Degrees of Freedom
In short: Degrees of freedom tally the independent pieces of information left after a restriction, like a known average, takes its toll, and they decide which exact t or chi-square curve applies.
AP StatisticsStatisticsAP Statistics Unit 4
What is Degrees of Freedom?
Degrees of freedom (df) measure the independent information left in a sample after you have used some of it. Suppose five numbers must average exactly 12. You may choose any four of them, but the fifth is then forced, so only four values were free.
That is the reason a one-sample t procedure uses \(df = n – 1\): the mean came out of the same data and consumed one piece of freedom. A goodness-of-fit setting uses \(df = \text{categories} – 1\), and a grid of counts uses \((\text{rows} – 1)(\text{columns} – 1)\). More degrees of freedom bring a t curve closer to the normal curve.
How to spot it
- A table of t or chi-square critical values asks which row to read.
- A calculator output prints df beside a test statistic.
- The problem asks how many values can vary once the total or the mean is known.
- A two-way table or a list of categories is given and a chi-square test is planned.
The method
- Name the procedure, such as a one-sample t or a grid of counts, since each has its own degrees of freedom rule.
- Apply the matching rule: \(n – 1\), \((r – 1)(c – 1)\) or \(k – 1\).
- Count the restrictions in words: how many parameters were estimated from the same data?
- Use the df to pick the correct curve or table row for the critical value or p-value.
Degrees of Freedom: solved example
Problem
Three analysts each need df. Ana studies 12 bus commute times with a t procedure. Ben asks whether four yogurt flavors sell equally often. Cleo cross-tabulates 300 students by grade level (3 levels) and favorite club (4 clubs).
Solution
Ana: \(df = n – 1 = 12 – 1 =\) 11.
Ben, four flavors: \(df = 4 – 1 =\) 3.
Cleo, 3 rows and 4 columns: \(df = (3 – 1)(4 – 1) = 2 \times 3 =\) 6.
Try it
Question 1
Four of five test scores are 70, 85, 90 and 75, and the mean of all five is 80. What is the fifth score?
Show the answer
The total is \(5 \times 80 = 400\), and the four known scores add to 320, so the fifth is 80. Only four scores were free to vary.
Question 2
What is the df for a one-sample t test with \(n = 25\)?
Show the answer
\(25 – 1 = 24\).
Degrees of Freedom vs. Population
Sample size \(n\) counts the observations collected. Degrees of freedom count how many of them still carry independent information after estimates have been made, so df is usually smaller than \(n\). Neither number describes the population itself.
Keep going
Lessons
Worksheets and quizzes
- t Intervals and t Tests for a Mean – Worksheet
- t Intervals and t Tests for a Mean – Quiz
- Chi-Square Tests – Worksheet
- Chi-Square Tests – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics