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How Histograms Work

A histogram looks very much like a bar chart. But there are three main differences between the two. There are intervals—not numbers—along the $x$ -axis, the columns are right next to each other with no space between, and the height must be calculated by dividing the frequency by the width of the interval. The area of the columns are the product of the frequency with the interval width.

You use a histogram when your data is sorted into groups of intervals.

Theory

Frequency for intervals of unequal length

$f = frequency$

$i = interval width$

$h = height of histogram$

f = i \cdot h \Rightarrow h = \frac{f}{i}

Example 1

Sarah is measuring how fast cars are driving on a road with a $50 mph$ limit:

Speed (mph) Number of cars
$[0, 20)$ $3$
$[20, 50)$ $15$
$[50, 75)$ $10$

Make a histogram that fits this table.

You begin by calculating the width of the intervals:

\begin{array}{l} 20 - 0 & = 20 \\ 50 - 20 & = 30 \\ 75 - 50 & = 25 \end{array}

The next step is to calculate the height of the histogram by dividing the frequency with the width of the intervals:

\begin{array}{l} \frac{3}{20} & = 0.15 \\ \frac{15}{30} & = 0.5 \\ \frac{10}{25} & = 0.4 \end{array}

You can now make the following histogram:

Histogram of car speeds

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