From Individual Responses to a Summary
In Mixed Practice: Units, Variables, and Types, you practiced identifying cases and classifying variables. Now we will summarize one categorical variable. A list of individual responses can be hard to scan; a frequency table organizes those responses by category and shows how often each category occurred.
A frequency is the count of responses in a category. A relative frequency is that category’s count divided by the total number of responses. Relative frequencies show the share of the responses in each category. They can be written as decimals, fractions, or percentages.
The method begins with the variable, not with the arithmetic. Use the variable-classification skills from Categorical Versus Quantitative Variables to confirm that the responses identify categories. Decide which response belongs in each category, then count. For the table to describe the full set of responses, the categories should be mutually exclusive (each response fits in only one category) and collectively exhaustive (every response fits somewhere).
A Reliable Way to Build the Table
When you receive raw responses, first read the question and identify the observational unit and categorical variable. As in Identifying the Observational Unit, be clear about what one response represents. Then make a list of the possible categories before counting. If a response could fit two categories, clarify the category definitions before building the table.
Write the category labels clearly. Use distinct, complete categories that match the recorded responses.
Scan the raw responses one at a time. Tally marks can help avoid losing your place; convert each set of marks into a count.
Add the category frequencies. The sum should equal the number of responses represented in the table. If it does not, check for a missed response, a duplicate count, or a response that was not assigned to a category.
Divide each category’s count by the same total number of responses. Keep unrounded values for checking, then round consistently for reporting.
A table may list just counts when the total is obvious and the goal is to show how many responses fall in each category. Relative frequencies are especially useful when the total is important to the interpretation, or when you want to compare groups with different numbers of responses. A relative frequency is not a new response or an additional count: it describes the portion of the total belonging to a category.
Worked Examples: Count and Convert
Worked Example: Summarizing 103 Snack Preferences
In a fictional school survey, 103 students each name one preferred after-school snack. The categorical variable is preferred snack, and the categories are fruit (F), yogurt (Y), crackers (C), and granola bar (G). Here are the responses in their recorded order. Each line contains 10 responses, except the last line, which contains three.
Raw responses:
F, Y, C, G, F, Y, C, G, F, C
Y, C, G, F, Y, C, G, F, Y, C
C, F, G, Y, C, F, G, Y, C, F
G, C, F, Y, G, C, F, Y, G, C
F, G, C, Y, F, G, C, Y, F, G
Y, F, C, G, Y, F, C, G, Y, F
C, Y, F, G, C, Y, F, G, C, Y
F, C, Y, G, F, C, Y, G, F, C
G, F, Y, C, G, F, Y, C, G, F
C, G, F, Y, C, G, F, Y, C, G
F, Y, G
Count the categories: Scanning the first ten lines gives 100 responses: 26 fruit, 23 yogurt, 27 crackers, and 24 granola bars. The last three responses are F, Y, and G, so the final counts are 27 fruit, 24 yogurt, 27 crackers, and 25 granola bars.
Check the frequency total: \(27+24+27+25=103\). This matches the number of responses, so the category counts account for all 103 students.
Calculate relative frequencies: Divide each count by 103. For fruit, \(27/103=0.262135\ldots\), which is 0.262 to three decimal places, or 26.2% to one decimal place. For yogurt, \(24/103=0.233009\ldots\), which is 0.233, or 23.3%. For crackers, \(27/103=0.262135\ldots\), which is 0.262, or 26.2%. For granola bars, \(25/103=0.242718\ldots\), which is 0.243, or 24.3%.
| Preferred snack | Frequency | Relative frequency | Percent |
|---|---|---|---|
| Fruit | 27 | 0.262 | 26.2% |
| Yogurt | 24 | 0.233 | 23.3% |
| Crackers | 27 | 0.262 | 26.2% |
| Granola bar | 25 | 0.243 | 24.3% |
| Total | 103 | 1.000 | 100.0% |
The rounded decimal relative frequencies sum to \(0.262+0.233+0.262+0.243=1.000\), and the percentages sum to 100.0%. The interpretation should stay tied to the surveyed group: 27 of the 103 responding students named fruit, which is about 26.2% of these responses. This table describes these 103 responses; by itself, it does not establish what all students prefer.
Worked Example: Counting Reusable-Bottle Choices
A fictional community event asks 40 attendees how they filled their drink: reusable bottle, disposable bottle, or drinking fountain. The responses are already tallied as 18 reusable bottle, 9 disposable bottle, and 13 drinking fountain. Make a frequency and relative-frequency table.
