How to Use a p-value Table

If you're dealing with statistics, you've probably encountered the "p-value" more than once. No matter if you're new to this field, trying to understand scientific papers, or need to perform your own statistics, this article will help you to:

• Understand how to find the p-value;
• Interpret a p-value table (or p-value chart);
• Determine when a p-value is significant; and
• Learn about the practical applications of a p-value table.

Before we come to the calculations of the p-value (probability value), let's first recap what this number actually means. As the name gives away, it has to do something with probability — but what probability exactly?

The p-value basically tells you the probability of getting such results as yours (or more extreme) under the null hypothesis (which states that there is no true effect). This might sound complicated, but all this means is that the p-value is simply a number that helps you decide whether your results are likely due to chance (null hypothesis accepted) or not (null hypothesis rejected). Let's move on to the next step — how do you find the p-value?

The p-value can be derived from several statistical tests, such as z-test, t-test, chi-square test, and F-test. This also means that there are different kinds of p-value tables, which are:

• t-tables;
• z-tables;
• chi-square-tables; and
• F-tables.

If you are performing a t-test, you receive the test-value (t-value) and degrees of freedom (df). You can then calculate the exact p-value using the t-test formula. Alternatively, you can estimate the p-value from a p-value table by finding your df and t-value and seeing which significance level it corresponds to.

If you are performing a z-test, you receive the z-value, from which you can directly calculate the p-value. You can also use the z-test formula or estimate it from a p-value table.

If you are performing a chi-square test, you receive the χ²-value and dfs. You can then calculate the exact p-value using the chi-square formula or estimate it from a p-value table based on your dfs and χ²-value.

If you are performing an F-test (e.g., ANOVA), you receive the F-value and two degrees of freedom. The p-value can be calculated via the F-distribution formula or estimated by finding your dfs and the F-value in the p-value table.

As the p-value originated from 1900, scientists didn't have a computer or a p-value calculator; they used p-value tables in textbooks. However, these days, p-value tables are still helpful for learning and quick checks. Below you can find an example of a p-value table, more precisely of a t-table:

This is how you read the t-table:

1. Find your t-value and degrees of freedom — these come from your t-test calculation.
2. Locate your df in the left-hand column — this will pick the correct row of the table.
3. Look across that row at the critical t-values shown for different significance levels (α) — the column headings show the p-values for one- or two-tailed tests.
4. Compare your calculated t-value with the critical t-values — see between which two critical values your t-value falls to estimate your p-value range.
5. Interpret your data — check whether the p-value (or its range) is less than your chosen significance level (commonly 0.05).
6. Draw a conclusion — if p < α, reject the null hypothesis (H₀); if p ≥ α, do not reject H₀.

As we've described above, the p-value is essential to assess the probability of the occurrence of your results under the null hypothesis. If the p-value is below your significance threshold (typically p < 0.05), then you can reject the null hypothesis. However, remember that this does not necessarily mean your alternative hypothesis is true. View it more as an indicator, whether your data just appears to have outstanding values or can actually be objectively considered significant by the p-value threshold.

Let's take a look at an example. Suppose you want to see if a new fertilizer helps wheat grow taller. Now, the null hypothesis assumes that the fertilizer has no effect on plant height. You measure the height of 15 plants with the new fertilizer and 15 without. You run the t-test with your values and receive:

• t-value= 1.3
• Degrees of freedom = 28

In the latter case, we can derive from the p-value table that the significance will be above 0.1. As this value is > 0.05, the result is not statistically significant at the 5% level. Therefore, we can conclude that the difference in height is probably random, which means that there isn’t strong evidence that the fertilizer really works.

Even though there is statistical software to perform statistical tests, a printed p-value table is still useful in some instances, even nowadays:

• Data analysis in research — quick checks for significant t-values without running software.
• Teaching — demonstrating the process of hypothesis testing and contrasting the outcomes of one-tailed and two-tailed tests.
• Fieldwork and practical settings — immediate evaluation of results in the absence of internet and computers to make quick decisions (e.g., quality control) before running full statistical analyses.