6.03 Two-way tables

Two-way tables allow us to display and examine the relationship between two sets of categorical data. The categories are labelled at the top and the left side of the table, and the frequency of the different combinations of characteristics appear in the interior of the table. Often the totals of each row and column also appear.

The following two-way table was made by surveying $100$100 people. They were asked two questions, if they are right or left-handed and if they are male or female, the results are as follows:

	Right-handed	Left-handed	Total
Male	43	9	52
Female	44	4	48
Total:	87	13	100

Note: The sum of the row totals equals the sum of the column totals and is the total number surveyed.

It's called a two-way table because we can read information from it in two directions. Here we have information about the two categories "gender" and "handedness". If read across each row, we can tell how many of each gender surveyed are right or left-handed. If we read down each column, we can tell how many of the right or left-handed people surveyed were male and how many were female. 

Where a particular row and column overlap, these are how many people satisfy both categories. For example, there were $9$9 left-handed males surveyed.

Worked example

The following are the statistics of the passengers and crew who sailed on the RMS Titanic on its fateful maiden voyage in 1912. 

	First Class	Second Class	Third Class	Crew	Total
Survived	$202$202	$118$118	$178$178	$212$212	$710$710
Died	$123$123	$167$167	$\editable{}$	$696$696	$1514$1514
Total:	$325$325	$285$285	$\editable{}$	$908$908	$\editable{}$

(a) What is the estimated total number of passengers and crew on-board the ship?

Think: Adding the totals in the final column will give the total number of people on-board the ship.

Do: 

Total passengers and crew	$=$=	$710+1514$710+1514
	$=$=	$2224$2224

(b) Find the missing values in the "Third class" column.

Think: We now know the total number of passengers and crew which will also be the sum of the values in the last row. We can use this to find the missing value of "the total number of third class passengers". Once we have this we can use it to find the "number of third class passengers who died".

Do: 

Total third class passengers	$=$=	$2224-908-285-325$2224−908−285−325
	$=$=	$706$706

Third class passengers who died	$=$=	$\text{Total}-\text{survivors}$Total−survivors
	$=$=	$706-178$706−178
	$=$=	$528$528

(c) What percentage of first class passengers survived?

Think: We need the fraction of first class survivors out of total number of first class passengers written as a percentage.

Do:

Percentage first class passengers that survived	$=$=	$\frac{\text{first class survivors}}{\text{total number first class passengers}}\times100%$first class survivorstotal number first class passengers​×100%
	$=$=	$\frac{202}{325}\times100%$202325​×100%
	$\approx$≈	$62.2%$62.2%

(d) What percentage of total passengers and crew survived?

Think: We need the fraction of total survivors out of total number of passengers and crew written as a percentage.

Do:

Percentage passengers and crew that survived	$=$=	$\frac{\text{total survivors}}{\text{total number passengers and crew}}\times100%$total survivorstotal number passengers and crew​×100%
	$=$=	$\frac{710}{2224}\times100%$7102224​×100%
	$\approx$≈	$31.9%$31.9%

Reflect: This type of table leads to examining if relationships exist between the categories. For example, for the given statistics above can you use supporting calculations to show that the survival rate is dependent on the class of the passenger?

 

Practice questions

QUESTION 1

QUESTION 2

Question 3

QUESTION 4