Construct histograms

A histogram represents quantitative data by grouping numerical values into contiguous, usually equal-width intervals and displaying the frequency in each interval with adjacent bars; the horizontal axis shows the intervals and the vertical axis shows their counts. Constructing one requires placing values according to interval boundaries, maintaining a consistent scale, and distinguishing a histogram from a bar graph, whose separated bars represent categorical categories rather than numerical ranges.

Detailed Explanation: Construct histograms

A histogram shows how often numerical values fall within equal-width intervals.

Example

The scores on a quiz were:

12,\ 18,\ 21,\ 24,\ 27,\ 31,\ 33,\ 35,\ 41,\ 44,\ 46,\ 48

1. Choose equal-width intervals

Use intervals of width $10$ :

$10$ – $19$
$20$ – $29$
$30$ – $39$
$40$ – $49$

These intervals are numerical ranges, not separate categories.

2. Count the values in each interval

Score interval	Scores in the interval	Frequency
$10$ – $19$	$12, 18$	$2$
$20$ – $29$	$21, 24, 27$	$3$
$30$ – $39$	$31, 33, 35$	$3$
$40$ – $49$	$41, 44, 46, 48$	$4$

3. Draw and label the axes

Label the horizontal axis Quiz score.
Label the vertical axis Frequency.
Mark the score intervals $10$ – $19$ , $20$ – $29$ , $30$ – $39$ , and $40$ – $49$ on the horizontal axis.
Use a consistent vertical scale: $0,1,2,3,4$ .

4. Draw the bars

Draw one bar for each interval:

The bar for $10$ – $19$ has height $2$ .
The bar for $20$ – $29$ has height $3$ .
The bar for $30$ – $39$ has height $3$ .
The bar for $40$ – $49$ has height $4$ .

The bars should touch because the intervals represent ranges of numerical values. A histogram uses touching bars, while a bar graph usually has separated bars for categories such as fruits or colors.

Learn by doing: Construct histograms

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Statistics - Histograms - Frequency Table to Histogram

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