You have worked with one-way tables (even though you may not have called them by that name). A one-way table is simply the data from a bar graph put into table form. In a one-way table, you are only working with one categorical variable.

Two-Way Frequency Table: (displays "counts")

You can probably guess that a two-way frequency table will deal with two variables (referred to as bivariate data). In so doing, these tables examine the relationships between the two categorical variables. Two-way frequency tables are especially important because they are often used to analyze survey results. Two-way frequency tables are also called contingency tables.

bullet The Basics of a Two-Way Frequency Table

Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data. The categories are labeled at the top and the left side of the table, with the frequency (count) information appearing in the four (or more) interior cells of the table. The "totals" of each row appear at the right, and the "totals" of each column appear at the bottom.
Note: the
"sum of the row totals" equals the "sum of the column totals" (the 240 seen in the lower right corner). This value (240) is also the sum of all of the counts from the interior cells.

A survey asked, "If you could have a new vehicle, would you want a sport utility vehicle or a sports car?


Let's take a look at the vocabulary used to identify cell locations in two-way frequency tables. Entries in the body of the table (the blue cells where the initial counts appear) are called joint frequencies. The cells which contain the sum (the orange "Totals" cells) of the initial counts by row and by column are called marginal frequencies. Note that the lower right corner cell (the total of all the counts) is not labeled as a marginal frequency.


hint gal
Take a look at the Sports Car column. This table shows 45 women chose Sports Car, while 39 men chose Sports Car. Would this information answer the question, "In this survey, do more women or men prefer Sports Car? Not really! Read on to discover why this would be misleading information if interpreted in this manner.



Two-Way Relative Frequency Table: (displays "percentages")

When a two-way table displays percentages or ratios (called relative frequencies), instead of just frequency counts, the table is referred to as a two-way relative frequency table. These two-way tables can show relative frequencies for the whole table, for rows, or for columns. Notice that the relative frequencies may be displayed as a ratio, a decimal (to nearest hundredth), or percent (to nearest percent).

bullet Relative Frequency for Whole Table:
If the two-way relative frequency is for the whole table, each entry in the table is divided by the total count (found in the lower right corner). The ratio of "1", or 100%, occurs only in the cell in the lower right corner.
Each of the main body cells (blue) is telling you the percentage of people surveyed that gave that response (based upon the total number of people responding).

hint gal
Under the Sports Car column, are we again seeing more women choosing a sports car than men (19% of the women and 16% of the men)??
Not really! Read on to discover why this can still be misleading information even when interpreted as a percentage.


So what's up with more women, than men, choosing a sports car?

It is certainly possible that women may love sporty cars just as much, or more, than men. While this may be possible, it is not the real situation regarding this survey data. The misleading information is that the frequency table and the relative frequency table shown above do not take into consideration how many women, and how many men, responded to this survey. There were only 60 men responding, while there were 180 women. There were three times more women responding to this survey, which presents misleading results when based upon the entire population.

To avoid such problems when comparing the categorical variables in a two-way frequency table, we need to exam the table by separate categories (rows or columns). When a relative frequency is determined based upon a row or column, it is called a "conditional" relative frequency. To obtain a conditional relative frequency, divide a joint frequency (count inside the table) by a marginal frequency total (outer edge) that represents the condition being investigated. You may also see this term stated as row conditional relative frequency or column conditional relative frequency.
Basically, we are going to look at the women and men separately, based upon how many women were surveyed, and how many men were surveyed.

bullet Conditional Relative Frequency for Rows:
If the two-way relative frequency is for rows, the entries in each row of the table are divided by the total for that row (on the right hand side). The ratio of "1", or 100%, occurs in all right hand "total" cells.

If we want to answer the question, "In this survey, do more men, or more women, prefer a sports car?", we need to set up a row conditional relative frequency. The listings of men and women are row headings, and we want to examine these categories separately to determine the answer to our question. By choosing a row method, we are comparing men and women in relation to car type.

hint gal
Do you see how this changes our previous interpretation of the data?
Using a row conditional relative frequency, we can see that 65% of the 60 men responding chose Sports Car, while only 25% of the 180 women responding chose Sports Car. This method takes into account the count of men and women separately, giving us a
more realistic view of the relationship between the variables.



bullet Conditional Relative Frequency for Columns:
If the two-way relative frequency is for columns, the entries in each column of the table are divided by the total for that column (at the bottom). The ratio of "1", or 100%, occurs in all of the "total" cells at the bottom.

hint gal
So, what the heck is this method showing us? Are we back to more women than men choosing the Sports Car? The problem is that a column approach does not address the issue of which car men and women prefer. In the column method, we are comparing an SUV to a sports car in relation to gender. An appropriate question would be, "Were SUVs or sports cars chosen more often by females?

What we have seen, by examining all of these tables, is that different tables answer different types of questions about the data. If you want to look for a relationship between the categorical variables, you will need to prepare a conditional relative frequency table. You will then need to decide if a "row" method or a "column" method will address the situation you wish to examine.


What Two-Way Tables Tell Us:

A variety of questions can be answered by examining a two-way frequency table.
Let's look at some possibilities:

Two-way frequency table


How many people responded to the survey? 240

How many males responded to the survey? 60

How many people chose an SUV? 156

How many females chose a sports car? 45

How many males chose an SUV? 21

Two-way relative frequency table
(whole table)

What percentage of the survey takers was female? 75%

What is the relative frequency of males choosing a sports car?

Was there a higher percentage of males or females
choosing an SUV?
higher percentage of females

bullet Associations Based on Conditional Relative Frequency:


An "association" exists between two categorical variables if the row (or column) conditional relative frequencies are different for the rows (or columns) of the table. The bigger the differences in the conditional relative frequencies, the stronger the association between the variables. If the conditional relative frequencies are nearly equal for all categories, there may be no association between the variables. Such variables are said to be independent.

In our Sports Car and SUV example (above), the row conditonal relative frequencies showed a good degree of difference. Based upon that information, if we knew the gender of a survey respondent, we could make a good prediction as to whether he/she chose a sports car or an SUV. The statistical information is strong enough to support an "association" between gender and choice of vehicle. Now, this does not mean that there is always an association between gender and choice of vehicle. It just means that such an association is evident in the data from this survey.

hint gal
Variables can be associated in many ways and to different degrees. Sometimes the best way to tell whether two variables are associated is to ask yourself whether they are not associated. Think backward.
In a two-way frequency table, if the relative frequencies for one variable are the same (or close) for all categories of another variable, there is no (or little) association.


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