Statistics

Histograms

Change the widths. Keep track of frequency.

Predict, test and explain

Step 1 of 3

Predict

Two classes have frequency 20 each. Their widths are 10 and 20. Which bar has greater height?

\text{frequency density}=\frac{\text{frequency}}{\text{class width}}

Scroll the diagram sideways to see every label.

f_1+f_2=40

Continuous intervals: the lower boundary is included and the upper boundary is excluded. Bar area, not height alone, represents frequency.

Class intervals and frequencies
Interval	Width	Frequency	Density
$0\leq x<10$	10	20	$2$
$10\leq x<30$	20	20	$1$

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

A histogram class has width 6 and frequency 24. What is its frequency density?

$\text{width}=6,\quad \text{frequency}=24$

Hint

Frequency density is frequency divided by class width.

Reveal answer

The bar height is 4. Its area is 6 times 4, giving frequency 24.

$\text{density}=\frac{24}{6}=4$

Hint 1

Find each class width from its two boundaries.

Hint 2

Compare bar area rather than height when widths differ.

Hint 3

Frequency density = frequency/class width; bar area = width × density = frequency.

Worked example

This example uses fixed values, separate from the diagram controls.

The classes 0 ≤ x < 10 and 10 ≤ x < 30 each have frequency 20. Find their densities.

Answer: Densities 2 and 1; the bar areas are both 20.

Connect this idea

Watch out: Equal frequencies need equal bar areas, not equal heights, when class widths differ.