Frequency density $\text{FD}=\dfrac{\text{frequency}}{\text{class width}}$the HEIGHT of each bar; use it when the classes have UNEQUAL widths
Frequency $\text{frequency}=\text{FD}\times\text{class width}$rearranged – the frequency is the AREA of the bar
Area = frequency $\text{area of a bar}=\text{frequency}$so a wide, short bar can hold as many as a narrow, tall one
Class width $\text{width}=\text{upper}-\text{lower}$e.g. the class $10
Total frequency $\textstyle\sum(\text{FD}\times\text{width})$add the area of every bar to get the total number of items
Frequency density · class $10
width $=30-10=20$
$\text{FD}=\dfrac{50}{20}=2.5$
Frequency · FD $=3$, class $5
width $=12-5=7$
frequency $=3\times7=21$
Total · bars: FD $2$ (width $10$), FD $4$ (width $5$)
areas $=2\times10=20$ and $4\times5=20$
total $=20+20=40$
Frequency density: frequency ÷ class width – the height of a histogram bar.
Class width: upper bound − lower bound of a class interval.
Histogram: a diagram for continuous data where the AREA of each bar shows the frequency.
Unequal classes: class intervals of different widths – the reason we need frequency density.
Frequency: how many items are in a class = the AREA of its bar.
✗ Reading the bar height as the frequency
✓ height is frequency DENSITY; frequency $=$ height $\times$ width.
✗ Forgetting the class width, e.g. FD $=$ frequency
✓ divide by the width: $\text{FD}=\dfrac{\text{frequency}}{\text{width}}$.
✗ Using width $=$ upper bound
✓ width $=$ upper $-$ lower (e.g. $10
✗ Leaving gaps between bars
✓ continuous data – the bars touch.
• The bar HEIGHT is frequency density, NOT frequency.
• The AREA of a bar (height × width) gives the frequency.
• A histogram is for continuous data, so the bars touch – there are no gaps.
• Make a frequency-density column: FD $=$ frequency ÷ width, for every class.
• Going the other way (from a bar): frequency $=$ FD $\times$ width $=$ the area.
• To estimate part of a class, assume the values are evenly spread across it.
mathedup.co.uk · sheet L4TG