Chi Square

c2 Test of Independence

Nuances:

See why we square and add to summarize the differences between 0-E. If we didn't square, they would sum to zero.
You divide by E to make everything relative. It's really like a Z score.
The chi-square statistic (c2) is evaluated for significance with a degrees of freedom equal to the number of categories minus 1(k-1, if k = number of categories).
You look up in in the Chi-square table. If you exceed the tabled value, you have significance. In our case, the value is 3.84 at the .05 level. So the first example is not significant. What about the second with more people?
The more categories you have, the larger the Chi-square you need for significance. You don't always have 2 groups. If you did ethnic identity in our example, you could have 5 or 6. You can have more than 2 categories.
What about Goodness of Fit? Now that we explained it, this particular Chi-square is sometimes called the the {and extra the}Goodness of Fit test. We tested whether our "Observed" frequencies "fit" against our "Expected "

Wait - Here's Another Kind of
Chi -Square:

The Test of Independence

The Big Idea:

Does the breakdown of proportions of frequencies for one category difference significantly from that of another category?

What this means is:

You got Men - You got Women.
They are now grown-up married yuppies.
They came into some money on their job.
They want to buy a new car.
They can buy a sports car or a SUV. {an SUV instead of a SUV?}

Do the proportions of men Vs women who want a SUV vs. a sports car differ?

{OR} Is car choice independent of gender?

Which means: DO men and women:
want the same kind of cars.

How are you going to figure this out? You will:

Ask men and women what kind of car they want.
Set up a contingency table that lays out the observed choices by gender and car type (Gender X Car).
Figure out the preference for cars if gender didn't matter.
Apply these preferences to men and women separately to see if the observed frequencies for men and women for each car type differ from what would be expected if car type didn't count.

Let's Look at Some Pictures & Do a Visual Chi-square

Picture 1:

20 Men (50%) like SUV

20 Women (50%) like SUVs

20 Men like (50%) Sports Cars

20 Women (50%) like Sports Cars

A simple idea! Men and women don't differ on car preference. You can see it.

Fifty percent of people like SUVs. Fifty percent like sports cars.

Gender doesn't matter. If we computed Chi-square (discussed later) it would not be significant.

Official Chi-square language follows.

Car preference is Independent of Gender.

It's the Test of Independence, remember!