Expected Counts

With a goodness-of-fit the expected counts are often simply the total sample size divided by the number of categories. For example if there were five categories, and 200 respondents, we would simply expect 200/5 = 40 respondents in each category. With a test of independence this is a little less obvious.

Let's take a look at the following example in Table 1.

Table 1.
*Example data*
Brand	Red	Blue	Total
Nike	..	..	100
Adidas	..	..	100
Total	100	100	200

If the brand has no influence on the color (and the other way around), we would expect each cell to be the same, since the totals are the same as well, i.e. 50 in each cell:

Table 2.
*Example data*
Brand	Red	Blue	Total
Nike	50	50	100
Adidas	50	50	100
Total	100	100	200

But what if we had twice as many people going for Red...

Table 3.
*Example data*
Brand	Red	Blue	Total
Nike	..	..	75
Adidas	..	..	75
Total	100	50	150

Note that in table 3 we have twice as many people who opted for red, while still an equal amount for Nike and Adidas. We would then expect (if the two variables are independent):

Table 4.
*Example data*
Brand	Red	Blue	Total
Nike	50	25	75
Adidas	50	25	75
Total	100	50	150

Note that the ratio of Red:Blue is the same in each row, including the total row, and the same goes for Nike:Adidas in the columns. The tricky part is when also the column totals are not evenly split, i.e. when the total of Nike and Adidas were also not the same.

Table 5.
*Example data*
Brand	Red	Blue	Total
Nike	...	...	50
Adidas	...	...	100
Total	50	100	150

So we have a chance of a respondent having chosen 'Red' of 50/150. The chance of having chosen 'Nike' is also 50/150. If the two chances are independent the chance for having chosen 'Red' and 'Nike' is simply those two chances multiplied with each other: 50/150 × 50/150. We simply multiply this chance with the total number of respondents to obtain how many people we'd expect to choose Nike and Red based on the given totals, i.e. 50/150 × 50/150 × 150 = 16.67.

We can repeat the steps from the previous paragraph for each cell. For Nike-Blue we get: 50/150 × 100/150 × 150 = 33.33. For Adidas-Red we get 100/150 x 50/150 × 150 = 33.33, and for Adidas-Blue we get 100/150 × 100/150 × 150 = 66.67.

Table 6.
*Example data*
Brand	Red	Blue	Total
Nike	16.67	33.33	50
Adidas	33.33	66.67	100
Total	75	75	150

These are the expected counts. We would expect these counts if the color has no influence on the brand, and the brand no influence on the color (i.e. the two are independent, there is no association).

If you look at the last calculation we did:

\(\frac{\text{Row Total}}{\text{Grand Total}}\times\frac{\text{Column Total}}{\text{Grand Total}}\times{\text{Grand Total}}\)

We can make this a little easier by writing:

\(\frac{\text{Row Total}\times\text{Column Total}}{\text{Grand Total}}\)

Yes, we could have just used the totals and subtract an already calculated cell value, but this might not work if you have a table larger than two by two.

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