In hypothesis testing, we aim to determine if there's enough evidence in a sample of data to reject a null hypothesis (H₀). The significance level, often denoted by the Greek letter alpha (α), is a threshold we set beforehand to decide whether to reject H₀. Common alpha levels are 0.05 (5%), 0.01 (1%), and 0.10 (10%).
The p-value is the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is true. It's a crucial measure used in conjunction with the significance level to make decisions.
How to Interpret the Results:
If the p-value is less than or equal to the significance level (α), we reject the null hypothesis (H₀). This suggests that the observed data is statistically significant and unlikely to have occurred by random chance alone.
If the p-value is greater than the significance level (α), we fail to reject the null hypothesis (H₀). This means the observed data is not sufficiently unlikely to have occurred by random chance to warrant rejecting H₀.
Inputs Explained:
Observed Value (O): In the context of calculating a p-value, this often refers to the probability associated with your calculated test statistic. For example, if you have a z-score, you would look up the area under the standard normal curve that corresponds to this score. This calculator uses the observed value as a direct input for simplicity in some scenarios, but for most statistical tests, the Test Statistic is the primary input used to find the p-value.
Expected Value (E) / Alpha Level (α): This is your predefined threshold for statistical significance. It represents the maximum probability of making a Type I error (rejecting a true null hypothesis) that you are willing to tolerate.
Test Statistic (z or t): This is the calculated value from your sample data using a specific statistical test (e.g., z-test, t-test, chi-squared test). It quantifies how far your sample result deviates from the null hypothesis.
Degrees of Freedom (df): This parameter is crucial for certain statistical tests, particularly the t-distribution and chi-squared distribution. It generally relates to the sample size and the number of independent pieces of information used to estimate a parameter. Many online calculators and statistical software will use the test statistic and df to directly compute the p-value.
Calculator Logic:
This calculator simplifies the p-value calculation. For common scenarios involving z-scores or t-scores:
If degreesOfFreedom is provided, it attempts a more accurate p-value calculation (approximated or using a lookup concept).
If only testStatistic is provided (and often assuming it's a z-score), it calculates the probability of observing a value as extreme or more extreme. For a two-tailed test (most common), the p-value is twice the probability of observing a value in one tail.
The observedValue input is more conceptually related to alpha, but in a typical statistical test, the p-value is derived from the test statistic. For demonstration, this calculator primarily uses the testStatistic and degreesOfFreedom to estimate the p-value. A direct calculation from observedValue and expectedValue is less standard for hypothesis testing's p-value.
Note: Accurately calculating p-values for specific test statistics and distributions often requires complex statistical functions (like CDFs – Cumulative Distribution Functions) which are not natively available in basic JavaScript. This calculator provides an approximation based on common z-score scenarios and uses the degreesOfFreedom to inform the calculation, but for precise results with specific tests (like complex t-tests or ANOVAs), dedicated statistical software or libraries are recommended.
Example:
Let's say you performed a t-test and obtained a t-statistic of 2.5 with 20 degrees of freedom. You set your significance level (α) to 0.05.
Input Observed Value (O): (Not directly used in the primary calculation logic for test statistics, but could represent alpha) e.g., 0.05
Input Expected Value (E) / Alpha Level (α): 0.05
Input Test Statistic: 2.5
Input Degrees of Freedom: 20
Running this through a proper statistical function (or a more advanced calculator) would yield a p-value of approximately 0.021. Since 0.021 is less than 0.05, you would reject the null hypothesis.
