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

Contents:

• What is a T Test?
• The T Score
• T Values and P Values
• Calculating the T Test
• What is a Paired T Test (Paired Samples T Test)?

What is a T test?

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:

• An Independent Samples t-test compares the means for two groups.
• A Paired sample t-test compares means from the same group at different times (say, one year apart).
• A One sample t-test tests the mean of a single group against a known mean.

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: µ_d = 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:

Step 1: Subtract each Y score from each X score.

Step 2: Add up all of the values from Step 1.
Set this number aside for a moment.

Step 3: Square the differences from Step 1.

Step 4: Add up all of the squared differences from Step 3.

Step 5: Use the following formula to calculate the t-score:

• 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.

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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References

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

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