.t-statistic-calculator-wrapper {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
max-width: 700px;
margin: 20px auto;
padding: 25px;
border: 1px solid #e0e0e0;
border-radius: 10px;
background-color: #fdfdfd;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.05);
}
.t-statistic-calculator-wrapper h2 {
text-align: center;
color: #2c3e50;
margin-bottom: 25px;
font-size: 1.8em;
}
.t-statistic-calculator-wrapper .input-group {
margin-bottom: 18px;
display: flex;
flex-direction: column;
}
.t-statistic-calculator-wrapper label {
margin-bottom: 8px;
color: #34495e;
font-weight: bold;
font-size: 0.95em;
}
.t-statistic-calculator-wrapper input[type="number"] {
width: 100%;
padding: 12px;
border: 1px solid #ccc;
border-radius: 6px;
box-sizing: border-box;
font-size: 1em;
transition: border-color 0.3s ease;
}
.t-statistic-calculator-wrapper input[type="number"]:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 2px rgba(0, 123, 255, 0.25);
}
.t-statistic-calculator-wrapper button {
display: block;
width: 100%;
padding: 14px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 6px;
font-size: 1.1em;
cursor: pointer;
transition: background-color 0.3s ease, transform 0.2s ease;
margin-top: 25px;
}
.t-statistic-calculator-wrapper button:hover {
background-color: #0056b3;
transform: translateY(-1px);
}
.t-statistic-calculator-wrapper button:active {
transform: translateY(0);
}
.t-statistic-calculator-wrapper .result-output {
margin-top: 30px;
padding: 20px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 8px;
font-size: 1.15em;
color: #155724;
text-align: center;
font-weight: bold;
line-height: 1.6;
}
.t-statistic-calculator-wrapper .result-output strong {
color: #0a3622;
}
.t-statistic-calculator-wrapper .error-message {
color: #dc3545;
background-color: #f8d7da;
border: 1px solid #f5c6cb;
padding: 10px;
border-radius: 5px;
margin-top: 15px;
text-align: center;
}
function calculateTStatistic() {
var sampleMean = parseFloat(document.getElementById('sampleMean').value);
var hypothesizedMean = parseFloat(document.getElementById('hypothesizedMean').value);
var sampleStdDev = parseFloat(document.getElementById('sampleStdDev').value);
var sampleSize = parseInt(document.getElementById('sampleSize').value);
var resultDiv = document.getElementById('result');
resultDiv.style.display = 'none';
resultDiv.classList.remove('error-message');
resultDiv.style.backgroundColor = '#e9f7ef';
resultDiv.style.borderColor = '#d4edda';
resultDiv.style.color = '#155724';
if (isNaN(sampleMean) || isNaN(hypothesizedMean) || isNaN(sampleStdDev) || isNaN(sampleSize)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
resultDiv.style.display = 'block';
resultDiv.classList.add('error-message');
resultDiv.style.backgroundColor = '#f8d7da';
resultDiv.style.borderColor = '#f5c6cb';
resultDiv.style.color = '#dc3545';
return;
}
if (sampleSize < 2) {
resultDiv.innerHTML = "Sample Size (n) must be at least 2.";
resultDiv.style.display = 'block';
resultDiv.classList.add('error-message');
resultDiv.style.backgroundColor = '#f8d7da';
resultDiv.style.borderColor = '#f5c6cb';
resultDiv.style.color = '#dc3545';
return;
}
if (sampleStdDev <= 0) {
resultDiv.innerHTML = "Sample Standard Deviation (s) must be greater than 0.";
resultDiv.style.display = 'block';
resultDiv.classList.add('error-message');
resultDiv.style.backgroundColor = '#f8d7da';
resultDiv.style.borderColor = '#f5c6cb';
resultDiv.style.color = '#dc3545';
return;
}
var standardError = sampleStdDev / Math.sqrt(sampleSize);
var tStatistic = (sampleMean – hypothesizedMean) / standardError;
var degreesOfFreedom = sampleSize – 1;
resultDiv.innerHTML = "
" + degreesOfFreedom;
resultDiv.style.display = 'block';
}
Understanding the Calculated T-Statistic
The t-statistic is a fundamental concept in inferential statistics, primarily used in hypothesis testing to determine if there is a significant difference between a sample mean and a hypothesized population mean, or between the means of two samples. It is particularly useful when dealing with small sample sizes or when the population standard deviation is unknown, which is often the case in real-world research.
What Does the T-Statistic Tell Us?
In essence, the t-statistic measures the size of the difference between the sample mean and the hypothesized population mean relative to the variability within the sample. A larger absolute t-statistic value suggests a greater difference between the observed sample mean and the expected population mean, making it less likely that the observed difference occurred by chance.
When to Use a T-Test?
T-tests are typically employed under the following conditions:
- When comparing a sample mean to a known or hypothesized population mean (one-sample t-test).
- When comparing the means of two independent groups (independent samples t-test).
- When comparing the means of two related groups (paired samples t-test).
- When the sample size is small (generally n < 30), and the population standard deviation is unknown.
- When the data is approximately normally distributed.
The One-Sample T-Test Formula
This calculator focuses on the one-sample t-test, which is used to compare a single sample mean to a known or hypothesized population mean. The formula is:
t = (x̄ - μ) / (s / √n)
- x̄ (x-bar): The sample mean, which is the average of your observed data points.
- μ (mu): The hypothesized population mean, which is the value you are testing against (e.g., a known standard, a previous average, or a theoretical value).
- s: The sample standard deviation, a measure of the spread or variability of your sample data.
- n: The sample size, which is the number of observations in your sample.
- s / √n: This term is known as the standard error of the mean, which estimates the standard deviation of the sampling distribution of the sample mean.
Interpreting the T-Statistic and Degrees of Freedom
Once you calculate the t-statistic, you compare it to a critical t-value from a t-distribution table or use statistical software to obtain a p-value. This comparison helps you decide whether to reject or fail to reject your null hypothesis.
The Degrees of Freedom (df) are also crucial for this comparison. For a one-sample t-test, the degrees of freedom are calculated as n - 1. The degrees of freedom indicate the number of independent pieces of information available to estimate a parameter. They influence the shape of the t-distribution, with higher degrees of freedom making the t-distribution more closely resemble a normal distribution.
Example Scenario:
Imagine a pharmaceutical company claims their new drug reduces blood pressure by an average of 10 mmHg. A researcher conducts a study on a sample of 30 patients (n) who took the drug and finds their average blood pressure reduction (sample mean, x̄) was 12.5 mmHg, with a sample standard deviation (s) of 3.2 mmHg. The hypothesized population mean (μ) is 10 mmHg.
Using the calculator:
- Sample Mean (x̄): 12.5
- Hypothesized Population Mean (μ): 10
- Sample Standard Deviation (s): 3.2
- Sample Size (n): 30
The calculator would yield a t-statistic that can then be used to determine if the observed reduction of 12.5 mmHg is significantly different from the claimed 10 mmHg.