This tutorial explains what t-test is, and the difference between `independent sample t-test`

and `paired sample t-test`

. It also explains what two-sample and one-sample t-test are.

## What is independent sample t-test?

`Indepdent sample t-test `

examines whether the means from `2 separate groups of people or objects`

are statistically significantly different. That is, we calculate two means from two groups of , and we can see that these two means are statistically different.

For instance, we want to compare the height of two groups of people. We can calculate the average height of both groups and get two mean numbers, such as 6 feet vs. 5 feet 6 inch. So, yes, we can see that these two means are different. However, without further tests, we do not know whether such a difference is statistically significant. In this case, a t-test can tell us whether these two means are really statistically different or just from some random errors.

Name | Height |
---|---|

Jack | 6 feet |

John | 6 feet 6 inch |

Tommy | 6 feet 6 inch |

Amy | 5 feet |

Eddie | 7 feet |

Mean of Group 1 | 6 feet |

Name | Height |
---|---|

Andy | 6 feet |

Luke | 6 feet |

Anderson | 5 feet |

Kim | 5 feet |

Cindy | 5 feet 6 inch |

Mean of Group 2 | 5 feet 6 inch |

## What is paried sample t-test?

`Paired sample t-test`

is used for the situation where `X`

and _{1}`X`

are about the same group of human respondents or non-human objects (e.g., cities, products). Typically, _{2}`X`

and _{1}`X`

are from two different time points. The following is the illustraction, which shows the heights of the same group people, but at two different time points. Then, paired sample t-test is going to tell us whether the means of these two time points are significantly different from one another._{2}

Name | Height of Time 1 | Height of Time 2 |
---|---|---|

Jack | 5 feet 6 inch | 6 feet |

John | 5 feet 6 inch | 6 feet 6 inch |

Tommy | 6 feet | 6 feet 6 inch |

Amy | 5 feet | 5 feet |

Eddie | 5 feet 6 inch | 6 feet |

Mean | 5 feet 6 inch | 6 feet |

## Difference between Independent and Paired Sample t-test

In particular, independent sample t-test is used for a situation where `X`

and _{1}`X`

are from two independent, different groups of respondents. For instance, if you want to understand whether women and men differ in their attitudes toward drinking coffee, in this case, women (_{2}`X`

) and men (_{1}`X`

) are the two levels of _{2}`X`

, and attitudes toward drinking coffee are the dependent measure (`Y`

).

In contrast, paired sample t-test is used for the situation where `X`

and _{1}`X`

are about the same group of respondents. For instance, you want to compare Exam 1 (_{2}`X`

) and Exam 2 (_{1}`X`

) in terms of studentsâ€™ grade performance. In this case, the students are the same group of people for both _{2}`X`

and _{1}`X`

, but just measured at two different time points. Thus, that is why some people also call paried sample t-test as one-sample t-test._{2}

## Two-Sample t-test vs. One Sample t-test

### Two-sample t-test

Independent sample t-test is also called two-sample t-test, because Independent sample t-test involves two separate, independent samples. That is, independent sample t-test and two-sample t-test are the same thing.

`Independent sample t-test = Two-sample t-test`

### One-sample t-test

One-sample t-test is different from independent sample t-test or paired-sample t-test. In particular, one-sample t-test is to determine whether an unknown population mean is different from a specific value. For instance, you measure the height of a group of 100 students and want to infer all the 3000 students’ height in the same school. Then, you get the mean of these 100 students is 6 ft, and then you want to compare whether the mean if 6 ft is different from 6 ft 1 in. In this case, it is a one-sample t-test.

The following table summarizes all these different t-tests, with alternative names as well as examples.

Other names | Examples | |
---|---|---|

Independent sample t-test | Two-sample t-test, Unpaired samples t-test | Test how men (`Mean` _{1}) and women (`Mean` _{2}) differ in their attitudes towards drinking coffee |

Paired sample t-test | Matched sample t-test | Compare Exam 1 (`Mean` _{1}) and Exam 2 (`Mean` _{2}) for the same group of students |

One-sample t-test | Single sample t-test | Compare a group of student’s average height () to a fixed, constant number (`Mean` `A Number` ) |