From Cell Counts to Shares of the Whole
A two-way table organizes individuals according to the categories of two categorical variables. Its cell counts tell us how many individuals fall into each combination of categories. But counts alone can be hard to compare: a count of 12 might represent a small share of a large group or a large share of a small one. Dividing each cell count by the total number of individuals represented in the table expresses every combination as a share of the same whole.
That share is called a joint relative frequency. “Joint” refers to the combination of one category from each variable; “relative frequency” means the count is expressed relative to a specified total. In this tutorial, the denominator is always the grand total of the table. Thus, every cell answers the same kind of question: what proportion of all individuals in the table belong to this combination of categories?
The table must represent a clearly defined set of individuals. Each individual included in that set should contribute to exactly one cell: that person or object has one category for each of the two variables. The grand total is the number of individuals represented across all cells. If records are incomplete and cannot be assigned to a cell, specify which individuals the table includes; do not silently use a denominator that includes records absent from the cells.
Calculating a Joint Relative-Frequency Table
The calculation is the same for every cell. First identify the grand total, then divide each cell count by that total. To express the result as a percentage, multiply the proportion by 100 and attach the percent sign. Keep enough decimal places to make the reported percentages informative; rounded entries may not add to exactly 100%.
A joint relative-frequency table can use the same row and column categories as the count table, with a proportion or percentage in place of each count. It is often useful to present both tables, so the reader can see the observed counts as well as their shares of the total.
Worked Example: Two Categories for Each Variable
Suppose a hypothetical survey records whether 80 students brought lunch from home and whether they bought a drink at school. The count table is:
| Bought a drink | Did not buy a drink | |
|---|---|---|
| Brought lunch | 18 | 22 |
| Did not bring lunch | 12 | 28 |
The grand total is \(n=80\). For the first cell, the joint relative frequency is \(18/80\). The other cells use that same grand total:
These calculations give the following joint relative-frequency table.
| Bought a drink | Did not buy a drink | |
|---|---|---|
| Brought lunch | 0.225 (22.5%) | 0.275 (27.5%) |
| Did not bring lunch | 0.150 (15.0%) | 0.350 (35.0%) |
The first cell means that 22.5% of all 80 students in this table both brought lunch and bought a drink. The last cell means that 35.0% of all 80 students did neither. These are shares of the entire group represented by the table, not percentages calculated only among students in a particular row or column.
Why the Joint Relative Frequencies Sum to One
When the categories account for every individual in the table, the cells divide the full group into non-overlapping combinations. Each individual appears in one and only one cell. Therefore, adding all the cell counts recovers the grand total. Dividing each count by the grand total gives a set of shares that adds to one.
To justify this result, let \(n_{ij}\) denote the count in each cell and let \(n\) be the sum of all cell counts. Adding the joint relative frequencies means adding each cell count after division by \(n\). Since the counts sum to \(n\), the sum of the relative frequencies is \(n/n=1\).
This gives a useful check on a calculation. If the unrounded proportions add to substantially more or less than 1, look for an arithmetic error, an incorrect grand total, or a table that does not account for its individuals as intended. With rounded values, a small discrepancy is normal. For example, rounding several entries to two decimal places can make their displayed sum differ slightly from 1.
Worked Example: A Table with Three Categories
In a hypothetical group of 120 adults, a questionnaire records whether each person uses a fitness app (yes or no) and their preferred exercise setting (home, outdoors, or gym). The observed counts are shown below.
| Exercise setting | Uses an app | Does not use an app |
|---|---|---|
| Home | 14 | 26 |
| Outdoors | 18 | 22 |
| Gym | 12 | 28 |
The grand total is \(n=120\). Divide each count by 120. For example, the joint relative frequency for using an app and preferring to exercise at home is \(14/120=0.1167\), or \(11.67\%\). Applying the same calculation to every cell gives:
| Exercise setting | Uses an app | Does not use an app |
|---|---|---|
| Home | 14/120 = 0.1167 (11.67%) | 26/120 = 0.2167 (21.67%) |
| Outdoors | 18/120 = 0.1500 (15.00%) | 22/120 = 0.1833 (18.33%) |
| Gym | 12/120 = 0.1000 (10.00%) | 28/120 = 0.2333 (23.33%) |
The cell for outdoors and uses an app represents 15.00% of all 120 adults in the table. It does not mean that 15.00% of app users prefer outdoors, because that wording describes a proportion within the app-user group, not within the full group of 120. The denominator determines the question being answered.
