Skill: Construct and interpret histograms

A histogram shows how quantitative data are distributed. To make one:

1. Choose intervals that are:
   • Equal in width
   • Contiguous, with no overlaps or gaps between intervals
   • Appropriate for the data
2. Count how many data values fall in each interval.
3. Draw bars whose heights equal the frequencies.
4. Make the bars touch, because the intervals represent numerical values on a continuous scale.

Worked example

The following data show the number of minutes 20 students spent exercising in one week:

$$12,\ 14,\ 15,\ 17,\ 18,\ 19,\ 21,\ 22,\ 24,\ 25,\ 26,\ 27,\ 29,\ 31,\ 33,\ 34,\ 42,\ 44,\ 46,\ 65$$

Step 1: Choose equal-width intervals

Use intervals of width $10$:

• $10$ to less than $20$
• $20$ to less than $30$
• $30$ to less than $40$
• $40$ to less than $50$
• $50$ to less than $60$
• $60$ to less than $70$

Writing intervals with “less than” avoids ambiguity. For example, a value of $20$ belongs in $20$ to less than $30$, not in $10$ to less than $20$.

Step 2: Count the values in each interval

Exercise time (minutes)	Frequency
$10$ to less than $20$	$6$
$20$ to less than $30$	$7$
$30$ to less than $40$	$3$
$40$ to less than $50$	$3$
$50$ to less than $60$	$0$
$60$ to less than $70$	$1$

Check that the frequencies add to the total number of observations:

$$6+7+3+3+0+1=20$$

Step 3: Draw the histogram

Put the intervals on the horizontal axis and frequency on the vertical axis. Draw rectangles with these heights:

• Height $6$ over $10$ to less than $20$
• Height $7$ over $20$ to less than $30$
• Height $3$ over $30$ to less than $40$
• Height $3$ over $40$ to less than $50$
• Height $0$ over $50$ to less than $60$
• Height $1$ over $60$ to less than $70$

The bars should touch each other. The empty interval from $50$ to less than $60$ appears as a gap, and the bar from $60$ to less than $70$ has height $1$.

Step 4: Interpret the histogram

From the histogram, we can say:

• The data have one main cluster from about $10$ to $30$ minutes.
• The typical exercise time is in the $20$-minute interval.
• The distribution is right-skewed because a few larger values extend toward $65$ minutes.
• There is a gap from $50$ to less than $60$ minutes.
• The value $65$ may be a possible outlier because it is separated from most of the data.

A histogram is used for quantitative data, such as time, height, or test score. A bar graph is used for categorical data, such as favorite sports or eye color. Histogram bars touch; bar graph bars usually have spaces between them.