5 Frequency Distribution
A frequency distribution is a table that shows how many observations, the frequency, fall into each class interval. It is the single most common way raw data gets summarized before any further analysis, and it is the foundation for histograms, measures of central tendency, and measures of dispersion covered later in this book. This chapter builds a complete frequency distribution from raw data, extends it with cumulative and relative frequencies, and introduces the standard charts used to visualize it.
5.1 Constructing a Frequency Distribution Table
Building a frequency distribution from raw data follows directly from the previous chapter: once the class intervals are decided, every observation is sorted, traditionally with tally marks, into the class it belongs to, and the tally marks in each class are counted to give the frequency for that class.
Example
The marks out of 100 scored by 30 students in a statistics test are:
45, 62, 71, 38, 55, 80, 91, 67, 58, 49, 73, 84, 52, 66, 95, 41, 59, 77, 63, 88, 46, 70, 54, 68, 82, 39, 61, 75, 90, 57
Using the class intervals 30–40, 40–50, 50–60, 60–70, 70–80, 80–90, 90–100 (exclusive method), tallying each observation gives:
| Class Interval | Tally | Frequency |
|---|---|---|
| 30–40 | || | 2 |
| 40–50 | |||| | 4 |
| 50–60 | |||| | | 6 |
| 60–70 | |||| | | 6 |
| 70–80 | |||| | 5 |
| 80–90 | ||| | 4 |
| 90–100 | ||| | 3 |
| Total | 30 |
5.2 Cumulative Frequency
Cumulative frequency shows the running total of frequencies up to a given class, and comes in two forms.
- Less-Than Cumulative Frequency: The total number of observations with a value less than the upper boundary of each class, built by adding each class’s frequency to the running total of all classes before it.
- More-Than Cumulative Frequency: The total number of observations with a value greater than or equal to the lower boundary of each class, built by working backward from the total.
Example
Extending the marks frequency table with both cumulative columns:
| Class Interval | Frequency | Less-Than Cumulative Frequency | More-Than Cumulative Frequency |
|---|---|---|---|
| 30–40 | 2 | 2 | 30 |
| 40–50 | 4 | 6 | 28 |
| 50–60 | 6 | 12 | 24 |
| 60–70 | 6 | 18 | 18 |
| 70–80 | 5 | 23 | 12 |
| 80–90 | 4 | 27 | 7 |
| 90–100 | 3 | 30 | 3 |
Reading the less-than column: 12 students scored less than 60. Reading the more-than column: 18 students scored 60 or more.
5.3 Relative Frequency and Percentage Frequency
Absolute frequencies are hard to compare across datasets of different sizes. Relative frequency and percentage frequency solve this by expressing each class’s frequency as a share of the total.
\[\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Number of Observations}}\]
\[\text{Percentage Frequency} = \text{Relative Frequency} \times 100\]
Relative and percentage frequencies are what make it possible to compare a distribution of 30 students’ marks with a distribution of 3,000 students’ marks on equal footing, and they are exactly what most dashboards and reports show instead of raw counts.
5.4 Graphical Representation of Frequency Distributions
A frequency table is precise, but a chart communicates its shape instantly. Three charts are used most often with grouped frequency data.
- Histogram: Adjacent bars, one per class, with height proportional to frequency and no gaps between bars, since the underlying variable is continuous. This is the most common way to visualize a frequency distribution’s shape.
- Frequency Polygon: A line graph connecting the class marks (midpoints) plotted against their frequencies, useful for comparing two or more distributions on the same chart.
- Ogive: A cumulative frequency curve, plotted from either the less-than or more-than cumulative frequency table, used to read off the approximate median and other percentiles graphically.
Choose the class width for a histogram the same way you chose it for the frequency table itself, using Sturges’ Rule as a starting guideline and adjusting for readability. A histogram with too few bars smooths away real patterns, such as a second peak in the data, while one with too many bars looks noisy and jagged.
When class widths are unequal, plotting raw frequency as bar height is misleading, because a wider class will appear to have “more” data simply due to its width, not because it is genuinely more common. In that situation, plot frequency density, frequency divided by class width, instead of raw frequency, so that bar area, not just height, represents the count fairly.
Recap
A frequency distribution takes a long list of raw values, sorts them into class intervals, and counts how many fall into each one, and then cumulative, relative, and percentage frequencies, along with histograms, polygons, and ogives, all build on that single table. Everything in the rest of this book’s descriptive statistics content, starting with measures of central tendency, works from a dataset organized exactly this way.
Summary
| Concept | Description |
|---|---|
| Building the Table | |
| Frequency Distribution | A table showing how many observations fall into each class interval |
| Tally Marks | A traditional counting method used to sort raw observations into their class as the table is built |
| Frequency | The count of observations that fall within a given class interval |
| Cumulative Frequency | |
| Less-Than Cumulative Frequency | The running total of frequencies for all values less than a class's upper boundary |
| More-Than Cumulative Frequency | The running total of frequencies for all values greater than or equal to a class's lower boundary |
| Relative and Percentage Frequency | |
| Relative Frequency | A class's frequency expressed as a share of the total number of observations |
| Percentage Frequency | Relative frequency expressed as a percentage by multiplying by 100 |
| Charts | |
| Histogram | Adjacent bars with height proportional to frequency, used to visualize the shape of a distribution |
| Frequency Polygon | A line graph connecting class midpoints to their frequencies, useful for comparing distributions |
| Ogive | A cumulative frequency curve used to read off the median and other percentiles graphically |
| Practical Cautions | |
| Choosing Histogram Bin Width | Follows the same guideline as choosing class width for the frequency table, typically Sturges' Rule |
| Unequal Class Widths | Makes raw frequency bar height misleading, since wider classes appear to have more data |
| Frequency Density | Frequency divided by class width, used instead of raw frequency when class widths are unequal |