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.

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