2.5: Organizing Bivariate Data

Section 1: Scatterplots

Earlier we discussed the methods of summarizing and organizing the data obtained from observing one variable. Such data are called univariate data.

The mileages of a sample of 5 cars.

Car	Mileage (mi)
1	78524
2	12574
3	24914
4	65813
5	39824

When we are interested in studying the relationship between a pair of variables we must collect and organize the data on two variables at the same time. Such data itself is called bivariate data and there are several ways to organize such data depending on the types of the variables.

The ages and mileages of a sample of 5 cars.

Car	Age (years)	Mileage (mi)
1	7	78524
2	2	12574
3	1	24914
4	5	65813
5	3	39824

When both variables are quantitative, we can construct a scatter plot to organize the data. To construct a scatterplot, we use a horizontal axis for the observations of one variable and a vertical axis for the observations of the other. When picking which axis to use for each variable consider whether you suspect that one variable depends on the other. The independent variable will be on the x-axis and dependent will be on the y axis. Each pair of observations is then plotted as a point. The example here demonstrates the summary of the observed relation between the age and the mileage of cars.

Figure \(\PageIndex{1}\): The scatterplot summarizing the mileage and conditions of a sample of 5 cars. (Copyright; author via source)

In the scatter plot above, each point is associated with a car, for example, car 1 is the point whose x-coordinate is 7 and the y-coordinate is 78.5 thousand, car 2 is the point whose x-coordinate is 2 and the y-coordinate is 12.5 thousand, etc.

Here is another summary that demonstrates the observed relation between the third exam score and the final exam score. Notice that the third exam is on the x-axis as we may suspect that it may have an effect on the final exam scores.

Figure \(\PageIndex{2}\): The scatterplot summarizing the third exam scores and the final exam scores of a sample of 10 students. (Copyright; author via source)

When one of the variables is time and there is only one observation for each moment of time, we can construct the scatter plot:

CO2 emissions in the United States between 2003-2009

Year	CO2 emissions in the United States
2003	5,681,664
2004	5,790,761
2005	5,826,394
2006	5,737,615
2007	5,828,697
2008	5,656,839
2009	5,299,563

Figure \(\PageIndex{3}\): The scatterplot summarizing the CO2 emissions in the US between 2003 and 2009. (Copyright; author via source)

Try It Yourself! \(\PageIndex{1.1}\)

And then we can connect the points on the scatter plot with lines - such graph is called time series. The example here demonstrates the summary of the observed relation between the year and the CO₂ emissions for the United States:

Figure \(\PageIndex{4}\): The time series summarizing the CO2 emissions in the US between 2003 and 2009. (Copyright; author via source)

Section 2: Contingency Tables

When we are interested in studying the relationship between a pair of variables we must collect and organize the data on two variables at the same time. Such data itself is called bivariate data and there are several ways to organize such data depending on the types of the variables.

Univariate Data: The colors of a sample of 5 cars.

Car	Color
1	Light
2	Dark
3	Dark
4	Light
5	Dark

Bivariate Data: The colors and conditions of a sample of 5 cars.

Car	Color	Condition
1	Light	Like new
2	Dark	Used
3	Dark	Like New
4	Light	Used
5	Dark	Used

A table called contingency table can be used to organize bivariate data in which one or both variables are qualitative. To construct the table, we arrange the observed frequencies into rows and columns. The intersection of a row and a column of a contingency table is called a cell. Each cell shows the frequency of observations that fit the description in the corresponding row and column.

Contingency table summarizing the colors and conditions of a sample of 5 cars.

Color\Condition	Like new	Used
Light	1	1
Dark	1	2

Try It Yourself! \(\PageIndex{2.1}\)

Many times, it makes sense to add the column and row with the totals for each column and row and for the entire table.

Contingency table summarizing the colors and conditions of a sample of 5 cars with totals.

Color\Condition	Like new	Used	Total
Light	1	1	2
Dark	1	2	3
Total	2	3	5

Try It Yourself! \(\PageIndex{2.2}\)

Alternatively, this table can be referred to as a two-way frequency table. Why? Cover the second and third columns and what's left can be easily seen as the frequency table for the colors of cars! Cover the second and third rows instead and what’s left can be easily seen as the frequency table for whether car conditions! Also note that every contingency table can be easily broken down into two frequency tables, but the two frequency tables cannot be combined into a contingency table if the original data is lost!

Here is another example of a contingency table. For example, we can see that there are 280 people out of 755 that are cell phone users and had no speeding violation in the last year:

Contingency table with totals summarizing whether a driver is a cell phone user and whether they received a speeding violation in the last year.

	Speeding violations in the last year	No speeding violations in the last year	Total:
Cell phone user	25	280	305
Not a cell phone user	45	405	450
Total:	70	665	755

Try It Yourself! \(\PageIndex{2.3}\)