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*Lead Author(s): ErikGregory*

The simple definition of degrees of freedom is "the number of items free to vary in the final calculation of a test statistic ¹."

Another convenient way to think of degrees of freedom is as "the minimum number of items we need to determine the rest of the items, given some already-determined parameter which is calculated using those items."

So how many variables are used in the final calculation of the variance? We can interpret the sample mean as a constant in this case since it is calculated using the same observations as the variance is. Suppose we had N-1 of the "x's" and knowledge of the variance. In this case then, we could use the already-determined information to find what the final "x" value is (by doing some simple algebra). Thus, there are N-1 degrees of freedom since knowing any fewer of the observations would make it impossible to calculate the sample variance.

NOTE: There is an underlying assumption that the variances of the populations are equal in this case.

So our hypotheses are:

And we calculate our test statistic as follows:

So we have assumed that our sample means are independent, and we know their values. In determining the number of degrees of freedom, we observe that to calculate the sample mean heights (and standard deviations) for New York City and Turlock there are N-1 and T-1 height measurements that are free to vary (degrees of freedom), respectively. So in total, we have (N-1)+(T-1) = N+T-2 measurements that are free to vary in our calculation of "t." So there are N+T-2 degrees of freedom.

Suppose we are doing a chi-square and need to determine what to put in the "expected" portion of our table. We already have observed values, and so we have the "Totals" entries filled in but need to figure out what to put in the "expected" entries.

Category A | Category B | Category C | Totals | |

Condition 1 | 33 | |||

Condition 2 | 33 | |||

Condition 3 | 34 | |||

Totals | 25 | 25 | 50 | 100 |

Category A | Category B | Category C | Totals | |

Condition 1 | 8 | 20 | 33 | |

Condition 2 | 8 | 33 | ||

Condition 3 | 6 | 34 | ||

Totals | 25 | 25 | 50 | 100 |

It may seem like we have little to work with at this point, but it turns out we have all of the information we need to solve the problem. For example, Category C/Condition 2 must be 50-(20+6)= 24, and Category A/Condition 3 must be 25-(8+8) = 9 Similarly, we can fill in all of the entries in the table- using this information only- as follows.

Category A | Category B | Category C | Totals | |

Condition 1 | 8 | 5 | 20 | 33 |

Condition 2 | 8 | 1 | 24 | 33 |

Condition 3 | 9 | 19 | 6 | 34 |

Totals | 25 | 25 | 50 | 100 |

In any table such as this, the degrees of freedom can be determined by simply using the formula:

Degrees of freedom = (R-1)*(C-1)

Where "R" is the number of rows representing different categories (not counting the "totals" row) we have to fill in, and "C" is the number of columns (not counting "totals") that we have to fill in. In our case this is (3-1)*(3-1) = 2*2 = 4, which is what we expect.

[¹] http://www.animatedsoftware.com/statglos/sgdegree.htm

I | Attachment | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|

png | Hypothesis.png | manage | 2.4 K | 28 Jul 2010 - 13:23 | ErikGregory | Hypotheses |

png | Variance.png | manage | 6.2 K | 28 Jul 2010 - 12:13 | ErikGregory | Variance formula |

png | tstat.png | manage | 20.0 K | 28 Jul 2010 - 13:42 | ErikGregory | |

png | tstat3.png | manage | 22.3 K | 28 Jul 2010 - 14:11 | ErikGregory | tstat3 |

Topic revision: r5 - 03 Oct 2010 - 20:42:05 - MaryBanach

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