I. What is variability?

II. What is between groups variability?

III. What causes between groups variability?

A. Random error

B. Treatment effects?

IV. What is within groups variability?

V. What causes within groups variability?

Random error

VI. If the treatment has an effect, then

between groups variance (treatment effects + random error)

should be bigger than

within groups variance (random error)

VI. Two things you can quickly determine by looking at an ANOVA summary table.

A. How many levels of treatment there were (because treatment levels -1= treatmentdf)

B. How many participants there were (because total number of participants - 1 = total

df)

Start of ANOVA Summary Table:

Source | df |

Between groups (Treatment) | 3 |

Within groups (Error) | 26 |

Total | 29 |

How many groups were used?

How many participants?

**Example 1: Results from a three-group experiment**

GROUP 1 | GROUP 2 | GROUP 3 |
---|---|---|

5 | 5 | 5 |

5 | 5 | 5 |

5 | 5 | 5 |

Mean= | Mean= | Mean= |

Source | Sum of Squares | df | Mean Square |
---|---|---|---|

Between
groups Treatment | _ | 2 | _ |

Within groups S/A (Error) | _ | 6 | _ |

Total | _ | 8 | _ |

**Example 2: Results from a three-group experiment**

GROUP 1 | GROUP 2 | GROUP 3 |
---|---|---|

3 | 6 | 6 |

3 | 6 | 6 |

3 | 6 | 6 |

Mean=3 | Mean=6 | Mean=6 |

Note: In this case, *SS* Treatment =
**18**

ANOVA Summary Table

Source | Sum of Squares | df | Mean Square |
---|---|---|---|

Between
groups Treatment | _ | _ | _ |

Within groups S/A (Error) | _ | _ | _ |

Total | _ | _ | _ |

**Example 3: The results from a three-group experiment**

GROUP 1 | GROUP 2 | GROUP 3 |
---|---|---|

2 | 5 | 5 |

3 | 6 | 6 |

4 | 7 | 7 |

Mean=3 | Mean=6 | Mean=6 |

Note: To compute the within groups sum of squares, start with group 1. Find the mean (3). Subtract each of the scores in that group from that mean, square those differences, and then sum them. Do this for each group.

Thus, in this example:

GROUP 1 | GROUP 2 | GROUP 3 |
---|---|---|

2-3=1 -1^{2}=1 |
5-6=-1 -1^{2}=1 |
5-6=-1 -1^{2}=1 |

3-3=0 0^{2}=0 |
6-6=0 0^{2}=0 |
6-6=0 0^{2}=0 |

4-3=1 1^{2}=1 |
7-6=1 1^{2}=1 |
7-6=1 1^{2}=1 |

Mean=3; SSG1=2 | Mean=6; SSG2=2 | Mean=6; SSG3=2 |

SS within = 2 + 2 + 2 = **6**

ANOVA Summary Table based on Example 3

Source | Sum of Squares | df | Mean Square | F |
---|---|---|---|---|

Between
groups Treatment | _ | _ | _ | _ |

Within groups S/A (Error) | _ | _ | _ | |

Total | _ | _ | _ |

**Example 4: The results from a three-group experiment**

GROUP 1 | GROUP 2 | GROUP 3 |
---|---|---|

2 | 5 | 3 |

3 | 6 | 6 |

4 | 7 | 9 |

Mean=3 | Mean=6 | Mean=6 |

ANOVA Summary Table

Source | Sum of Squares | df | Mean Square | F |
---|---|---|---|---|

Between
groups Treatment | _ | _ | _ | 2.46 |

Within groups S/A (Error) | _ | _ | _ | |

Total | _ | _ | _ |