Degrees of Freedom

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Degrees of Freedom in a Chi-Square Test

Degrees of freedom (df) represent the number of independent pieces of information in your data that can vary while still satisfying certain constraints. It’s an important value used to calculate the P-value in a Chi-Square test.

Formulas for Degrees of Freedom:

1. Goodness-of-Fit Test (One Variable):

   • df = (number of categories - 1)

2. Contingency Table (Two Variables):

   • df = (rows - 1) × (columns - 1)

Example 1: Categories with Candies

Imagine you have a bag of candies with two categories: Red and Blue. You want to test if the candies are evenly distributed.

1. Number of Categories:

   • Red and Blue = 2.

2. Degrees of Freedom:

   • df = (number of categories - 1)
   • df = 2 - 1 = 1.

Insight:
If there are only two categories, knowing the count of one category determines the other. That’s why there’s only 1 degree of freedom.

Example 2: Contingency Table

You survey kids and adults about their preferences for Red or Blue candies. Here’s the data:

1. Rows:

   • Kids, Adults = 2.

2. Columns:

   • Red, Blue = 2.

3. Degrees of Freedom:

   • df = (rows - 1) × (columns - 1)
   • df = (2 - 1) × (2 - 1)
   • df = 1 × 1 = 1.

Insight:
In contingency tables, degrees of freedom reflect how many cells can vary independently while keeping the row and column totals fixed.

Key Takeaways:

• For a single variable (e.g., Red vs. Blue candies):
  • df = (number of categories - 1).
• For a contingency table (e.g., candy preferences by age group):
  • df = (rows - 1) × (columns - 1).

Degrees of freedom are crucial for adjusting your test to the structure of your data and ensuring valid results.

• P Value is Luck Meeter (Large is Luck, Small is Not Chance)