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Percentages and Sums in Two-way Frequency Tables

Maths Applications (Year 12) - Bivariate Data Analysis

Bhakti Sharma

When looking at bivariate data, often times we are interested in looking at the relationship between two variables. We can do this by plotting the information at hand into a table, using percentages or proportions.


Example

Let’s suppose that the number of male and female members of Gym A and Gym B, are shown on the table below:

If we wanted to compare the number of males and females, or how many members there were in each gym, we would find the various sums within this table. The table below depicts this:

We can make a couple of observations from the table above, for example:

  • Gym B has more memberships than Gym A (37 more memberships)

  • Gyms A and B (combined) have more female members than male members


Percentages and Proportions

Other observations might be more apparent if this data was presented as percentages or proportions of the total (207). In this case, we would divide each number by the total to find the proportion (e.g. 46/207 = 0.22). To find the percentage values, multiply this by 100 (0.22 x 100 = 22%). The tables below show this:

The following observations can be more easily made from the tables above:

  • 33% of all memberships are female memberships from Gym B

  • 48% of all gym members are males


Rows and Columns

In addition to total percentages, we can also consider row percentages:


Or column percentages:


Now we can make observations such as:

  • 54% of the male gym members from the two gyms are from Gym A

  • 56% of the female gym memberships (collectively) come from Gym B

  • 54% of Gym A memberships are male memberships compared to 44% from Gym B


Using the data shown above, we can create a stacked 100% column graph (as shown below). We can see from this diagram, that as we move from Gym A to Gym B, there is a difference in the proportion of male gym memberships to female gym memberships. The proportion of male gym memberships in Gym A is higher than those in Gym B. This suggests to us that there is an association between the two categorical variables of Gym and Gender of the member.


As shown in the examples above, two-way diagrams can be converted to show percentages and proportions instead of just raw data. Having the data presented in such a way can help us make additional observations, such as associations between the variables.

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