Two-Way Frequency Tables
Standard
S.ID.5 Summarize categorical data for two categoeries in two-way frequency tables. Interpret relative frequencies in the context of the data... Recognize possible associations and trends in the data.
Goals for this section
Students should be able to read and interpret a two-way frequency table and should be able to create one.
Two-way Frequency Table
These types of tables are great to compare categorical data. Here's an example:
Joint Frequencies
- These are the entries that are in the middle i.e. 16, 4, 18, 12.
- For example, the entry with 12 is a joint frequency. It corresponds to the "Girls" row and the "Do Not Own a Bicycle" column. So there are 12 girls who do not own a bicycle.
Marginal Frequency
- These are the entries that are on the margin, or the total row/column.
- For example, the 20 entry corresponds to adding up all the numbers its row (16 + 4). There are 20 boys.
- For example, the 50 entry corresponds to the total of all boys and girls.
Making a Two-Way Frequency Table
Example
A survey asked 150 freshmen and sophomores whether they preferred Math or English. Of the 83 freshmen, 15 preferred Math. Of the 67 sophomores, 25 preferred English. Make a two-way frequency table for the data.
Step 1: Identify Variables
We have two variables of two types each:
- Grade level
- Freshmen
- Sophomores
- Subject
- Math
- English
Set these as your table headers. Where you put them is up to you so long as they're consistent!
In this step, we will also include the marginal frequency (total row/column).
Step 2: Input the info you have
Our given information are:
- 150 freshmen and sophomores
- 83 freshmen
- Of the freshmen, 15 preferred Math
- 67 sophomores
- Of the sophomores, 25 preferred English
Input these into the table.
Step 3: Fill in the rest of the entries
If there's a total of 83 freshmen and 15 of them like Math, then the rest must like English. 83 - 15 = 68.
Similarly for Sophomores.
The totals will be adding the Math together and the English together.
- Math. 15 + 42 = 57
- Sophomoers. 68 + 25 = 93
Notice that 57 + 93 = 150.
Relative Frequencies
Joint Relative Frequency
The ratio of the frequency of the category to the total.
Recall that joint frequencies are in the middle. These are the entries with 16, 4, 18, and 12. You take those entries and divide by the overall total, which is 50.
- Example: (16 / 50) = 0.32 = 32%
- This means that of the people surveyed, 32% were male and owned a bicycle.
Marginal Relative Frequency
The ratio of the marginal frequency to the total.
Recall that marginal frequencies are the outside entries that are on the "margin", so entries 20, 30, 34, 16, and 50. You take those numbers and divide by the overall total, which is 50.
Two-Way Relative Frequency Table
We can create a table of joint/marginal relative frequencies.
Notice that we're just dividing by the overall total.
Conditional Relative Frequency
The ratio of the joint frequency and the related marginal frequency.
Conditional relative frequencies are dependent on what the question is asking for. You will not be using the overall total.
Conditional problems may generally say, "What is the probability that a person owns a bicycle GIVEN that they're a boy?" The word "given" is your keyword!
Example
What is the conditional relative frequency that a person owns a bicycle given that they're a boy?
Answer: Given that they're a boy, 80% own a bicycle.
Step 1: Look at the "second part" of the statement to determine related marginal frequency.
"Second part" meaning "given that they're a boy". These means that we're looking at just the boys row, so how many boys are there in total? 20. This will be our related marginal frequency.
Step 2: Find the joint frequency.
So since we're just looking at the boys row, how many boys own a bike? 16. This will be our joint frequency.
Step 3: Create your ratio.
joint frequency / related marginal frequency = 16/20 = 0.8
Example
What is the conditional relative frequency that a person is a girl given that they do not own a bicycle?
Answer: 75% of those who do not own a bicycle are girls.
Step 1: Look at the "second part" of the statement to determine related marginal frequency.
We're looking at just the column labeled "do not own a bicycle". The total or related marginal frequency is 16.
Step 2: Find the joint frequency.
So how many who do not own a bicycle is a girl? 12. This will be our joint frequency.
Step 3: Create your ratio.
joint frequency / related marginal frequency = 12/16 = 0.75
Tip!
It might be confusing that we have these terms:
- Joint frequency
- Marginal frequency
- Joint relative frequency
- Marginal relative frequency
Basically, if you see the word "relative", then think "ratio" or "fraction"! If you don't see it, then we're just thinking of the quantity or amount of.