Frequency Distribution Tables
A frequency distribution table is the simplest way to organise data: list each distinct value (or class), and write how many times it occurs. This compact summary is the basis for every graph and statistic that follows. This lesson teaches both the ungrouped form (one frequency per value) and the grouped form (one frequency per interval).
Definitions
A frequency is the number of times a particular value or class occurs in a data set.
An ungrouped frequency distribution lists each distinct value separately with its frequency.
A grouped frequency distribution groups values into classes (intervals) and lists the frequency of each class.
The total frequency is the sum of all frequencies , equal to the total number of observations.
Ungrouped frequency table
Best for discrete data with few distinct values.
Example. The number of children per family in a sample of 20 families: 1, 2, 2, 1, 3, 0, 2, 1, 2, 4, 1, 2, 1, 0, 3, 2, 1, 1, 2, 0.
Frequency table:
| Number of children | Frequency |
|---|---|
| 0 | 3 |
| 1 | 7 |
| 2 | 7 |
| 3 | 2 |
| 4 | 1 |
Total: 20. (Always verify by summing the frequencies.)
Grouped frequency table
Best for continuous data or discrete data with many distinct values. Group into intervals (classes) and count.
Example. Marks of 30 students: 12, 15, 22, 23, 25, 28, 31, 33, 35, 38, 40, 41, 43, 44, 45, 47, 50, 52, 55, 58, 60, 61, 62, 66, 70, 72, 75, 78, 82, 88.
Classes of width 10 starting at 10:
| Class | Frequency |
|---|---|
| 10-20 | 2 |
| 20-30 | 4 |
| 30-40 | 5 |
| 40-50 | 6 |
| 50-60 | 4 |
| 60-70 | 4 |
| 70-80 | 3 |
| 80-90 | 2 |
Total: 30. ✓
Cumulative frequency
The cumulative frequency at a class is the total of all frequencies up to and including that class. Useful for finding percentiles, medians, and quartiles.
For the table above:
| Class | Frequency | Cumulative frequency |
|---|---|---|
| 10-20 | 2 | 2 |
| 20-30 | 4 | 6 |
| 30-40 | 5 | 11 |
| 40-50 | 6 | 17 |
| 50-60 | 4 | 21 |
| 60-70 | 4 | 25 |
| 70-80 | 3 | 28 |
| 80-90 | 2 | 30 |
The cumulative frequency of the last class always equals the total frequency.
Worked examples
Example 1. The number of goals scored by a team in 12 matches: 2, 1, 0, 3, 1, 2, 0, 1, 2, 1, 3, 0. Make a frequency distribution.
| Goals | Frequency |
|---|---|
| 0 | 3 |
| 1 | 4 |
| 2 | 3 |
| 3 | 2 |
Total: 12.
Example 2. Heights of 20 plants (cm): 23, 28, 31, 33, 35, 36, 38, 40, 42, 45, 47, 48, 50, 52, 55, 56, 58, 60, 62, 65. Group into classes of width 10.
| Class | Frequency |
|---|---|
| 20-30 | 2 |
| 30-40 | 5 |
| 40-50 | 5 |
| 50-60 | 5 |
| 60-70 | 3 |
Total: 20.
Example 3. A frequency distribution has classes 0-10, 10-20, 20-30, 30-40 with frequencies 3, 7, 5, 5. Find the cumulative frequencies.
3, 10, 15, 20.
Example 4. From the table in Example 3, what is the total number of observations?
Total = (or read the final cumulative frequency).
Example 5. In a frequency distribution, the modal class has frequency 12 and the next class has frequency 9. Could the modal class change if we recount one observation?
The modal class is the class with the highest frequency. If we miscount and reduce the frequency of the modal class to 10 or fewer, then another class might become modal. So yes, careful counting matters.
Try it yourself
- The number of TV sets in 15 households: 1, 1, 2, 0, 1, 3, 1, 2, 1, 0, 2, 1, 1, 4, 2. Make an ungrouped frequency table.
- Weights of 25 students (kg): 32, 35, 38, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 64, 66, 68, 70, 71, 73, 74, 76, 78, 80, 85, 90. Group into classes of width 10 starting at 30.
- A frequency distribution has cumulative frequencies 5, 12, 20, 28, 30. Find the individual frequencies.
- Define cumulative frequency.
- Why might one prefer a grouped frequency distribution over an ungrouped one?
- Make a frequency table for the daily temperature data: 28, 30, 31, 32, 33, 29, 30, 31, 32, 28 (10 days). Use ungrouped form.
- In a class of 40 students, marks distributed as: 0-20: 5, 20-40: 10, 40-60: 15, 60-80: 7, 80-100: 3. Find the cumulative frequencies.
- State which form (grouped or ungrouped) is better for: (a) shoe sizes, (b) heights in cm.
- From a grouped frequency table with class width 10 and frequencies 3, 5, 8, 9, 6, 4 in classes 0-10, 10-20, ..., 50-60, find the modal class.
- A grouped distribution has frequencies 2, 5, 8, 7, 3 with classes 10-20, 20-30, 30-40, 40-50, 50-60. Find the total and the cumulative frequencies.
Pitfalls / Insight
- Always check the total. Sum of frequencies should equal the number of observations.
- Be consistent with class boundaries. Use either inclusive or exclusive throughout.
- Cumulative frequencies are running totals. The last one equals the total number of observations.
Insight. A frequency distribution is half the analysis. Once organised, the data tells a story by itself: where the values cluster, how spread out they are, whether there is a clear peak. The next lessons turn this story into pictures (graphs) and summaries (mean, median, mode).