Chi Square Statistics
Test method. Use the chi-square test for independence to determine whether there is a significant relationship between two categorical variables. A chi-square statistic is one way to show a relationship between two categorical variables. Tip: The Chi-square statistic can only be used on numbers. This is what is tested by the chi squared (χ²) test (pronounced with a hard ch as in "sky"). . The following recommendations may be regarded as a sound guide. Half the total of the four values is then subtracted from that the difference to.
The sampling method is simple random sampling. The variables under study are each categorical.
If sample data are displayed in a contingency tablethe expected frequency count for each cell of the table is at least 5. This approach consists of four steps: The null hypothesis states that knowing the level of Variable A does not help you predict the level of Variable B.
That is, the variables are independent. Variable A and Variable B are independent. Variable A and Variable B are not independent. The alternative hypothesis is that knowing the level of Variable A can help you predict the level of Variable B. Support for the alternative hypothesis suggests that the variables are related; but the relationship is not necessarily causal, in the sense that one variable "causes" the other. Formulate an Analysis Plan The analysis plan describes how to use sample data to accept or reject the null hypothesis.
However, once you have determined the probability that the two variables are related using the Chi-square testyou can use other methods to explore their interaction in more detail.
For a fairly simple way of discussing the relationship between variables, I recommend the odds ratio.
Some further considerations are necessary when selecting or organizing your data to run a Chi-square test. The variables you consider must be mutually exclusive; participation in one category should not entail or allow participation in another. In other words, the data from all of your cells should add up to the total count, and no item should be counted twice.
You should also never exclude some part of your data set. If your study examined males and females registered as Republican, Democrat, and Independent, then excluding one category from the grid might conceal critical data about the distribution of your data. It is also important that you have enough data to perform a viable Chi-square test.
If the estimated data in any given cell is below 5, then there is not enough data to perform a Chi-square test. In a case like this, you should research some other techniques for smaller data sets: There are also tests written specifically for smaller data sets, like the Fisher Exact Test.
Degrees of Freedom A broader description of this topic can be found here. So plus 14 minus 20 squared over 20 plus 34 minus 30 squared over 30 plus-- I'll continue over here-- 45 minus 40 squared over 40 plus 57 minus 60 squared over 60, and then finally, plus 20 minus 30 squared over I just took the observed minus the expected squared over the expected.
I took the sum of it, and this is what gives us our chi-square statistic. Now let's just calculate what this number is going to be. So this is going to be equal to-- I'll do it over here so you don't run out of space.
So we'll do this a new color. We'll do it in orange. This is going to be equal to 30 minus 20 is 10 squared, which is divided by 20, which is 5. I might not be able to do all of them in my head like this.
Plus, actually, let me just write it this way just so you can see what I'm doing. This right here is over 20 plus 14 minus 20 is negative 6 squared is positive So plus 36 over Plus 34 minus 30 is 4, squared is So plus 16 over Plus 45 minus 40 is 5 squared is So plus 25 over Plus the difference here is 3 squared is 9, so it's 9 over Plus we have a difference of 10 squared is plus over And this is equal to-- and I'll just get the calculator out for this-- this is equal to, we have divided by 20 plus 36 divided by 20 plus 16 divided by 30 plus 25 divided by 40 plus 9 divided by 60 plus divided by 30 gives us So let me write that down.
Chi-Square Test for Independence
So this right here is going to be This is my chi-square statistic, or we could call it a big capital X squared. Sometimes you'll have it written as a chi-square, but this statistic is going to have approximately a chi-square distribution. Anyway, with that said, let's figure out, if we assume that it has roughly a chi-square distribution, what is the probability of getting a result this extreme or at least this extreme, I guess is another way of thinking about it.
So let's do it that way. Let's figure out the critical chi-square value.
Pearson's chi square test (goodness of fit) (video) | Khan Academy
And if this is more extreme than that, then we will reject our null hypothesis. So let's figure out our critical chi-square values. And actually the other thing we have to figure out is the degrees of freedom. The degrees of freedom, we're taking one, two, three, four, five, six sums, so you might be tempted to say the degrees of freedom are six. But one thing to realize is that if you had all of this information over here, you could actually figure out this last piece of information, so you actually have five degrees of freedom.
When you have just kind of n data points like this, and you're measuring kind of the observed versus expected, your degrees of freedom are going to be n minus 1, because you could figure out that nth data point just based on everything else that you have, all of the other information.
So our degrees of freedom here are going to be 5.
It's n minus 1.