function calculateSignificanceLevel() {
var observedValue = parseFloat(document.getElementById("observedValue").value);
var expectedValue = parseFloat(document.getElementById("expectedValue").value);
var testStatistic = parseFloat(document.getElementById("testStatistic").value);
var degreesOfFreedom = parseFloat(document.getElementById("degreesOfFreedom").value);
var resultValueElement = document.getElementById("result-value");
var interpretationElement = document.getElementById("interpretation");
var resultDiv = document.getElementById("result");
resultValueElement.textContent = "";
interpretationElement.textContent = "";
resultDiv.style.display = 'none';
var pValue = NaN;
var interpretation = "";
// Basic validation for critical inputs
if (isNaN(testStatistic) || testStatistic === null) {
interpretationElement.textContent = "Please enter a valid Test Statistic.";
resultDiv.style.display = 'block';
return;
}
if (isNaN(expectedValue) || expectedValue = 1) {
interpretationElement.textContent = "Please enter a valid Alpha Level (between 0 and 1, exclusive).";
resultDiv.style.display = 'block';
return;
}
// — Simplified p-value approximation —
// This is a highly simplified approximation. Real p-value calculation
// involves complex CDFs (Cumulative Distribution Functions) and often
// requires statistical libraries.
// Approximation for z-scores (ignoring df for simplicity if not provided)
if (isNaN(degreesOfFreedom) || degreesOfFreedom === null || degreesOfFreedom = 3.5, p-value is very small.
var absZ = Math.abs(testStatistic);
if (absZ > 6) { pValue = 0; }
else {
// Approximation based on polynomial fitting for standard normal CDF
// Source: Abramowitz and Stegun, Handbook of Mathematical Functions
// This is a common approximation used in many contexts.
var t = absZ;
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var erf = 1.0 – (((((a5*t+a4)*t)+a3)*t+a2)*t+a1)*t*Math.exp(-t*t);
var cdf = 0.5 * (1.0 + erf);
pValue = 2 * (1 – cdf); // Two-tailed test
}
interpretation = "Approximated p-value based on Z-score (two-tailed).";
} else {
// Basic handling for t-distribution (highly simplified)
// A precise t-distribution CDF is complex. This is a placeholder
// for demonstration that df *can* be used.
// For a proper calculation, use a statistical library.
// This simple check assumes larger df approaches normal distribution.
// A real implementation would use a more sophisticated algorithm.
// Placeholder logic: if df is large, it's like a z-score.
// If df is small, p-value is generally larger for the same statistic.
// This is NOT a precise calculation.
var absT = Math.abs(testStatistic);
if (degreesOfFreedom > 100) { // Approximates z-score for large df
if (absT > 6) { pValue = 0; }
else {
var t = absT;
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var erf = 1.0 – (((((a5*t+a4)*t)+a3)*t+a2)*t+a1)*t*Math.exp(-t*t);
var cdf = 0.5 * (1.0 + erf);
pValue = 2 * (1 – cdf); // Two-tailed test
}
interpretation = "Approximated p-value based on t-statistic (large df ~ Z-score, two-tailed).";
} else {
// Very crude approximation for smaller df
// For a given t-statistic, p-value increases as df decreases.
// This is extremely basic and just illustrates the concept.
// Real calculation requires specialized functions.
var basePValueApprox = 2 * (1 – normalCdfApprox(absT)); // Approximation based on z-score logic
// Adjust upwards slightly for lower df – very rough estimate!
pValue = basePValueApprox * (1 + 10 / (degreesOfFreedom + 10));
if (pValue > 1) pValue = 1; // Cap at 1
interpretation = "Highly approximated p-value based on t-statistic (small df, two-tailed). Use with caution.";
}
}
// — Determine final result and interpretation —
if (!isNaN(pValue)) {
resultValueElement.textContent = pValue.toFixed(5); // Display p-value with reasonable precision
if (pValue <= expectedValue) {
interpretation += " Since p-value alpha, fail to reject the null hypothesis.";
}
interpretationElement.textContent = interpretation;
resultDiv.style.display = 'block';
} else {
interpretationElement.textContent = "Could not calculate p-value. Please check inputs.";
resultDiv.style.display = 'block';
}
}
// Helper function for approximate standard normal CDF (used in z-score approximation)
// This is a common polynomial approximation.
function normalCdfApprox(x) {
var t = x;
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = 1;
if (t < 0) {
t = -t;
sign = -1;
}
var erf = 1.0 – (((((a5*t+a4)*t)+a3)*t+a2)*t+a1)*t*Math.exp(-t*t);
var cdf = 0.5 * (1.0 + sign*erf);
return cdf;
}