.calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
max-width: 700px;
margin: 20px auto;
padding: 25px;
border-radius: 12px;
background-color: #f9f9f9;
box-shadow: 0 6px 15px rgba(0, 0, 0, 0.1);
border: 1px solid #e0e0e0;
}
.calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 25px;
font-size: 28px;
font-weight: 600;
}
.calculator-container .input-group {
margin-bottom: 18px;
display: flex;
flex-direction: column;
}
.calculator-container .input-group label {
margin-bottom: 8px;
color: #555;
font-size: 16px;
font-weight: 500;
}
.calculator-container .input-group input[type="number"],
.calculator-container .input-group select {
padding: 12px 15px;
border: 1px solid #ccc;
border-radius: 8px;
font-size: 16px;
width: 100%;
box-sizing: border-box;
transition: border-color 0.3s ease;
}
.calculator-container .input-group input[type="number"]:focus,
.calculator-container .input-group select:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25);
}
.calculator-container button {
display: block;
width: 100%;
padding: 14px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 8px;
font-size: 18px;
font-weight: 600;
cursor: pointer;
transition: background-color 0.3s ease, transform 0.2s ease;
margin-top: 25px;
}
.calculator-container button:hover {
background-color: #0056b3;
transform: translateY(-2px);
}
.calculator-container .result {
margin-top: 30px;
padding: 20px;
border-radius: 10px;
background-color: #e9f7ff;
border: 1px solid #b3e0ff;
color: #004085;
font-size: 17px;
line-height: 1.6;
}
.calculator-container .result h3 {
color: #004085;
margin-top: 0;
margin-bottom: 15px;
font-size: 22px;
font-weight: 600;
}
.calculator-container .result p {
margin-bottom: 8px;
}
.calculator-container .result strong {
color: #0056b3;
}
.calculator-container .info-text {
margin-top: 20px;
font-size: 14px;
color: #777;
text-align: center;
}
// Helper function for log gamma
function gammaLn(z) {
var p = [
0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7
];
var g = 7;
if (z < 0.5) {
return Math.log(Math.PI / (Math.sin(Math.PI * z) * Math.exp(gammaLn(1 – z))));
}
z -= 1;
var a = p[0];
var t = z + g + 0.5;
for (var i = 1; i < p.length; i++) {
a += p[i] / (z + i);
}
return 0.5 * Math.log(2 * Math.PI) + (z + 0.5) * Math.log(t) – t + Math.log(a);
}
// Helper function for incomplete beta function (regularized)
// This implementation uses series expansion for x < (a+1)/(a+b+2) and continued fraction otherwise.
function incompleteBeta(a, b, x) {
if (x 1.0) return NaN;
if (x == 0.0) return 0.0;
if (x == 1.0) return 1.0;
if (a <= 0.0 || b <= 0.0) return NaN;
var lbeta = gammaLn(a + b) – gammaLn(a) – gammaLn(b);
if (x < (a + 1.0) / (a + b + 2.0)) {
// Use series expansion
var bt = Math.exp(lbeta + a * Math.log(x) + b * Math.log(1.0 – x));
var sum = 0.0;
var term = 1.0;
for (var n = 0; n < 200; n++) { // Limit iterations
sum += term;
term = term * (a + n) * (1.0 – x) / ((a + b + n) * (n + 1.0));
if (Math.abs(term) < 1e-10 * Math.abs(sum)) break; // Convergence check
}
return bt * sum / a;
} else {
// Use continued fraction expansion
var bt = Math.exp(lbeta + b * Math.log(1.0 – x) + a * Math.log(x));
var c = 1.0;
var d = 1.0 – (a + b) * x / (b + 1.0);
if (Math.abs(d) < 1e-30) d = 1e-30; // Avoid division by zero
d = 1.0 / d;
var h = d;
for (var m = 1; m < 200; m++) { // Limit iterations
var m2 = 2 * m;
var aa = m * (b – m) * x / ((b + m2 – 1.0) * (b + m2));
d = 1.0 + aa * d;
if (Math.abs(d) < 1e-30) d = 1e-30;
c = 1.0 + aa / c;
if (Math.abs(c) < 1e-30) c = 1e-30;
d = 1.0 / d;
h *= d * c;
aa = -(a + m) * (a + b + m) * x / ((a + m2) * (a + m2 + 1.0));
d = 1.0 + aa * d;
if (Math.abs(d) < 1e-30) d = 1e-30;
c = 1.0 + aa / c;
if (Math.abs(c) < 1e-30) c = 1e-30;
d = 1.0 / d;
h *= d * c;
if (Math.abs(h) < 1e-10) break; // Convergence check
}
return 1.0 – bt * h / b;
}
}
// Student's t-distribution Cumulative Distribution Function (CDF)
// Returns P(T <= t | df)
function studentTCdf(t, df) {
if (df 0) {
// For t > 0, P(T <= t) = 1 – 0.5 * I_x(df/2, 0.5)
return 1.0 – 0.5 * incompleteBeta(df / 2.0, 0.5, x);
} else { // t < 0
// For t < 0, P(T <= t) = 0.5 * I_x(df/2, 0.5)
return 0.5 * incompleteBeta(df / 2.0, 0.5, x);
}
}
// Main calculation function
function calculateTDistribution() {
var tScore = parseFloat(document.getElementById("tScore").value);
var degreesOfFreedom = parseInt(document.getElementById("degreesOfFreedom").value);
var tailType = document.getElementById("tailType").value;
if (isNaN(tScore) || isNaN(degreesOfFreedom) || degreesOfFreedom |tScore|) or 2 * P(T < -|tScore|)
// Which is 2 * (1 – studentTCdf(Math.abs(tScore), degreesOfFreedom))
pValue = 2 * (1.0 – studentTCdf(Math.abs(tScore), degreesOfFreedom));
}
// Ensure pValue is within [0, 1] due to potential floating point inaccuracies
pValue = Math.max(0, Math.min(1, pValue));
var resultHtml = "
T-Distribution P-value Calculator
Two-tailed
One-tailed (Right)
One-tailed (Left)
Calculation Result:
Enter your values and click "Calculate P-value" to see the result.
