Histograms
Section: Statistics | Syllabus: Cambridge IGCSE Mathematics (0580)
Frequency Density and Histograms with Unequal Class Widths
Extended Only
- When class widths are unequal, plotting raw frequency as bar height is misleading - a wider bar would appear to represent more data even if its actual density of values is lower
- Instead, histograms use frequency density on the vertical axis: frequency density=(frequency/class width)
- The area of each bar (not its height) represents the frequency for that class
Worked Example: Calculating Frequency Density
- Question: The table shows the time, t minutes, taken by 90 runners to finish a race. Calculate the frequency density for each class.
Time, t (min) 0 20 30 40 Frequency 18 24 30 18 - Step 1: Class widths are 20,\,10,\,10,\,30
- Step 2: Frequency density = frequency ÷ class width for each class: (18/20)=0.9,(24/10)=2.4,(30/10)=3.0,(18/30)=0.6
- Answer: frequency densities are 0.9,\,2.4,\,3.0,\,0.6
Common Mistakes
MistakePlotting the raw frequency as the bar height instead of the frequency density, when class widths are unequal
Fixalways divide frequency by class width first - raw frequency should never be plotted directly on a histogram with unequal class widths
MistakeDividing class width by frequency instead of frequency by class width
Fixfrequency density = frequency ÷ class width - the frequency is always the numerator
Finding Frequency from a Histogram (Using Area)
Extended Only
- Since area = frequency density × class width = frequency, the number of items in any class can be found by multiplying its frequency density (bar height) by its class width
- Sometimes a histogram's bars are drawn to a scaled height (e.g. in cm) rather than the frequency density value itself - in this case, use one known bar to find the scale factor between frequency density and height before finding any others
Worked Example: Finding Bar Heights Using a Scale Factor
- Question: The table shows the mass, m grams, of 80 items. A histogram is drawn to show this information. The height of the bar for 10 is 8 cm. Calculate the height of the bar for each of the other intervals.
Mass, m (g) 0 10 25 35 Frequency 8 30 24 18 - Step 1: For 10: frequency density =(30/15)=2.0; this corresponds to a height of 8 cm, so the scale factor is 82.0=4
- Step 2: 0: frequency density =(8/10)=0.8, height =40.8=3.2 cm
- Step 3: 25: frequency density =(24/10)=2.4, height =42.4=9.6 cm
- Step 4: 35: frequency density =(18/25)=0.72, height =40.72=2.88 cm
- Answer: heights are 3.2 cm, 9.6 cm and 2.88 cm
Common Mistakes
MistakeAssuming a bar's height (in cm) is exactly equal to its frequency density, without checking whether the histogram uses a scale factor
Fixalways find the scale factor from one known bar first (height ÷ frequency density), then apply it consistently to every other bar
MistakeMultiplying frequency density by the wrong class width, or using the width of a different interval by mistake
Fixalways match each frequency density to its own class's width when finding an area (frequency) or scaled height
Interactive revision notes, videos and practice questions load below.