Key takeaway: Degrees of freedom count how many values in a calculation are free to vary after the constraints are applied, and they determine which t or chi-square curve to use.
Grade 12 – AP StatisticsAP Statistics Unit 4Statistics and Data
Definition of Degrees of Freedom
The degrees of freedom (df) of a statistic equal the amount of independent information that went into it. Estimating a mean from \(n\) values uses up one piece of information, because once the mean is fixed, the last value is forced.
Fewer degrees of freedom mean less information and more uncertainty, so t curves and chi-square curves change shape with df. Each procedure has its own rule for counting them.
Notation
Common formulas:
- one-sample t procedures: \(df=n-1\);
- chi-square goodness of fit with \(k\) categories: \(df=k-1\);
- chi-square for a table with \(r\) rows and \(c\) columns: \(df=(r-1)(c-1)\).
The count is always the number of values minus the number of independent constraints.
Key facts
- Every estimated parameter used in a calculation takes away one degree of freedom.
- As df grows, the t distribution gets closer to the standard normal curve.
- The df value tells you which row of a t or chi-square table to read.
- Technology often reports fractional df for two-sample t procedures.
Degrees of Freedom example
Five values have a mean of 12. The first four are 9, 15, 11 and 14. What must the fifth value be, and how many degrees of freedom does the standard deviation have? Then count the degrees of freedom for a 4-row by 3-column table.
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The total must be \(5\times 12=60\). The first four add to \(9+15+11+14=49\).
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The fifth value is forced: \(60-49=11\). Only four values were free, so \(df=5-1=4\).
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For the table, \(df=(4-1)(3-1)=3\cdot 2=6\).
The fifth value is 11; the sample has 4 degrees of freedom, and the 4 by 3 table has 6.
Common mistake
Watch out
A student uses the sample size, 16, as the degrees of freedom when looking up a one-sample t critical value.
Why it is wrong
One value of information was spent on estimating the mean, so the correct count is \(n-1=15\). Using 16 reads the wrong row of the table and gives a slightly smaller critical value than the correct one, making intervals too narrow.
Related terms and lessons
- AP Statistics · Inference for MeansLessonWorksheetQuiz
- AP Statistics · Chi-Square Tests for Homogeneity and IndependenceLessonWorksheetQuiz
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