Use this calculator to determine the p-value for a two-sample t-test, comparing the means of two independent groups. The p-value helps you assess the statistical significance of your observed results.
Two-tailed
One-tailed (Left)
One-tailed (Right)
Understanding P-Values and the Two-Sample T-Test
The p-value is a fundamental concept in inferential statistics, used to quantify the evidence against a null hypothesis. In hypothesis testing, we typically start with a null hypothesis (H₀), which states there is no effect or no difference, and an alternative hypothesis (H₁), which states there is an effect or a difference.
What is a P-Value?
A p-value (probability value) is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your sample data, assuming that the null hypothesis is true. A small p-value suggests that your observed data would be very unlikely if the null hypothesis were true, thus providing strong evidence against H₀.
Interpreting the P-Value
If p-value < α (significance level): You reject the null hypothesis. This suggests that the observed difference is statistically significant and unlikely to have occurred by random chance.
If p-value ≥ α (significance level): You fail to reject the null hypothesis. This means there isn't enough evidence to conclude a statistically significant difference. It does not mean the null hypothesis is true, only that your data doesn't provide sufficient evidence to reject it.
The Two-Sample T-Test
This calculator specifically uses a two-sample t-test (also known as an independent samples t-test) to compare the means of two independent groups. It's commonly used when you want to determine if there's a statistically significant difference between the average values of two separate populations based on samples drawn from them. For instance, comparing the average test scores of students taught by two different methods, or the average yield of two different fertilizer types.
Assumptions of the T-Test
While robust, the t-test relies on certain assumptions:
Independence: Observations within each sample and between samples must be independent.
Normality: The data in each group should be approximately normally distributed. This assumption becomes less critical with larger sample sizes due to the Central Limit Theorem.
Homogeneity of Variances (for pooled t-test): The variances of the two populations should be equal. This calculator uses Welch's t-test, which does not assume equal variances, making it more robust for many real-world scenarios.
One-tailed vs. Two-tailed Tests
Two-tailed test: Used when you are interested in detecting a difference in either direction (e.g., Group A mean is different from Group B mean). The alternative hypothesis is H₁: μ₁ ≠ μ₂.
One-tailed test (Left): Used when you are specifically interested in detecting if one mean is significantly *less than* the other (e.g., Group A mean is less than Group B mean). The alternative hypothesis is H₁: μ₁ < μ₂.
One-tailed test (Right): Used when you are specifically interested in detecting if one mean is significantly *greater than* the other (e.g., Group A mean is greater than Group B mean). The alternative hypothesis is H₁: μ₁ > μ₂.
Choosing the correct test type is crucial as it affects the calculated p-value.
Example Calculation
Let's say a researcher wants to compare the effectiveness of two different teaching methods on student test scores. They randomly assign 30 students to Method A and 35 students to Method B.
Method A (Sample 1): Mean score = 10.5, Standard Deviation = 2.1, Sample Size = 30
Method B (Sample 2): Mean score = 9.8, Standard Deviation = 1.9, Sample Size = 35
Significance Level (α): 0.05
Test Type: Two-tailed (since they just want to know if there's a difference, not a specific direction)
Using the calculator with these values:
Sample 1 Mean: 10.5
Sample 1 Standard Deviation: 2.1
Sample 1 Size: 30
Sample 2 Mean: 9.8
Sample 2 Standard Deviation: 1.9
Sample 2 Size: 35
Significance Level: 0.05
Test Type: Two-tailed
The calculator would output:
t-statistic: Approximately 1.40
Degrees of Freedom (df): Approximately 59.14
P-value: Approximately 0.166
Conclusion: Since 0.166 ≥ 0.05, we fail to reject the null hypothesis. There is not enough statistical evidence to conclude a significant difference in test scores between the two teaching methods at the 0.05 significance level.
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// Helper functions for Gamma and Incomplete Beta functions, necessary for t-distribution CDF
// Adapted from common statistical library patterns for pure JS implementation.
// Log Gamma function (Lanczos approximation)
function logGamma(z) {
if (z < 0.5) {
return Math.log(Math.PI) – Math.log(Math.sin(Math.PI * z)) – logGamma(1 – 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;
var x = z;
var y = x + g + 0.5;
var arg = (x + 0.5) * Math.log(y) – y;
var sum = p[0];
for (var i = 1; i < p.length; i++) {
sum += p[i] / (x + i);
}
return arg + Math.log(Math.sqrt(2 * Math.PI) * sum / x);
}
// Regularized Incomplete Beta Function I_x(a, b)
// This function is crucial for the Student's t-distribution CDF.
