Histograms

When bar widths matter

• A bar chart always has equal-width bars. But what if your data classes have different widths?
• A histogram 直方图 solves this: the area (not the height) of each bar represents the frequency.

Frequency density 频数密度

• When class widths 组距 vary, the vertical axis shows frequency density:

A pictogram uses a symbol to stand for a number of items

Worked examples

• Frequency $25$, width $10$ → density $= 2.5$.
• Frequency $30$, width $5$ → density $= 6$.
• Frequency $40$, width $20$ → density $= 2$.

Reading a histogram

• To find the frequency of a class: $\text{density} \times \text{width} = \text{frequency}$.
• To compare classes: look at the area of each bar, not the height.

Why histograms look like bar charts

• When all classes have the same width, the density is proportional to the frequency — so the histogram looks like a normal bar chart.
• Use a labelled frequency-density axis consistently, including when all class widths are equal.

Histogram bars are like proportional blocks: the area (width × height) gives the frequency, not just the height.

Reconstruct a whole histogram

• Classes $0\le x<5,5\le x<15,15\le x<35$ have frequencies 10,30,20. Widths 5,10,20 give heights $D=f/w=2,3,1$; draw touching bars over their actual intervals.
• Reverse the method for a missing frequency: width 8 and density 2.5 give $f=Dw=2.5(8)=20$. The tallest bar need not hold the most observations if widths differ.

You've got it

Write from a density of 4

• A class has frequency 24 and class width 6.
• Find the frequency density.