Histograms & Frequency Polygons
The fundamental principle of a histogram is that the area of each bar is proportional to the frequency of the class interval it represents.
When class widths are equal, the height of the bar is directly proportional to the frequency, allowing the y-axis to be labeled simply as 'Frequency'.
For datasets with unequal class widths, the y-axis must represent Frequency Density to maintain the area-frequency relationship.
The formula for frequency density is defined as:
To construct a frequency polygon, identify the midpoint of each class interval using the formula: .
Plot a point at the height corresponding to the frequency (or frequency density) directly above the calculated midpoint on the x-axis.
Connect these points sequentially with straight line segments; unlike a curve, the polygon uses linear paths to show the change between intervals.
Standard convention dictates that the polygon should not be joined back to the x-axis at the ends unless the frequency of the adjacent theoretical class is zero.
Check for Gaps: Always ensure bars in a histogram are touching; leaving gaps is a common error that suggests discrete data rather than continuous data.
Labeling the X-axis: Ensure class boundaries are marked precisely at the edges of the bars, not in the center (unless you are drawing a polygon).
Midpoint Accuracy: When drawing a frequency polygon, double-check that points are plotted at the exact midpoint of the interval, not at the start or end boundaries.
Frequency vs. Density: Always check if class widths are equal; if they are not, you must calculate and plot frequency density instead of raw frequency.