Frequency is area, not height
On a histogram the bar's area equals the frequency. The height is the frequency density .
Two formulas, rearranged as needed
and .
Drawn from real examiner reports.
Reading the bar height as the frequency
On a histogram the vertical axis is FREQUENCY DENSITY, not frequency. Reading a bar's height straight off as the frequency ignores the class width. To get the frequency you must multiply the height (density) by the class width — the bar's AREA is the frequency.
A commonly reported error, especially reading the vertical scale.
Totalling bar heights instead of areas
To find a total frequency you add the AREAS of the bars, not their heights. Each area is height × width, so simply summing the frequency-density heights (e.g. ) undercounts everything. Work out each bar's frequency first, then add those.
A reported error on histogram questions.
Not dividing by the class width
To PLOT a histogram bar you need the frequency density, which is frequency ÷ class width — you must divide. Plotting the raw frequency as the height is wrong whenever class widths are unequal. For a class of width containing values, the density is , not .
Miscounting the class width
The class width is the upper boundary minus the lower boundary. For the width is (not , and not the frequency). Getting the width wrong feeds a wrong density into every calculation, so read the class boundaries carefully.
Confusing a histogram with a bar chart
A histogram is for CONTINUOUS data with (often unequal) class widths: the bars touch, the vertical axis is frequency density, and area = frequency. A bar chart is for categories with equal-width, separated bars where height = frequency. Do not read a histogram as if height were the frequency.
Read the vertical scale carefully
Misreading the frequency-density axis is a frequent, avoidable error — check the gridline values.
Make a frequency-density column
Before plotting, add a column for frequency density = frequency ÷ class width and fill it in for every class. Plot each bar at that height across its class width. Working in a table makes unequal widths easy and keeps the method marks.
Find a frequency by area
To read a class frequency from a histogram, multiply the bar's height (frequency density) by its class width — that product is the bar's area, which equals the frequency. For a total, add the areas of the relevant bars.
Histograms are used when class intervals are unequal. The vertical axis is frequency density, not frequency: Rearranged: — this is the area of the bar.
What does the height of a histogram bar represent?
On a histogram, the bar for the class has a frequency density of . Find the frequency for this class.
