Student's t-Distribution Calculator
Results:
Cumulative Probability P(T ≤ t):
Two-tailed Probability P(|T| > |t|):
// Helper function for Lanczos approximation of logGamma(z)
function logGamma(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(logGamma(1 – z))));
}
z -= 1;
var x = p[0];
for (var i = 1; i < p.length; i++) {
x += p[i] / (z + i);
}
var t = z + g + 0.5;
return 0.5 * Math.log(2 * Math.PI) + (z + 0.5) * Math.log(t) – t + Math.log(x);
}
// Helper function for incompleteBeta: continued fraction for I_x(a,b)
function betacf(x, a, b) {
var ITMAX = 100;
var EPS = 3.0e-7;
var FPMIN = 1.0e-30;
var m = 1;
var aa, c, d, del, h;
var qab = a + b;
var qap = a + 1;
var qam = a – 1;
c = 1.0;
d = 1.0 – qab * x / qap;
if (Math.abs(d) < FPMIN) d = FPMIN;
d = 1.0 / d;
h = d;
for (m = 1; m <= ITMAX; m++) {
var m2 = 2 * m;
aa = m * (b – m) * x / ((qam + m2) * (a + m2));
d = 1.0 + aa * d;
if (Math.abs(d) < FPMIN) d = FPMIN;
c = 1.0 + aa / c;
if (Math.abs(c) < FPMIN) c = FPMIN;
d = 1.0 / d;
h *= d * c;
aa = -(a + m) * (qab + m) * x / ((a + m2) * (qap + m2));
d = 1.0 + aa * d;
if (Math.abs(d) < FPMIN) d = FPMIN;
c = 1.0 + aa / c;
if (Math.abs(c) < FPMIN) c = FPMIN;
d = 1.0 / d;
del = d * c;
h *= del;
if (Math.abs(del – 1.0) < EPS) break;
}
return h;
}
// Regularized Incomplete Beta Function I_x(a, b)
function incompleteBeta(x, a, b) {
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 bt = Math.exp(logGamma(a + b) – logGamma(a) – logGamma(b) + a * Math.log(x) + b * Math.log(1.0 – x));
if (x < (a + 1.0) / (a + b + 2.0)) {
return bt * betacf(x, a, b) / a;
} else {
return 1.0 – bt * betacf(1.0 – x, b, a) / b;
}
}
// Student's t-Distribution Cumulative Distribution Function (CDF)
function tDistributionCDF(t, df) {
if (df 0) {
return 1 – 0.5 * incompleteBeta(x, a, b);
} else { // t < 0
return 0.5 * incompleteBeta(x, a, b);
}
}
function calculateTDistribution() {
var degreesOfFreedomInput = document.getElementById("degreesOfFreedom").value;
var tScoreInput = document.getElementById("tScore").value;
var df = parseFloat(degreesOfFreedomInput);
var t = parseFloat(tScoreInput);
if (isNaN(df) || isNaN(t) || df <= 0 || !Number.isInteger(df)) {
document.getElementById("cumulativeProbability").textContent = "Please enter valid positive integer for Degrees of Freedom and a valid number for t-score.";
document.getElementById("twoTailedProbability").textContent = "";
return;
}
var cumulativeProb = tDistributionCDF(t, df);
var twoTailedProb = 2 * (1 – tDistributionCDF(Math.abs(t), df));
if (isNaN(cumulativeProb) || isNaN(twoTailedProb)) {
document.getElementById("cumulativeProbability").textContent = "Calculation error. Please check inputs.";
document.getElementById("twoTailedProbability").textContent = "";
} else {
document.getElementById("cumulativeProbability").textContent = cumulativeProb.toFixed(6);
document.getElementById("twoTailedProbability").textContent = twoTailedProb.toFixed(6);
}
}
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font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 20px;
max-width: 600px;
margin: 20px auto;
box-shadow: 0 4px 8px rgba(0, 0, 0, 0.05);
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color: #333;
text-align: center;
margin-bottom: 20px;
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transition: background-color 0.3s ease;
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background-color: #0056b3;
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Understanding the Student's t-Distribution
The Student's t-distribution, often simply called the 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 is a fundamental concept in inferential statistics, particularly for hypothesis testing and constructing confidence intervals.
What is the t-Distribution?
The t-distribution is similar in shape to the normal distribution, but it has heavier tails, meaning it is more prone to producing values that fall far from its mean. This characteristic accounts for the increased uncertainty when working with small sample sizes. As the sample size increases, the t-distribution approaches the standard normal distribution.
Key Parameters: Degrees of Freedom (df) and t-score (t)
- Degrees of Freedom (df): This parameter defines the shape of the t-distribution. It is typically calculated as
n - 1, where n is the sample size. A higher number of degrees of freedom means the t-distribution more closely resembles the normal distribution.
- t-score (t): Also known as the t-statistic, this is a standardized value that measures how far a sample mean is from the population mean in units of the standard error. It is calculated using the formula:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size)).
When is the t-Distribution Used?
The t-distribution is primarily used in situations where:
- The sample size is small (typically n < 30).
- The population standard deviation is unknown.
- The underlying population is assumed to be normally distributed (or approximately normal for larger sample sizes due to the Central Limit Theorem).
Common applications include:
- Hypothesis Testing: To test claims about population means (e.g., one-sample t-test, two-sample t-test, paired t-test).
- Confidence Intervals: To construct intervals that estimate the range within which a population mean is likely to fall.
Interpreting the Calculator Results
Our t-distribution calculator provides two key probabilities based on your input for degrees of freedom (df) and t-score (t):
- Cumulative Probability P(T ≤ t): This is the probability that a random variable from the t-distribution with the given degrees of freedom will be less than or equal to your specified t-score. In hypothesis testing, this can be used for one-tailed tests (e.g., testing if a mean is significantly less than a hypothesized value).
- Two-tailed Probability P(|T| > |t|): This is the probability that the absolute value of a random variable from the t-distribution will be greater than the absolute value of your specified t-score. This is commonly used for two-tailed hypothesis tests, where you are interested in whether the sample mean is significantly different from the population mean in either direction (greater or less). This value is often referred to as the p-value when comparing your calculated t-statistic to a critical region.
Example Usage:
Let's say you conduct a study with a sample size of 11 (so df = 10) and calculate a t-score of 2.228.
- Input: Degrees of Freedom (df) = 10, t-score (t) = 2.228
- Output:
- Cumulative Probability P(T ≤ 2.228) ≈ 0.975000
- Two-tailed Probability P(|T| > |2.228|) ≈ 0.050000
This means there is a 97.5% chance that a t-value with 10 degrees of freedom will be less than or equal to 2.228. For a two-tailed test, there is a 5% chance of observing a t-value with an absolute magnitude greater than 2.228. If your significance level (alpha) was 0.05, you would reject the null hypothesis in a two-tailed test because your p-value (0.05) is not greater than alpha (0.05).