Understanding Histograms
A histogram is a graphical representation of data grouped into continuous intervals called class intervals. Unlike bar charts, histograms display data where the horizontal axis represents these intervals without gaps between bars, reflecting the continuous nature of the data.
Each bar's width corresponds to the class interval width, and the area of the bar represents the frequency of data within that interval. This means that when class intervals have different widths, the height of the bars must adjust so that the area, not just the height, reflects frequency.
Histograms are particularly useful for showing the shape and spread of data distributions, such as identifying skewness, modality, or uniformity.
- Histograms display continuous data grouped in class intervals.
- Bars touch each other to indicate continuous intervals.
- Bar width equals class interval width.
- Bar area represents frequency, not just height.
Frequency Density: Definition and Importance
Frequency density is a measure that adjusts frequency by the width of the class interval. It is calculated by dividing the frequency of a class by the width of that class interval.
This adjustment is necessary because when class widths vary, simply plotting frequency as bar height would misrepresent the data. Using frequency density ensures that the area of each bar (height × width) corresponds exactly to the frequency.
Labelling the vertical axis as 'Frequency density' is essential to avoid confusion with simple frequency counts and to correctly interpret the histogram.
- Frequency density = Frequency ÷ Class width.
- Ensures bar area represents frequency accurately.
- Essential when class widths differ.
- Vertical axis must be labelled 'Frequency density'.
Calculating Frequency Density: Worked Example
Consider a data set grouped into two class intervals: 10–15 with frequency 20, and 15–25 with frequency 30.
Step 1: Calculate class widths: 10–15 has width 5 (15−10), 15–25 has width 10 (25−15).
Step 2: Calculate frequency density for each class: For 10–15, frequency density = 20 ÷ 5 = 4; for 15–25, frequency density = 30 ÷ 10 = 3.
- Class 10–15: Width = 5, Frequency = 20, Frequency density = 4.
- Class 15–25: Width = 10, Frequency = 30, Frequency density = 3.
- Use frequency density as bar height in histogram.
Constructing a Histogram Using Frequency Density
To draw a histogram: first, mark the class intervals on the horizontal axis with widths proportional to the class widths.
Next, calculate frequency density for each class and mark these values on the vertical axis labelled 'Frequency density'.
Draw bars for each class interval with the calculated frequency density as the height and the class width as the width. The area of each bar equals the frequency for that interval, allowing accurate visual comparison even when class widths differ.
- Horizontal axis: class intervals with correct widths.
- Vertical axis: frequency density labelled.
- Bars: height = frequency density, width = class width.
- Area of bar = frequency.
Interpreting Histograms with Unequal Class Widths
When class widths vary, the height of bars alone does not indicate frequency; the area does.
A taller bar with a narrow width may represent fewer data points than a shorter bar with a wider width.
Always consider the area of bars to understand the distribution and frequency of data in each interval.
- Bar height alone is insufficient if class widths differ.
- Area of bar represents frequency.
- Compare areas, not just heights, to interpret data correctly.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Calculate frequency density | Divide frequency by class width | Frequency = 20, class width = 5, frequency density = 4 | Show calculation clearly with units |
| Label axes | Horizontal: class intervals; Vertical: frequency density | Vertical axis labelled 'Frequency density' | Avoid losing marks for incorrect axis labels |
| Draw bars | Width = class width; Height = frequency density | Class 10–15 bar width 5, height 4 | Ensure area represents frequency accurately |
| Interpret histogram | Focus on bar areas, not just heights | A bar twice as tall but half as wide has same area | Explain reasoning when describing data |
Active recall
Quick check
1. What is frequency density if a class has frequency 15 and class width 3?
Frequency density = 15 ÷ 3 = 5.
2. Why must the vertical axis be labelled 'Frequency density' in a histogram with unequal class widths?
Because the bar height represents frequency density, not frequency, ensuring the area of each bar corresponds to frequency.
3. How do you find the area of a histogram bar?
Multiply the frequency density (height) by the class width (width).
4. If one bar is twice as tall but half as wide as another, how do their frequencies compare?
Their areas are equal, so the frequencies are the same.