Two-way tables

So far, we have looked at tables of frequency counts from experiments with a singe row, such as the number of rolls of each number on a fair or biased dice.

Number on dice	1	2	3	4	5	6
Frequency of rolls	20	17	16	15	20	12

Tables like this are called one-way tables because the table has information on one category only. In this case the number on a rolled dice.

If we wanted to display frequencies where we had two categories then we would need a two-way table.

Let’s look at a situation where a two-way table is produced.

Rosie and Jim are playing darts. They take it in turns to throw three darts each and then add the scores of the three darts. If they get 60 or more then that counts for one point. If they get below 60 then it doesn’t count for any points. They record the results of their throws in a two-way frequency table.

By the end of the game Rosie and Jim haven’t had the same number of attempts to score 60. (Jim was rather slow and kept going to the toilet.)

	under 60	60 or more
Rosie	23	36
Jim	16	22

Estimate the probability that Rosie scores 60 or more in three darts.

This is really just an extension of a one-way table

	under 60	60 or more
Rosie	23	36

The relative frequency for Rosie is

For Jim, what is the relative frequency that he scores 60 or more ?

	under 60	60 or more
Jim	16	22

The relative frequency for Jim is

From the information that we have, Rosie is more likely to score 60 or more.

Does the relative frequencies for Rosie or Jim give a more accurate statement of their chance of scoring 60 or more with three darts?

Rosie made 23 + 36 = 59 attempts, compared to Jim’s 16 + 22 = 28.

As Rosie has made more attempts than Jim, her relative frequency is a more accurate representation of her chances of scoring 60 or more than Jim’s.

If Rosie and Jim teamed up to play together in a tournament then what is the best estimate of the probability of them scoring 60 or more if you don’t know which of them is throwing?

	under 60	60 or more
Rosie	23	36
Jim	16	22

We can transform the table above into another one-way table

	under 60	60 or more
Rosie and Jim	23 + 16 = 38	36 + 22 = 58

The relative frequency for team Rosie and Jim to score 60 or more is

This makes sense because 0.604 is between Rosie’s relative frequency of 0.610 and Jim’s of 0.579.