// It uses a continued fraction expansion.
function incompleteBeta(a, b, x) {
if (x 1) return NaN;
if (x === 0) return 0;
if (x === 1) return 1;
// Log Beta function (ln(Beta(a,b)))
var logBeta = logGamma(a) + logGamma(b) – logGamma(a + b);
// Continued fraction for I_x(a, b)
var F = function(aa, bb, xx) {
var C = 1;
var D = 1 / C;
var H = D;
var m = 1;
var m2 = 2;
var delta = 0;
var term;
while (m < 1000) { // Max iterations for convergence
var numerator;
var denominator;
if (m % 2 === 0) { // Even terms
numerator = (m / 2) * (bb – (m / 2)) * xx / ((aa + m – 1) * (aa + m));
denominator = 1;
} else { // Odd terms
numerator = -((aa + (m – 1) / 2) * (aa + bb + (m – 1) / 2) * xx) / ((aa + m – 1) * (aa + m));
denominator = 1;
}
C = 1 + numerator * C;
if (Math.abs(C) < 1e-30) C = 1e-30; // Avoid division by zero
D = 1 + denominator / D;
if (Math.abs(D) < 1e-30) D = 1e-30; // Avoid division by zero
H *= C * D;
term = Math.abs(1 – H);
if (term < 1e-7) { // Convergence check
delta = term;
break;
}
m++;
}
return H;
};
var result;
// Use series expansion for small x, continued fraction for larger x (or 1-x for I_{1-x}(b,a))
// This conditional logic helps with numerical stability and convergence.
if (x < (a + 1) / (a + b + 2)) {
result = Math.exp(a * Math.log(x) + b * Math.log(1 – x) – logBeta) * F(a, b, x) / a;
} else {
result = 1 – Math.exp(b * Math.log(1 – x) + a * Math.log(x) – logBeta) * F(b, a, 1 – x) / b;
}
return result;
}
// Student's t-distribution Cumulative Distribution Function (CDF)
function studentTCDF(t, df) {
if (df <= 0) return NaN;
var x = df / (df + t * t);
if (t < 0) {
return 0.5 * incompleteBeta(df / 2, 0.5, x);
} else {
return 1 – 0.5 * incompleteBeta(df / 2, 0.5, x);
}
}
function calculatePValue() {
var sample1Mean = parseFloat(document.getElementById("sample1Mean").value);
var sample1StdDev = parseFloat(document.getElementById("sample1StdDev").value);
var sample1Size = parseInt(document.getElementById("sample1Size").value);
var sample2Mean = parseFloat(document.getElementById("sample2Mean").value);
var sample2StdDev = parseFloat(document.getElementById("sample2StdDev").value);
var sample2Size = parseInt(document.getElementById("sample2Size").value);
var significanceLevel = parseFloat(document.getElementById("significanceLevel").value);
var testType = document.getElementById("testType").value;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(sample1Mean) || isNaN(sample1StdDev) || isNaN(sample1Size) ||
isNaN(sample2Mean) || isNaN(sample2StdDev) || isNaN(sample2Size) ||
isNaN(significanceLevel)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (sample1StdDev <= 0 || sample2StdDev <= 0) {
resultDiv.innerHTML = "Standard deviations must be positive.";
return;
}
if (sample1Size <= 1 || sample2Size <= 1) {
resultDiv.innerHTML = "Sample sizes must be greater than 1.";
return;
}
if (significanceLevel = 1) {
resultDiv.innerHTML = "Significance level (α) must be between 0 and 1.";
return;
}
// Calculate t-statistic (Welch's t-test for unequal variances)
var s1_sq_n1 = (sample1StdDev * sample1StdDev) / sample1Size;
var s2_sq_n2 = (sample2StdDev * sample2StdDev) / sample2Size;
var t_statistic = (sample1Mean – sample2Mean) / Math.sqrt(s1_sq_n1 + s2_sq_n2);
// Calculate degrees of freedom (Welch-Satterthwaite equation)
var numerator_df = Math.pow(s1_sq_n1 + s2_sq_n2, 2);
var denominator_df = (Math.pow(s1_sq_n1, 2) / (sample1Size – 1)) + (Math.pow(s2_sq_n2, 2) / (sample2Size – 1));
var degrees_of_freedom = numerator_df / denominator_df;
// Calculate p-value
var p_value;
var cdf_t = studentTCDF(t_statistic, degrees_of_freedom);
if (testType === "two-tailed") {
p_value = 2 * Math.min(cdf_t, 1 – cdf_t);
} else if (testType === "one-tailed-left") {
p_value = cdf_t;
} else { // one-tailed-right
p_value = 1 – cdf_t;
}
// Determine conclusion
var conclusion;
if (p_value < significanceLevel) {
conclusion = "Reject the null hypothesis. There is statistically significant evidence of a difference.";
} else {
conclusion = "Fail to reject the null hypothesis. There is not enough statistical evidence of a difference.";
}
resultDiv.innerHTML =
"Calculated t-statistic: " + t_statistic.toFixed(4) + "" +
"Degrees of Freedom (df): " + degrees_of_freedom.toFixed(2) + "" +
"P-value: " + p_value.toFixed(5) + "" +
"Significance Level (α): " + significanceLevel.toFixed(2) + "" +
"Conclusion: " + conclusion + "";
}