Reading a Two-Way Table by Columns
A two-way table organizes counts for two categorical variables. The table can be read in more than one direction: a question about the makeup of a particular column calls for a column conditional distribution. For example, a table might record both a person’s travel mode and whether the person arrived on time. If we focus on people who travelled by car, we can ask what proportion of that column arrived on time and what proportion did not.
The key is to describe the group represented by the column, then express each row category as a share of that group. The resulting proportions describe the column’s composition. They do not describe what fraction of the entire sample is in each cell.
The denominator is the total for the column being described. Each cell in that column gets divided by the same column total. A proportion can be reported as a decimal, such as 0.75, or as a percentage, such as 75%. Both express the same share.
Worked Calculation
Worked Example: Arrival Status Within Travel Mode
In this hypothetical table, 200 commuters are classified by travel mode and arrival status. Find the column conditional distribution of arrival status for each travel mode.
| Arrival status | Walk | Transit | Car | Total |
|---|---|---|---|---|
| On time | 48 | 56 | 51 | 155 |
| Late | 12 | 24 | 9 | 45 |
| Total | 60 | 80 | 60 | 200 |
For walkers, the column total is 60. The proportion who arrived on time is \(48/60=0.800\), or 80.0%; the proportion who were late is \(12/60=0.200\), or 20.0%. Together these make 1.000, or 100%.
For transit users, the column total is 80. The on-time proportion is \(56/80=0.700\), or 70.0%, and the late proportion is \(24/80=0.300\), or 30.0%. For people travelling by car, the column total is 60: \(51/60=0.850\), or 85.0%, arrived on time, and \(9/60=0.150\), or 15.0%, were late.
Thus each column describes the arrival-status composition of one travel-mode group. For instance, 70.0% of the transit column arrived on time. This is not the statement that 70.0% of all 200 commuters were both transit users and on time; that joint proportion would use 200 as its denominator.
Why the Column Proportions Sum to One
Every observation in a nonempty column belongs to exactly one of the row categories, assuming the categories are defined to cover the observations in the table. Consequently, the cell counts in that column add to its column total. Dividing that equality by the column total gives the sum of its conditional proportions.
This is both a property of the calculation and a useful check. If a column’s proportions do not add to approximately 1 (or 100%), check the cell counts, the column total, and any rounding. Small discrepancies can occur when percentages are rounded; the unrounded proportions still sum to 1.
The same calculation can be understood as a conditional relative frequency. Among the observations known to be in column category \(j\), count how many are also in row category \(i\), then divide by the number known to be in category \(j\). The cell count represents both category memberships, while the column total represents the group on which the description is conditioned.
Worked Example: Preferred Study Location Within School Type
Worked Example: Study Location Within School Type
A hypothetical questionnaire records the preferred study location for students at two types of schools. The counts are shown below. Find the column conditional distribution of preferred location within each school type.
| Preferred location | Public school | Private school | Total |
|---|---|---|---|
| Library | 40 | 30 | 70 |
| Home | 30 | 45 | 75 |
| Café | 30 | 25 | 55 |
| Total | 100 | 100 | 200 |
The public-school column has 100 students. Its location proportions are \(40/100=0.400\) for the library, \(30/100=0.300\) for home, and \(30/100=0.300\) for a café. As percentages, this column is 40.0% library, 30.0% home, and 30.0% café. The check is \(0.400+0.300+0.300=1.000\).
The private-school column also has 100 students. Its proportions are \(30/100=0.300\) for the library, \(45/100=0.450\) for home, and \(25/100=0.250\) for a café. In context, 45.0% of the students in the private-school column preferred studying at home.
Each statement is restricted to the named column group. The calculation does not say what percentage of all students who preferred home attended private school; that question would condition on the home row instead.
A Reliable Calculation Process
When asked for a column conditional distribution, use the following sequence. The first step is to identify which column category the requested description is about; the other calculations follow from that choice.
State the category whose composition is being described, such as “among students in the private-school column.”
Add or read the counts down that column. This is the denominator for every row category in the column.
Calculate one proportion for every row category in the selected column.
Confirm that the proportions sum to approximately 1, then state each result as a share of the named column group.
A column total of zero is a special case. If no observations fall in a column category, its conditional proportions are undefined: division by zero is not possible, and there is no observed group whose composition can be summarized. Do not report zero proportions for that column.
Worked Example: Reminder Status Within Follow-Up Outcome
Worked Example: Reminder Status Within Follow-Up Outcome
In a hypothetical set of 100 clinic appointments, a table records whether a reminder was sent and whether the appointment was completed. Calculate the column conditional distribution of reminder status for completed and missed appointments.
| Reminder status | Completed | Missed | Total |
|---|---|---|---|
| Sent | 54 | 12 | 66 |
| Not sent | 18 | 16 | 34 |
| Total | 72 | 28 | 100 |
For completed appointments, the column total is 72. The proportion with a reminder sent is \(54/72=0.750\), or 75.0%. The proportion with no reminder sent is \(18/72=0.250\), or 25.0%. These values sum to 1.000.
For missed appointments, the column total is 28. The proportion with a reminder sent is \(12/28\approx0.4286\), or 42.9%. The proportion with no reminder sent is \(16/28\approx0.5714\), or 57.1%. The rounded percentages add to 100.0%.
The correct interpretation is that 75.0% of the completed appointments in this table had a reminder sent, while 42.9% of the missed appointments had a reminder sent. These are descriptions of the recorded groups, not evidence that reminders caused an appointment to be completed or missed.
Common Mistakes and Why the Direction Matters
The denominator controls what a proportion describes. In a column conditional distribution, it is the column total, so the interpretation begins “among observations in this column category.” A proportion based on the grand total instead describes the cell’s share of the whole table. A proportion based on a row total describes the composition within that row. Those quantities answer different questions even though they use the same cell count in the numerator.
- Using the grand total: Dividing a cell count by the total number of observations gives a joint relative frequency, not a column conditional proportion.
- Dividing by a row total: This describes the row composition rather than the composition of the selected column.
- Reporting counts as percentages: A cell count of 54 is not 54% unless the relevant denominator makes it so. In the completed column above, the percentage is \(54/72\times100\%=75.0\%\).
- Leaving out the group in the interpretation: Say “75.0% of completed appointments had a reminder sent,” not merely “75.0% had a reminder.” Naming the group makes the conditioning clear.
- Claiming a cause from a conditional distribution: A table describes how recorded categories occur together. The proportions alone do not establish that one category caused another.
Check Your Understanding
Use the tables and ideas in this tutorial to answer the following questions.
- A column contains counts 18, 27, and 15. Find the column total and the three conditional proportions.
- In the travel-mode example, what denominator is used to calculate the proportion of late transit users? Interpret that proportion in context.
- Why should the conditional proportions in a nonempty column add to 1 before rounding?
- A cell count is 24, its column total is 80, and the grand total is 200. Calculate the column conditional proportion and the joint relative frequency. What does each describe?
- What can be concluded about a column conditional distribution when its column total is zero?