P values are everywhere in statistics. They’re in all types of hypothesis tests. But how do you calculate a p-value? Unsurprisingly, the precise calculations depend on the test. However, there is a general process that applies to finding a p value.
In this post, you’ll learn how to find the p value. I’ll start by showing you the general process for all hypothesis tests. Then I’ll move on to a step-by-step example showing the calculations for a p value. This post includes a calculator so you can apply what you learn.
General Process for How to Find the P value
To find the p value for your sample, do the following:
- Identify the correct test statistic.
- Calculate the test statistic using the relevant properties of your sample.
- Specify the characteristics of the test statistic’s sampling distribution.
- Place your test statistic in the sampling distribution to find the p value.
Before moving on to the calculations example, I’ll summarize the purpose for each step. This part tells you the “why.” In the example calculations section, I show the “how.”
Identify the Correct Test Statistic
All hypothesis tests boil your sample data down to a single number known as a test statistic. T-tests use t-values. F-tests use F-values. Chi-square tests use chi-square values. Choosing the correct one depends on the type of data you have and how you want to analyze it. Before you can find the p value, you must determine which hypothesis test and test statistic you’ll use.
Test statistics assess how consistent your sample data are with the null hypothesis. As a test statistic becomes more extreme, it indicates a larger difference between your sample data and the null hypothesis.
Calculate the Test Statistic
How you calculate the test statistic depends on which one you’re using. Unsurprisingly, the method for calculating test statistics varies by test type. Consequently, to calculate the p value for any test, you’ll need to know the correct test statistic formula.
To learn more about test statistics and how to calculate them for other tests, read my article, Test Statistics.
Specify the Properties of the Test Statistic’s Sampling Distribution
Test statistics are unitless, making them tricky to interpret on their own. You need to place them in a larger context to understand how extreme they are.
The sampling distribution for the test statistic provides that context. Sampling distributions are a type of probability distribution. Consequently, they allow you to calculate probabilities related to your test statistic’s extremeness, which lets us find the p value!
For example, what does a t-value of two indicate? Is it significant? As you’ll see in the example, the t-distribution answers that question and allows us to calculate the p-value.
Like any distribution, the same sampling distribution (e.g., the t-distribution) can have a variety of shapes depending upon its parameters. For this step, you need to determine the characteristics of the sampling distribution that fit your design and data.
That usually entails specifying the degrees of freedom (changes its shape) and whether the test is one- or two-tailed (affects the directions the test can detect effects). In essence, you’re taking the general sampling distribution and tailoring it to your study so it provides the correct probabilities for finding the p value.
Each test statistic’s sampling distribution has unique properties you need to specify. At the end of this post, I provide links for several.
Learn more about degrees of freedom and one-tailed vs. two-tailed tests.
Placing Your Test Statistic in its Sampling Distribution to Find the P value
Finally, it’s time to find the p value because we have everything in place. We have calculated our test statistic and determined the correct properties for its sampling distribution. Now, we need to find the probability of values more extreme than our observed test statistic.
In this context, more extreme means further away from the null value in both directions for a two-tailed test or in one direction for a one-tailed test.
At this point, there are two ways to use the test statistic and distribution to calculate the p value. The formulas for probability distributions are relatively complex. Consequently, you won’t calculate it directly. Instead, you’ll use either an online calculator or a statistical table for the test statistic. I’ll show you both approaches in the step-by-step example.
In summary, calculating a p-value involves identifying and calculating your test statistic and then placing it in its sampling distribution to find the probability of more extreme values!
Let’s see this whole process in action with an example!
Step-by-Step Example of How to Find the P value for a T-test
For this example, assume we’re tasked with determining whether a sample mean is different from a hypothesized value. We’re given the sample statistics below and need to find the p value.
- Mean: 330.6
- Standard deviation: 154.2
- Sample size: 25
- Null hypothesis value: 260
Let’s work through the step-by-step process of how to calculate a p-value.
First, we need to identify the correct test statistic. Because we’re comparing one mean to a null value, we need to use a 1-sample t-test. Hence, the t-value is our test statistic, and the t-distribution is our sampling distribution.
Second, we’ll calculate the test statistic. The t-value formula for a 1-sample t-test is the following:
- x̄ is the sample mean.
- µ0 is the null hypothesis value.
- s is the sample standard deviation.
- n is the sample size
- Collectively, the denominator is the standard error of the mean.
Let’s input our sample values into the equation to calculate the t-value.
Third, we need to specify the properties of the sampling distribution to find the p value. We’ll need the degrees of freedom.
The degrees of freedom for a 1-sample t-test is n – 1. Our sample size is 25. Hence, we have 24 DF. We’ll use a two-tailed test, which is the standard.
Now we’ve got all the necessary information to calculate the p-value. I’ll show you two ways to take the final step!
One method is to use an online p-value calculator, like the one I include below.
Enter the following in the calculator for our t-test example.
- In What do you want?, choose Two-tailed p-value (the default).
- In What do you have?, choose t-score.
- In Degrees of freedom (d), enter 24.
- In Your t-score, enter 2.289.
The calculator displays a result of 0.031178.
There you go! Using the standard significance level of 0.05, our results are statistically significant!
Using a Statistical Table to Find the P Value
The other common method is using a statistical table. In this case, we’ll need to use a t-table. For this example, I’ll truncate the rows. You can find my full table here: T-Table.
This method won’t find the exact p value, but you’ll find a range and know whether your results are statistically significant.
Start by looking in the row for 24 degrees of freedom, highlighted in light green. We need to find where our t-score of 2.289 fits in. I highlight the two table values that our t-value fits between, 2.064 and 2.492. Then we look at the two-tailed row at the top to find the corresponding p values for the two t-values.
In this case, our t-value of 2.289 produces a p value between 0.02 and 0.05 for a two-tailed test. Our results are statistically significant, and they are consistent with the calculator’s more precise results.
Displaying the P value in a Chart
In the example above, you saw how to calculate a p-value starting with the sample statistics. We calculated the t-value and placed it in the applicable t-distribution. I find that the calculations and numbers are dry by themselves. I love graphing things whenever possible, so I’ll use a probability distribution plot to illustrate the example.
Using statistical software, I’ll create the graphical equivalent of calculating the p-value above.
This chart has two shaded regions because we performed a two-tailed test. Each region has a probability of 0.01559. When you sum them, you obtain the p-value of 0.03118. In other words, the likelihood of a t-value falling in either shaded region when the null hypothesis is true is 0.03118.
I showed you how to find the p value for a t-test. Click the links below to see how it works for other hypothesis tests:
Now that we’ve found the p value, how do you interpret it precisely? If you’re going beyond the significant/not significant decision and really want to understand what it means, read my posts, Interpreting P Values and Statistical Significance: Definition & Meaning.
If you’re learning about hypothesis testing and like the approach I use in my blog, check out my Hypothesis Testing book! You can find it at Amazon and other retailers.
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