p-value vs. t-value

When conducting statistical analyses, especially hypothesis testing, it is easy to get confused about the different parameters and how they relate to each other. In this article, you can find answers about the meaning and connection of:

• t-test and p-value;
• t-value vs. p-value;
• t-statistic vs. p-value; and
• p-value vs test statistic.

No need to wonder anymore — let's dive in and clear things up!

The t-test is used to test hypotheses and determine whether a statistically significant difference exists between the means of two data sets. In other words, it checks how different both groups are from one another, and whether that difference can be linked to a true effect. One of the results you receive after this test is a t-value, which we'll look at more closely in the next section, along with degrees of freedom.

Depending on the nature of the data and the purpose of statistical testing, there are three types of t-tests you can use:

1. One-sample t-test — comparing a single group to the population mean;
2. Two-sample t-test — comparing the mean of 2 unrelated groups; and
3. Paired t-test — comparing the mean within the group.

In order to run a t-test, you only need to have the following data:

1. The mean difference — the difference between the mean values from each group;
2. The standard deviation — of both groups; and
3. The sample size — of each dataset.

Then, you can enter your data into our t-test calculator. If you have a larger sample size, you may also use our z-test calculator.

The t-value shows how far your sample mean is from the hypothesized mean, measured in units of standard error. The t-value is calculated from the t-test, which, depending on the type, differs in its formula.

In the one-sample t-test, the t-value is calculated like so:

where:

• $\bar{x}$ — Sample mean;
• $\mu$ — Population mean;
• $s$ — Sample standard deviation; and
• $n$ — Sample size.

For the two sample t-test:

where:

• $\bar{x}_1,\ \bar{x}_2$ — Sample means;​
• $s_p$ — Pooled standard deviation; and
• ​$n_1,\ n_2$ — Sample sizes.

​
For the paired t-test:

where:

• $\bar{d}$ — Mean of the differences between paired observations;
• $s_d$ — Standard deviation of the differences; and
• $n$ — Number of pairs.

The bigger the t-value, the stronger the evidence that the groups are different, which usually results in a smaller p-value. A smaller t-value, on the other hand, suggests the groups are more similar. The sample size also impacts how small or large the p-value becomes.

While the t-value shows the difference from the group means, the p-value uses that measured distance to calculate the probability. With the t-value and degrees of freedom derived from the t-test, you can then determine the p-value. But what "extra" does the p-value give you? Well, it is an established threshold to determine whether you can or cannot dispute the null hypothesis.

The p-value, or probability value, tells you the probability of getting a result at least as extreme as yours if the null hypothesis were true. The null hypothesis (N₀) claims that the results occurred by mere chance and that there is no true effect attributed to your observed data. Let's see how that looks in practice:

• A small p-value (usually <0.05) — your results can be considered statistically significant (true effect likely);
• A larger p-value (usually >0.05) — the evidence is too weak to reject N₀ (the difference is not unusual enough to exclude chance).

You could therefore say that the t-value describes your data, and the p-value helps you interpret it.

Now that we have a good understanding of the t-value (or t-statistic) and the p-value, let's summarize the main points. Have a look at the comparison table of t-statistics vs. p-value below: