# Dependent t-test

**Student's t-test for paired samples (dependent)**

Comparison of two means from dependent samples.

**Examples for dependence:**

Repeated measure, parallelized samples, virtual dependence (e.g. twins)

**Requirements:**

Dependent variable is at least interval scaled and normal distributed.

**Hypothesis:**

Nondirectional:

H0: H1:

Directional (example):

H0: H1:

The dependence of two measures from the two samples is gathered from the differences of the measurement pairs.

Mean and standard deviation of these differences are defined as follows:

Whereas n is the number of pairs.

The decision between the hypothesis is made according to the sample distribution of .

If both populations are normal distributed, the sample distribution is normal, too, with mean

and standard deviation

If is estimated from the sample, we get the following t-distributed equation:

with degree of freedom df = n-1.

**Example of a paired samples t-test**

A psychologist is interested in his new learning program. 15 Subjects learn a list of 50 words. Learning performance is measured using a recall test. After the first test all subjects are instructed how to use the learning program and then learn a second list of 50 words. Learning performance is again measured with the recall test. In the following table the number of correct remembered words are listed for both tests.

Subject | Score 1 | Score 2 | ||

1 | 24 |
26 |
-2 |
0.36 |

2 | 17 |
24 |
-7 |
19.36 |

3 |
32 |
31 |
1 |
12.96 |

4 |
14 |
17 |
-3 |
0.16 |

5 |
16 |
17 |
-1 |
2.56 |

6 |
22 |
25 |
-3 |
0.16 |

7 |
26 |
25 |
1 |
12.96 |

8 |
19 |
24 |
-5 |
5.76 |

9 |
19 |
22 |
-3 |
0.16 |

10 |
22 |
23 |
-1 |
2.56 |

11 |
21 |
26 |
-5 |
5.76 |

12 |
25 |
28 |
-3 |
0.16 |

13 |
16 |
19 |
-3 |
0.16 |

14 |
24 |
23 |
1 |
12.96 |

15 |
18 |
22 |
-4 |
1.96 |

Σ | -37 |
78 |

n=15

The observed t-value is smaller than the critical t-value (1%, one-tailed). The learning program has a positive effect. Subjects remember more words after the training.

BrightStat output of paired samples t-test example

This is a fictitious example.