As a check, the unrounded cell proportions sum to \(1\). The displayed percentages add to approximately \(100.00\%\); any tiny difference after rounding is due to displaying only four decimal places for proportions.
Reading and Reporting a Joint Relative Frequency
A careful interpretation names both categories in the cell and the full group used as the denominator. A useful template is: “Of all [individuals represented in the table], [percentage] were in category [A] for the first variable and category [B] for the second variable.” This wording makes clear that the cell describes a category combination and that its percentage refers to the whole table.
Joint relative frequencies are descriptive summaries of the records in the table. They do not, by themselves, establish why individuals fall into particular categories or show that one variable causes the other. If the individuals were selected in a way that supports representing a broader population, the table may also provide information about that population, but the arithmetic of a joint relative frequency alone does not guarantee that the sample is representative.
It is also important to distinguish the joint relative frequency from other possible proportions in the same table. The joint relative frequency uses the grand total as its denominator. A proportion restricted to a particular row or column would use a different denominator and answer a different question. For the calculation in this tutorial, do not replace the grand total with a row or column count.
Worked Example: Interpreting a Cell in Context
In a hypothetical class survey of 50 students, each student reports their usual way of getting to campus (walk, bicycle, or bus) and whether they live on campus. The count table is:
| Usual way to campus | Lives on campus | Lives off campus |
|---|---|---|
| Walk | 10 | 4 |
| Bicycle | 6 | 8 |
| Bus | 3 | 19 |
The grand total is \(n=50\). For the cell representing students who take the bus and live off campus:
The correct interpretation is: 38% of all 50 students in this hypothetical survey usually take the bus to campus and live off campus. It would be incorrect to say that 38% of off-campus students take the bus, because that statement would require dividing by the number of off-campus students, not by all 50 students.
For completeness, the other cells are \(10/50=0.20\), \(4/50=0.08\), \(6/50=0.12\), \(8/50=0.16\), and \(3/50=0.06\). Together with \(19/50=0.38\), these proportions sum to \(0.20+0.08+0.12+0.16+0.06+0.38=1.00\). Thus, 100% of the students in the table are accounted for by the category combinations.
Common Mistakes
- Using the wrong denominator. Dividing a cell count by its row total or column total does not give a joint relative frequency. Divide by the grand total of the table.
- Reporting a count as a percentage. A cell count of 18 is not 18% unless the grand total happens to make \(18/n=0.18\). Calculate the fraction before expressing it as a percentage.
- Leaving out one of the categories. A joint interpretation must name the combination, such as “brought lunch and bought a drink,” rather than naming just one variable’s category.
- Ignoring what the table includes. Interpret the percentage as a share of the individuals represented in the table, not automatically as a percentage of a wider population.
Check Your Understanding
Answer each question using the grand total as the denominator for joint relative frequencies.
- A two-way table has a grand total of 200 and a cell count of 30. Calculate the cell’s joint relative frequency as a proportion and as a percentage.
- In a table of 60 people, 9 are in the cell for “owns a pet” and “lives in an apartment.” State what \(9/60\) means in context.
- Why should the joint relative frequencies sum to 1 when every individual belongs to exactly one cell?
- A student divides a cell count by its row total and calls the result a joint relative frequency. What is wrong with that reasoning?
- A table’s displayed joint relative frequencies sum to 0.99 after rounding. Must there be an error? Explain.