p-value Calculator

The p-value Calculator typically requires the user to input the following information:

1. Test Statistic: The value of the test statistic (e.g., t-value, z-value, or F-value) that was calculated from the data.
2. Degrees of Freedom: The number of degrees of freedom associated with the test statistic.
3. Significance Level: The desired level of significance, often denoted as α (alpha), which is typically set at 0.05 (5%) or 0.01 (1%).

p-value Examples Chart

p-value Formula

In the case of a z-test or t-test, the p-value can be calculated using the following formula:

p-value = 2 × (1 - Φ(|z|))

Where:

• Φ(|z|) is the cumulative distribution function of the standard normal distribution evaluated at the absolute value of the test statistic (z-score or t-statistic).

Suppose we conducted a t-test and obtained a t-statistic of 2.45 with 20 degrees of freedom. To calculate the p-value, we would use the formula:

p-value = 2 × (1 - Φ(|2.45|))

Where Φ(|2.45|) is the cumulative distribution function of the standard normal distribution evaluated at the absolute value of the t-statistic, which is 2.45.

Using a standard normal distribution table or calculator, we can find that Φ(2.45) ≈ 0.9928.

p-value = 2 × (1 - 0.9928) = 0.0144

Therefore, the p-value for this t-test with a t-statistic of 2.45 and 20 degrees of freedom is approximately 0.0144.

How is the p-value Calculated?

The p-value is calculated based on the test statistic (e.g., z-statistic, t-statistic, or F-statistic) and the underlying probability distribution of the test statistic under the null hypothesis.

Suppose we want to test the hypothesis that the mean of a population is equal to a certain value (the null hypothesis, H0). We take a sample from the population, calculate the sample mean, and then compute the t-statistic:

t = (Sample Mean - Hypothesized Mean) / (Standard Error of the Sample Mean)

The t-statistic follows a t-distribution with (n-1) degrees of freedom, where n is the sample size.

The p-value is the probability of obtaining a t-statistic at least as extreme as the one observed, given that the null hypothesis is true. In other words, it represents the likelihood of seeing the observed (or more extreme) results if the null hypothesis is correct.

To calculate the p-value, we follow these steps:

1. Determine the appropriate probability distribution (in this case, the t-distribution) based on the test statistic and the degrees of freedom.
2. Calculate the absolute value of the t-statistic, |t|.
3. Use the cumulative distribution function (CDF) of the t-distribution to find the probability of obtaining a t-statistic with an absolute value greater than or equal to |t|. This is the p-value.

The calculated t-statistic is 2.45 and the degrees of freedom are 20. Using a t-distribution table or calculator, we can find the p-value:

p-value = 2 × (1 - Φ(|2.45|))

Where Φ(|2.45|) is the CDF of the standard normal distribution evaluated at the absolute value of the t-statistic, which is 2.45.

p-value = 2 × (1 - 0.9928) = 0.0144

The p-value for this t-test with a t-statistic of 2.45 and 20 degrees of freedom is approximately 0.0144 or 1.44%.