Math Core

A list like $0, 2, 1, 1, 0, 3, 1, 2, 5, 0, 1, 2$ doesn't tell you much at a glance. Organize the same numbers into a picture or a table, and patterns jump out: where the data cluster, where the gaps are, and whether any value is unusual. In this lesson you'll read and build three common displays: dot plots, frequency tables (with histograms) and stem-and-leaf plots.

Dot plots

A dot plot (also called a line plot) shows each data value as a mark above a number line. Values that repeat stack up, so the tallest stack is the mode.

Worked example: Reading a dot plot

Twelve students were asked how many pets they have. The results are shown below.

Number of pets

Find the mode, the median and the mean.

Mode: the tallest stack is over $1$ , so the mode is $1$ pet.
Median: list the $12$ values in order by reading the stacks from left to right: $0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 5$ . The $6$ th and $7$ th values are both $1$ , so the median is $1$ pet.
Mean: multiply each value by the height of its stack and add: $0(3) + 1(4) + 2(3) + 3(1) + 5(1) = 18$ . The mean is $\dfrac{18}{12} = 1.5$ pets.

The student with $5$ pets sits apart from the others, so $5$ looks like an outlier. That's why the mean is higher than the median.

Dot plots work best for small data sets with only a few different values, because you can still see every data point.

Frequency tables and histograms

When there are many different values, group them into intervals of equal width and count how many values fall in each one.

Definition

Frequency table

A frequency table lists categories or intervals and the frequency of each, meaning how many data values fall there. A histogram is a bar graph of a frequency table for numerical intervals: the bars touch, and each bar's height is the frequency.

Worked example: Building a frequency table

These are the scores of $20$ students on a quiz:

72, 85, 91, 64, 78, 88, 95, 70, 83, 76, 67, 89, 74, 98, 81, 79, 62, 86, 75, 93

Make a frequency table with intervals $60$ – $69$ , $70$ – $79$ , $80$ – $89$ and $90$ – $99$ .

Go through the list once and sort each score into its interval.

$60$ – $69$ : $64, 67, 62$
$70$ – $79$ : $72, 78, 70, 76, 74, 79, 75$
$80$ – $89$ : $85, 88, 83, 89, 81, 86$
$90$ – $99$ : $91, 95, 98, 93$

Count each group to get the frequencies.

Score	Frequency
$60$ – $69$	$3$
$70$ – $79$	$7$
$80$ – $89$	$6$
$90$ – $99$	$4$

The frequencies add to $3 + 7 + 6 + 4 = 20$ , which matches the number of scores. A histogram of this table would have four touching bars with heights $3, 7, 6, 4$ , and the tallest bar shows that the most common interval is $70$ – $79$ .

Tip

Always check that the frequencies add up to the number of data values. If they don't, you skipped or double-counted a value.

Common mistake

Grouping data hides the individual values. From the table above you can say that $4$ students scored in the $90$ s, but you cannot tell the highest score or the exact median. If you need those, go back to the original data.

Stem-and-leaf plots

A stem-and-leaf plot groups data like a histogram but keeps every value. Each number is split into a stem (all digits except the last) and a leaf (the last digit). The leaves are written in order.

Worked example: Reading a stem-and-leaf plot

The plot shows the ages of the people at a family reunion. The key is $2 \mid 3 = 23$ years.

Stem	Leaves
$2$	$3 \; 7$
$3$	$1 \; 1 \; 5 \; 8$
$4$	$2 \; 4 \; 4 \; 4$
$5$	$1$

Find the number of people, the range and the median.

The data are $23, 27, 31, 31, 35, 38, 42, 44, 44, 44, 51$ : that's $11$ people. The range is $51 - 23 = 28$ years. The leaves are already in order, so the median is the $6$ th value, $38$ years.

Choosing a display

Which display should I use?

Dot plot: small data sets with few distinct values. Shows every value.
Stem-and-leaf plot: small to medium data sets with two-digit values. Shows every value and the shape.
Frequency table or histogram: large data sets or many distinct values. Shows the shape, but not individual values.