Statistics How To

T Test (Student’s T-Test): Definition and Examples

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What is a T test?

Watch the video for an overview of the T-test:

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students-normal t

Various degrees of freedom for Student’s T.

The t test tells you how significant the differences between groups are; In other words it lets you know if those differences (measured in means) could have happened by chance.

A very simple example: Let’s say you have a cold and you try a naturopathic remedy. Your cold lasts a couple of days. The next time you have a cold, you buy an over-the-counter pharmaceutical and the cold lasts a week. You survey your friends and they all tell you that their colds were of a shorter duration (an average of 3 days) when they took the homeopathic remedy. What you really want to know is, are these results repeatable? A t test can tell you by comparing the means of the two groups and letting you know the probability of those results happening by chance.

Another example: Student’s T-tests can be used in real life to compare averages. For example, a drug company may want to test a new cancer drug to find out if it improves life expectancy. In an experiment, there’s always a control group (a group who are given a placebo, or “sugar pill”). The control group may show an average life expectancy of +5 years, while the group taking the new drug might have a life expectancy of +6 years. It would seem that the drug might work. But it could be due to a fluke. To test this, researchers would use a Student’s t-test to find out if the results are repeatable for an entire population.

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The T Score.

The t score is a ratio between the difference between two groups and the difference within the groups. The larger the t score, the more difference there is between groups. The smaller the t score, the more similarity there is between groups. A t score of 3 means that the groups are three times as different from each other as they are within each other. When you run a t test, the bigger the t-value, the more likely it is that the results are repeatable.

  • A large t-score tells you that the groups are different.
  • A small t-score tells you that the groups are similar.

T-Values and P-values

How big is “big enough”? Every t-value has a p-value to go with it. A p-value is the probability that the results from your sample data occurred by chance. P-values are from 0% to 100%. They are usually written as a decimal. For example, a p value of 5% is 0.05. Low p-values are good; They indicate your data did not occur by chance. For example, a p-value of .01 means there is only a 1% probability that the results from an experiment happened by chance. In most cases, a p-value of 0.05 (5%) is accepted to mean the data is valid.

Calculating the Statistic / Test Types

There are three main types of t-test:

You probably don’t want to calculate the test by hand (the math can get very messy, but if you insist you can find the steps for an independent samples t test here.

Use the following tools to calculate the t test:
How to do a T test in Excel.
T test in SPSS.
T distribution on the TI 89.
T distribution on the TI 83.

What is a Paired T Test (Paired Samples T Test / Dependent Samples T Test)?

A paired t test (also called a correlated pairs t-test, a paired samples t test or dependent samples t test) is where you run a t test on dependent samples. Dependent samples are essentially connected — they are tests on the same person or thing. For example:

  • Knee MRI costs at two different hospitals,
  • Two tests on the same person before and after training,
  • Two blood pressure measurements on the same person using different equipment.

When to Choose a Paired T Test / Paired Samples T Test / Dependent Samples T Test

Choose the paired t-test if you have two measurements on the same item, person or thing. You should also choose this test if you have two items that are being measured with a unique condition. For example, you might be measuring car safety performance in vehicle research and testing and subject the cars to a series of crash tests. Although the manufacturers are different, you might be subjecting them to the same conditions.

With a “regular” two sample t test, you’re comparing the means for two different samples. For example, you might test two different groups of customer service associates on a business-related test or testing students from two universities on their English skills. If you take a random sample each group separately and they have different conditions, your samples are independent and you should run an independent samples t test (also called between-samples and unpaired-samples).

The null hypothesis for the independent samples t-test is μ1 = μ2. In other words, it assumes the means are equal. With the paired t test, the null hypothesis is that the pairwise difference between the two tests is equal (H0: µd = 0). The difference between the two tests is very subtle; which one you choose is based on your data collection method.

Paired Samples T Test By hand

Example question: Calculate a paired t test by hand for the following data:
paired t test example 2

Step 1: Subtract each Y score from each X score.
paired t test example 3

Step 2: Add up all of the values from Step 1.
Set this number aside for a moment.
paired t test example 4

Step 3: Square the differences from Step 1.
paired t test example 5

Step 4: Add up all of the squared differences from Step 3.
paired t test example 6

Step 5: Use the following formula to calculate the t-score:
paired samples t test example

  • ΣD: Sum of the differences (Sum of X-Y from Step 2)
  • ΣD2: Sum of the squared differences (from Step 4)
  • (ΣD)2: Sum of the differences (from Step 2), squared.

If you’re unfamiliar with Σ you may want to read about summation notation first.

paired t test example 7v3

Step 6: Subtract 1 from the sample size to get the degrees of freedom. We have 11 items, so 11-1 = 10.

Step 7: Find the p-value in the t-table, using the degrees of freedom in Step 6. If you don’t have a specified alpha level, use 0.05 (5%). For this example problem, with df = 10, the t-value is 2.228.

Step 8: Compare your t-table value from Step 7 (2.228) to your calculated t-value (-2.74). The calculated t-value is greater than the table value at an alpha level of .05. The p-value is less than the alpha level: p <.05. We can reject the null hypothesis that there is no difference between means.

Note: You can ignore the minus sign when comparing the two t-values, as ± indicates the direction; the p-value remains the same for both directions.

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Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New York: Wiley, pp. 50-55, 1956.

Stephanie Glen. "T Test (Student’s T-Test): Definition and Examples" From Elementary Statistics for the rest of us!

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