Check the count: \(18+9+13=40\), which matches the number of attendees. Calculate each relative frequency using 40 as the denominator. Reusable bottle: \(18/40=0.45\), or 45%. Disposable bottle: \(9/40=0.225\), or 22.5%. Drinking fountain: \(13/40=0.325\), or 32.5%.
| Drink source | Frequency | Relative frequency | Percent |
|---|---|---|---|
| Reusable bottle | 18 | 0.450 | 45.0% |
| Disposable bottle | 9 | 0.225 | 22.5% |
| Drinking fountain | 13 | 0.325 | 32.5% |
| Total | 40 | 1.000 | 100.0% |
The relative-frequency check is \(0.450+0.225+0.325=1.000\). In context, 45% of the 40 event attendees reported using a reusable bottle. It would be inaccurate to say that 45 attendees used one: 45% is a share, while 18 is the count.
Worked Example: Comparing Relative Frequencies for Two Groups
Two fictional library workshops ask participants which session time they prefer: morning, afternoon, or evening. Workshop A has 20 responses: 8 morning, 7 afternoon, and 5 evening. Workshop B has 40 responses: 12 morning, 16 afternoon, and 12 evening. Compare the relative frequencies rather than comparing counts alone.
Check each group: Workshop A’s counts sum to \(8+7+5=20\). Its relative frequencies are \(8/20=0.400\), \(7/20=0.350\), and \(5/20=0.250\). Workshop B’s counts sum to \(12+16+12=40\). Its relative frequencies are \(12/40=0.300\), \(16/40=0.400\), and \(12/40=0.300\).
| Preferred time | Workshop A frequency | Workshop A relative frequency | Workshop B frequency | Workshop B relative frequency |
|---|---|---|---|---|
| Morning | 8 | 0.400 (40%) | 12 | 0.300 (30%) |
| Afternoon | 7 | 0.350 (35%) | 16 | 0.400 (40%) |
| Evening | 5 | 0.250 (25%) | 12 | 0.300 (30%) |
| Total | 20 | 1.000 (100%) | 40 | 1.000 (100%) |
Workshop B has more afternoon responses by count, 16 compared with 7. But Workshop B also has twice as many responses overall. Relative frequencies put the groups on a common scale: 40% of Workshop B respondents preferred afternoon, compared with 35% of Workshop A respondents. These percentages summarize the workshop responses; they do not explain why the preferences differ.
Choose Clear Categories and Report Carefully
Frequency tables are only as useful as their category definitions. If a survey allows more than one answer but a table treats responses as though each person gave exactly one, the count total may not equal the number of people. State what the responses represent and use a denominator that matches the data being summarized. In the 103-response example, each student named one snack, so each response belongs to exactly one category and the total is 103.
Category order should make the table easy to read. For categories with a natural order, such as levels of agreement or ordered ratings, preserve that order. For categories without a natural order, a sensible consistent order—such as the order in which categories were listed in the question—works well. This organization does not change the frequencies or relative frequencies.
Common Mistakes and AP Exam Tips
- Using the wrong denominator. Divide by the total number of responses represented in the table, not by the category count or by a different group’s total.
- Stopping after counting. A frequency is a count; a relative frequency is a share. Label the columns so a reader can tell which is which.
- Leaving responses out or counting them twice. Check that the frequencies sum to the stated total. A mismatch is a signal to review the raw responses and category rules.
- Mixing percentages and decimals. A relative frequency of 0.262 is about 26.2%, not 0.262%. Multiply a decimal by 100 to convert it to a percent.
- Rounding too early. Keep the full fraction or calculator value until the final reported result. Small rounding differences are expected, so use a consistent number of decimal places.
- Interpreting a share as a count. “About 26.2% of the 103 respondents” describes a proportion; “27 respondents” gives the frequency.
- Comparing counts across different totals without context. Counts alone can be misleading when groups have different sizes. Compare relative frequencies when the goal is to compare the distribution of responses across those groups.
For a full-credit explanation, identify the category and the denominator in words. For example: “27 of the 103 responding students preferred fruit, so the relative frequency is \(27/103\), or about 0.262 (26.2%).” This connects the calculation to the context and makes clear what the reported percentage describes.
Check Your Understanding
Use the counts or raw responses in each question to build or interpret a frequency or relative-frequency summary.
- A fictional class has 30 responses about favorite study location: 12 library, 9 home, 6 classroom, and 3 another location. Find each relative frequency as a decimal and as a percent. Check both totals.
- A poll receives 50 responses: 15 choose option A, 20 choose option B, and 15 choose option C. What is the frequency and relative frequency for option B? Interpret the relative frequency in context.
- A second group has 100 responses, with 30 choosing option A, 45 option B, and 25 option C. Compare the relative frequency for option B with the poll in question 2. Why are the counts alone not the best comparison?
- For a question that asks each person to choose exactly one category, a table’s frequencies sum to 47 even though 50 people responded. Name two possible counting or category problems to check.
- A table reports relative frequencies of 0.375, 0.250, and 0.375. What is their sum, and what percentage of responses belongs to the second category?