P-Value Calculator

A p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It measures the strength of evidence against the null hypothesis. A smaller p-value indicates stronger evidence against the null hypothesis.

P-values are typically compared to a significance level (α): • p < 0.001: Very strong evidence against null hypothesis • p < 0.01: Strong evidence against null hypothesis • p < 0.05: Moderate evidence against null hypothesis • p < 0.10: Weak evidence against null hypothesis • p ≥ 0.10: Insufficient evidence to reject null hypothesis

There are three main types of hypothesis tests: • **Left-tailed test**: Tests if a parameter is less than a specified value (H₁: parameter < value) • **Right-tailed test**: Tests if a parameter is greater than a specified value (H₁: parameter > value) • **Two-tailed test**: Tests if a parameter is different from a specified value (H₁: parameter ≠ value)

This calculator supports four common statistical distributions: • **Z-score**: Standard normal distribution N(0,1) • **T-score**: Student's t-distribution (requires degrees of freedom) • **Chi-square score**: Chi-square distribution (requires degrees of freedom) • **F-score**: F-distribution (requires two degrees of freedom)

Degrees of freedom are required for: • **T-distribution**: Always required (typically n-1 for sample mean tests) • **Chi-square distribution**: Always required (depends on test type) • **F-distribution**: Always requires two degrees of freedom • **Z-distribution**: Not required (standard normal distribution)

The p-value is calculated from your data and represents the probability of observing your results under the null hypothesis. The significance level (α) is a threshold you choose before conducting the test (commonly 0.05, 0.01, or 0.10). You reject the null hypothesis if p-value ≤ α.

No, p-values cannot be negative. P-values are probabilities and must be between 0 and 1. A p-value of 0 would indicate that the observed result is impossible under the null hypothesis, while a p-value of 1 would indicate that the observed result is exactly what would be expected under the null hypothesis.

A p-value of exactly 0.05 means that there is a 5% chance of observing results at least as extreme as yours if the null hypothesis is true. At the α = 0.05 significance level, you would reject the null hypothesis. However, p = 0.05 is considered borderline significant and should be interpreted with caution.

The choice of significance level depends on your field and the consequences of errors: • **α = 0.01**: Very strict, used when false positives are costly • **α = 0.05**: Standard in most fields, balances Type I and Type II errors • **α = 0.10**: More lenient, used when false negatives are costly Consider the practical implications of your decision when choosing α.

In hypothesis testing: • **Type I error (α)**: Rejecting the null hypothesis when it's true (false positive) • **Type II error (β)**: Failing to reject the null hypothesis when it's false (false negative) The significance level α controls the probability of Type I error. The power of the test (1-β) is the probability of correctly rejecting a false null hypothesis.

Use a **one-tailed test** when you have a specific directional hypothesis (e.g., 'the new drug is better than the old one'). Use a **two-tailed test** when you're testing for any difference (e.g., 'the new drug is different from the old one'). Two-tailed tests are more conservative and are often preferred unless you have strong theoretical reasons for a directional hypothesis.

This calculator uses mathematical approximations for the cumulative distribution functions. For most practical purposes, these approximations are very accurate. However, for critical research applications, you may want to verify results using statistical software packages like R, SPSS, or SAS.

Very large or small test statistics typically result in very small p-values, indicating strong evidence against the null hypothesis. However, extremely large values may approach the limits of numerical precision. The calculator handles this by capping p-values between 0 and 1.