sample of 30 students (n = 30) and finds their average IQ to be 105 (x̄ = 105). Assume the population standard deviation for IQ scores is known to be 15 (σ = 15).
Using the formula:
Z = (105 – 100) / (15 / √30)
Z = 5 / (15 / 5.4772)
Z = 5 / 2.7386
Z ≈ 1.8250
In this example, the calculated Z-score is approximately 1.8250. If we were conducting a two-tailed test at a 0.05 significance level, the critical Z-values would be ±1.96. Since 1.8250 falls between -1.96 and 1.96, we would fail to reject the null hypothesis, meaning there isn't enough statistical evidence to conclude that the sample's average IQ is significantly different from the population average of 100.
Use the calculator above to quickly compute the Z-score for your own data by inputting the sample mean, population mean, population standard deviation, and sample size.