A Two-Way Table of Counts

The following table displays a partial listing of the Titanic data. Each row corresponds to the values of the four categorical variables for a passenger.

CLASS	AGE	SEX	SURVIVED
First	Adult	male	Yes
First	Adult	male	Yes
First	Adult	male	Yes
First	Child	female	No

Crew	Adult	female	Yes
Crew	Adult	female	No
Crew	Adult	female	No
Crew	Adult	female	No

The Titanic accident always will be remembered for the large number of fatalities. Let's first focus on the "Survived" variable in the dataset and construct a frequency table. The table below shows the counts of people who survived and who did not survive; this table also shows the corresponding percentages. From this table, we see that about 2/3 of the passengers did not survive this trip.

	Survived		Total
	no	yes
Count	1490	711	2201
%	67.7	32.3	100

Let's consider the question posed earlier -- were passengers of certain economic classes more or less likely to survive the accident?

To answer this question, we classify the passengers in the below table by the survival status and the economic status (crew, or first, second, or third class). We see, for example, there were 122 people in first class who did not survive, and 203 people in the first class who survived.

		Survived		Total
		no	yes
Class	Crew	673	212	885
	First	122	203	325
	Second	167	118	285
	Third	528	178	706
	Total	1490	711	2201

Although this table is useful in understanding the numbers of people who died and survived in different classes, it is difficult to compare the distributions since there were many more non-survivors in the dataset. For comparison purposes, it is better to compute the percentage of survivors and non-survivors for each class.

We can get these percentages from the table by dividing each basic count by the corresponding row total. For example, the percentage of first-class people who survived is 203 / 325 = 62%. If we compute these row percentages for all of the table entries,

		Survived		Total
		No	yes
Class	crew	673/885	212/885	885
	first	122/325	203/325	325
	second	167/285	118/285	285
	third	528/706	178/706	706
	TOTAL	1490/2201	711/2201	2201

we obtain the row percentages.

		Survived		TOTAL
		no	yes
Class	Crew	76%	24%	100%
	First	38%	62%	100%
	Second	59%	41%	100%
	Third	75%	25%	100%
	Total	68%	32%	100%

We see from the row percentages that there is a strong association between class and survival. Overall, 32% of the passengers survived. However, a majority (62%) of first-class passengers survived compared to a survival rate of 41% for second-class passengers and a survival rate of 25% for third-class passengers and crew. To display this association graphically we can use a set of stacked bar charts. Each bar corresponds to a passenger class, and the bar is divided by the survival status. This graph clearly shows the high survival rate of the first-class passengers.

In the above analysis, we divided each table count by the row total and found the row percentages. This told us the percentage of each class group that survived and did not survive. But we can look at the association in the table a different way. To compare the class of the non-survivors with the class of the survivors, we can construct side-by-side bar charts of the two sets of column frequencies, as shown below.

This graph is informative, but it is difficult to compare the two sets of bars, since there are different numbers of passengers in the two classes. It is easier to compare the two sets of frequencies if they are converted to percentages. Again we start with our basic count table

		Survived		Total
		no	yes
Class	Crew	673	212	885
	First	122	203	325
	second	167	118	285
	Third	528	178	706
	Total	1490	711	2201

and divide the counts by the column totals

		Survived		TOTAL
		No	yes
CLASS	crew	673/1490	212/711	885/2201
	first	122/1490	203/711	325/2201
	second	167/1490	118/711	285/2201
	third	528/1490	178/711	706/2201
	TOTAL	1490	711	2201

to obtain the column percentages

		Survived		TOTAL
		no	yes
CLASS	Crew	45%	30%	40%
	First	8%	29%	15%
	Second	11%	17%	13%
	Third	35%	25%	32%
	TOTAL	100%	100%	100%

Let's interpret these column percentages. Of the 2201 passengers in the Titanic, 40%, 15%, 13%, and 32% were respectively crew, first-class, second-class, and third-class passengers. However, there was a different composition if you consider only the 711 people who survived the accident. Of this group, there were about equal percentages of crew and first-class passengers, 25% third class, and 17% that were second class. But more importantly, if the passengers who died, almost half of them were crew and only 8% were first-class.

We can display these two sets of column percentages in several ways. We can use side-by-side bar charts, where the percentage (instead of a frequency) is plotted.

Alternately, we can use two stacked bar charts, where each bar chart contains the percentages of class memberships for a survival group.

One can study the association in a two-way table by computing row percentages or column percentages. Which way to go depends on the particular application and ease of interpretation.

In this example, is it easier to talk about

·the survival rate of different classes of passengers

or is it easier to discuss

the class status of the group of passengers who survived and the status of the passengers who did not survive?

Here the author thinks it makes more sense to talk about how survival depends on the economic status of the passenger, so I would prefer computing row percentages and using a corresponding graph (such as a set of segmented bar charts) to communicate the difference in row percentages across class.

Practice: Looking at the Age and Gender of the Titanic Passengers

It is also interesting to relate the survival of the Titanic passengers with respect to their age and their gender.

1. The below table classifies the passengers with respect to their survival (no or yes) against their age (child or adult). For each age, compute the percentage of survivors and percentages of nonsurvivors and place your results in the second table. Were children more or less likely to survive than adults? Explain.

		Survived?
		No	Yes
Age	Child	52	57
Age	Adult	1438	654

		Survived?
		No	Yes
Age	Child
Age	Adult

2. Using two stacked bar charts, show how the survival of the passenger depends on age.

3. In a similar fashion, compute relevant percentages to see if the survival status of the passengers depended on their gender. Place your percentages in the second table and describe your conclusions.

		Survived?
		No	Yes
Gender	Female	126	344
Gender	Male	1364	367

		Survived?
		No	Yes
Gender	Female
Gender	Male

4. Use stacked bar charts to show how the survival depends on gender.

5. We have seen that the survival status of the Titanic passengers was dependent on their economic status and also dependent on their gender. That raises the question: was there any relationship between economic status and gender? By using the below table and computing relevant row or column percentages, answer this question.

		Class
		Crew	First	Second	Third
Gender	Female	23	145	106	196
Gender	Male	862	180	179	540

6. Does it make sense that children and women were more likely to survive this disaster? Explain.