The chi–square independence test is a procedure for testing if two categorical variables are related in some population.

Here is handwritten example : https://github.com/arcarchit/datastories/blob/master/notes/chi2.pdf

Chi square distribution

• Chi square distribution
  • Squaring samples from standard normal distribution [0]
  • Distribution changes with degrees of freedom
  • When DoF = 1 it is more concentrated around 0
• It is distribution is sum of squares
  • When dice is biased sum of squares will be higher. Hence more significant.
  • When it is fair it will be closed to zero. Difference is with expected value.

Chi Square Test for Equality of Proportions

• H0 : Distribution of some variable is same in all population
• Example of testing coin fairness
  • https://github.com/arcarchit/datastories/blob/master/notes/chi2.pdf
• Multiple choice questions (A, B, C, D)
  • H0 = Equal probability of correct choices (P(A) = P(B) = P(C) = P(D) = 0.25)
• Degrees of Freedom is 3 here

Chi square vs T test

• When to use which one
• T-test is used to compare mean of two distributions
• Chi square is used to check whether observation gathered of categorical data meets the assumption

Chi Square for goodness of fit testing

• Chi Square Goodness of fit
  • Restaurant example
  • H0 = Percentage given by customer is correct
• We calculate expected for each cell and calculate chi^2

Chi Square for relationship testing

• H0 : Variables are independent of each other
• It helps testing if two categorical variables are related
• Calculate Chi square statistics by summing all cells and check against degree’s of freedom
• Examples
  • Hypothesis testing :
    • H0 = Herbs1, Herb2, placebo are same
    • H0 = Herbs do nothing
    • We can’t say herb does nothing
      • We are working on accumulated data here
      • Whereas ANOVA is about variance
  • Homogeneity testing  :
    • H0 = Left and Right handed people have same preference for arts, science
    • H0 = Preference of arts/science is independent of natural hand left/right
    • H0 = Variables are independent
    • Filling up table
      • P(STEM | right) = P(STEM)
      • x / 60 = 40/100 => x = 40 * 60 / 100 = 24
      • We can also say that value of cell is product of marginals divide by total
    • Degrees of freedom = (r-1)*(c-1)  = 2 * 1 = 2

References

[0] : https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests#chi-square-goodness-of-fit-tests

[1] : https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests#chi-square-goodness-of-fit-tests

[2] : https://biology.stackexchange.com/questions/13486/deciding-between-chi-square-and-t-test

[3] : https://fhssrsc.byu.edu/SitePages/ANOVA,%20t-tests,%20Regression,%20and%20Chi%20Square.aspx