Histograms with Unequal Class Widths and Frequency Density Cheat Sheet

Histograms with unequal class widths are used when data is grouped into intervals that are not all the same size. This cheat sheet helps students avoid reading bar height as frequency when the widths differ. It explains how frequency density connects class width, frequency, and bar area.

These ideas are important for interpreting real data shown in grouped frequency tables and histograms.

The key rule is that frequency density is found using $\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$ . In a histogram, the area of each bar represents the frequency, so $\text{frequency} = \text{class width} \times \text{frequency density}$ . Wider classes may have lower bar heights but still represent large frequencies.

Correct scales, interval widths, and area comparisons are essential for accurate histogram work.

Key Facts

The class width is found by subtracting the lower class boundary from the upper class boundary, so $\text{class width} = \text{upper boundary} - \text{lower boundary}$ .
Frequency density is calculated using $\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$ .
The frequency for a histogram bar is calculated using $\text{frequency} = \text{class width} \times \text{frequency density}$ .
In a histogram, the area of each bar represents the frequency, not just the height of the bar.
When all class widths are equal, bar heights are proportional to frequencies, but when widths are unequal, frequency density must be used.
A class interval such as $10 \leq x < 20$ has width $20 - 10 = 10$ .
The total frequency is the sum of all class frequencies, written as $\text{total frequency} = f_1 + f_2 + f_3 + \cdots$ .
To compare two bars with unequal widths, compare their areas using $\text{area} = \text{width} \times \text{height}$ .

Vocabulary

Histogram: A histogram is a graph for grouped numerical data where bars touch and each bar represents a class interval.
Class Interval: A class interval is a range of values such as $20 \leq x < 30$ used to group data.
Class Width: Class width is the size of a class interval, found using $\text{upper boundary} - \text{lower boundary}$ .
Frequency: Frequency is the number of data values that fall inside a class interval.
Frequency Density: Frequency density is the frequency per unit of class width, calculated by $\frac{\text{frequency}}{\text{class width}}$ .
Bar Area: Bar area is the product of class width and frequency density, and it represents frequency in a histogram.

Common Mistakes to Avoid

Using frequency as the bar height for unequal class widths is wrong because histogram bar height must be frequency density, not frequency.
Forgetting to calculate class width from the interval boundaries is wrong because $0 \leq x < 5$ has width $5$ , while $5 \leq x < 20$ has width $15$ .
Comparing only bar heights is wrong because a shorter but wider bar can have a larger area and therefore a larger frequency.
Leaving gaps between histogram bars is wrong because grouped numerical intervals are continuous and adjacent bars should touch unless there is a true gap in the data range.
Using inconsistent vertical scale is wrong because frequency densities such as $1.5$ , $3$ , and $6$ must be plotted on one accurate shared scale.

Practice Questions

1 A class interval $0 \leq x < 10$ has frequency $25$ . Find its frequency density.
2 A histogram bar has class width $8$ and frequency density $3.5$ . Find the frequency represented by the bar.
3 The grouped data table has intervals $0 \leq x < 5$ with frequency $12$ , $5 \leq x < 15$ with frequency $30$ , and $15 \leq x < 25$ with frequency $20$ . Calculate the frequency density for each interval.
4 A bar for $0 \leq x < 5$ is taller than a bar for $5 \leq x < 25$ . Explain why the taller bar does not necessarily represent a greater frequency.

Understanding Histograms with Unequal Class Widths and Frequency Density

Unequal intervals often appear because the data are spread unevenly. A survey of journey times might have small groups for short trips, where many values occur, then wider groups for long trips, where fewer values occur. This keeps the graph readable without making a very long list of bars.

It is useful in data about income, house prices, waiting times, test scores, rainfall, and population age. The grouping choice affects how the shape looks. A very wide interval can hide variation inside it, so a histogram gives a summary rather than every individual value.

A reliable method starts with the horizontal scale. Read the two ends of each class and find how far apart they are. Then use the bar height together with that width to recover the number of data values in the class.

For example, a class running from forty to fifty has width ten. If its density is three, its frequency is thirty. A class from fifty to seventy is twice as wide.

If its density is two, its frequency is forty, even though the bar is shorter. This is the central visual idea. A lower bar can contain more data when it covers more horizontal distance.

Class boundaries need careful attention. Histogram classes represent continuous ranges, so neighbouring bars should touch with no gaps. A measurement recorded as whole numbers may still describe a continuous quantity.

For example, heights recorded to the nearest centimetre can be placed in boundary intervals that begin halfway between whole numbers. This prevents the same recorded value from belonging to two groups. Students sometimes use the printed class labels as though they were separate categories.

That creates incorrect widths. Check whether the question gives class boundaries directly or gives rounded class limits that need interpreting.

When reading a completed graph, first identify the vertical axis. It should be labelled frequency density, not frequency, when widths differ. Next, use the area of every relevant bar for comparisons or totals.

Adding heights is not meaningful unless all widths match. A useful accuracy check is to calculate each class frequency and add them. The result should equal the stated number of observations when one is given.

Histograms can support estimates of the modal class, median, and proportion in a range, but they cannot reveal the exact original values. Any answer taken from part of a bar is an estimate because the graph does not show how values are distributed within that interval.

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