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# 'Student's' t Test (For Paired Samples)

Use this test to compare two small sets of quantitative data when data in each sample set are related in a special way.

### Criteria

• The number of points in each data set must be the same, and they must be organized in pairs, in which there is a definite relationship between each pair of data points
• If the data were taken as random samples, you must use the independent test even if the number of data points in each set is the same
• Even if data are related in pairs, sometimes the paired t is still inappropriate
• Here's a simple rule to determine if the paired t must not be used - if a given data point in group one could be paired with any data point in group two, you cannot use a paired t test

### Examples

The paired t test is generally used when measurements are taken from the same subject before and after some manipulation such as injection of a drug. For example, you can use a paired t test to determine the significance of a difference in blood pressure before and after administration of an experimental pressor substance. You can also use a paired t test to compare samples that are subjected to different conditions, provided the samples in each pair are identical otherwise. For example, you might test the effectiveness of a water additive in reducing bacterial numbers by sampling water from different sources and comparing bacterial counts in the treated versus untreated water sample. Each different water source would give a different pair of data points.

The value of the paired t test is best demonstrated in an example. Suppose patient 1 responds to a drug with a 5 mm Hg rise in mean blood pressure from 100 to 105. Patient 2 has a 30 mm Hg rise, from 90 to 120. Likewise for several other subjects. The response to the drug varied widely, but all patients had one thing in common - there was always a rise in blood pressure. Some of that experimental error is avoided by the paired t test, which likely will pick up a significant difference. The independent test, which would be improperly applied in this case, would not be able to reject the null hypothesis.

Be certain that use of the paired t test is valid before applying it to real data. An applied statistics course or supervision of a qualified mentor may provide the experience you need.

Some spreadsheet programs include the paired t test as a built-in option. Even without a built-in option, is is so easy to set up a spreadsheet to do a paired t test that it may not be worth the expense and effort to buy and learn a dedicated statistics software program, unless more complicated statistics are needed.