Calculation Result:
"; resultHtml += "Calculated P-value: " + pValue.toFixed(6) + ""; resultHtml += "This P-value represents the probability of observing a t-statistic as extreme as, or more extreme than, your input t-score, given the specified degrees of freedom and tail type. A smaller P-value typically indicates stronger evidence against the null hypothesis."; document.getElementById("result").innerHTML = resultHtml; }Understanding the T-Distribution and P-values
The t-distribution, often referred to as Student's t-distribution, is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset, who published under the pseudonym "Student."
When to Use the T-Distribution
The t-distribution is primarily used in hypothesis testing and for constructing confidence intervals under specific conditions:
- Small Sample Sizes: It is particularly useful when the sample size (n) is small, typically less than 30. For larger sample sizes, the t-distribution closely approximates the normal distribution.
- Unknown Population Standard Deviation: When the population standard deviation is unknown, and you must estimate it using the sample standard deviation, the t-distribution is the appropriate choice.
- Hypothesis Testing: It forms the basis for t-tests, which are used to compare means (e.g., comparing a sample mean to a hypothesized population mean, or comparing means of two independent samples).
Key Concepts
To use the t-distribution effectively, it's important to understand its core components:
- T-Score (t-statistic): This value measures how many standard errors a sample mean is from the population mean (or hypothesized population mean). It's calculated using the formula:
t = (sample_mean - population_mean) / (sample_std_dev / sqrt(sample_size))
A larger absolute t-score indicates a greater difference between the sample mean and the hypothesized population mean. - Degrees of Freedom (df): This parameter defines the shape of the t-distribution. For a single sample t-test, the degrees of freedom are calculated as
n - 1, wherenis the sample size. As the degrees of freedom increase, the t-distribution becomes more similar to the standard normal (Z) distribution. - P-value: The p-value is the probability of observing a test statistic (like your t-score) as extreme as, or more extreme than, the one calculated from your sample data, assuming that the null hypothesis is true. It helps you determine the statistical significance of your results.
Interpreting the P-value
After calculating the p-value, you compare it to a predetermined significance level (alpha, commonly 0.05 or 0.01):
- If P-value < Alpha: You reject the null hypothesis. This suggests that your observed results are statistically significant and unlikely to have occurred by random chance alone.
- If P-value ≥ Alpha: You fail to reject the null hypothesis. This means there isn't enough statistical evidence to conclude that your observed results are significantly different from what the null hypothesis predicts.
Tail Types in Hypothesis Testing
The choice of tail type depends on the alternative hypothesis you are testing:
- One-tailed (Right): Used when your alternative hypothesis predicts that the true mean is greater than a hypothesized value. You are interested in extreme positive t-scores.
- One-tailed (Left): Used when your alternative hypothesis predicts that the true mean is less than a hypothesized value. You are interested in extreme negative t-scores.
- Two-tailed: Used when your alternative hypothesis predicts that the true mean is simply different from a hypothesized value (it could be either greater or less). You are interested in extreme t-scores in both positive and negative directions.
How to Use This Calculator
This T-Distribution P-value Calculator helps you quickly find the p-value associated with a given t-score and degrees of freedom:
- Enter T-Score: Input the t-statistic you calculated from your sample data.
- Enter Degrees of Freedom: Input the degrees of freedom (sample size minus 1).
- Select Tail Type: Choose whether your hypothesis test is one-tailed (left or right) or two-tailed, based on your alternative hypothesis.
- Click "Calculate P-value": The calculator will instantly display the corresponding p-value.
Use this p-value to make informed decisions about your hypothesis tests.