Math Core

A quantitative variable can take many different values, so its distribution needs a display that shows where the values fall along a number line. AP Statistics uses three main graphs for one quantitative variable: dot plots, stemplots and histograms. Each one keeps a different amount of detail, and choosing the right one depends on how much data you have and what you want to see.

Dot plots

A dot plot places a mark above a number line for every individual. Identical values stack up. Because every value is shown, you can read off the exact data, count how many values fall in a range, and spot gaps and unusual values immediately.

Worked example: Summer reading

A teacher asked $22$ students how many books they read over the summer.

Books read over the summer by 22 students

What percent of the students read at least $3$ books?

"At least $3$ " means $3$ or more: $4 + 2 + 1 + 1 = 8$ students. $\dfrac{8}{22} \approx 0.364$ , so about $36.4\%$ of the students read at least $3$ books.

You can also see features that a table would hide: most students read between $0$ and $3$ books, and one student who read $8$ books stands apart from everyone else.

Dot plots are ideal for small data sets. With hundreds of values, the stacks get too tall to draw.

Stemplots

A stemplot (stem-and-leaf plot) splits each value into a stem (all but the final digit) and a leaf (the final digit). Stems are listed in a column in increasing order, with no stems skipped, and each value's leaf is written to the right of its stem. Leaves are usually put in increasing order. Every stemplot needs a key that tells the reader how to read a stem and leaf.

Worked example: Building a stemplot

Here are the resting pulse rates, in beats per minute, of $18$ students:

56, \ 58, \ 61, \ 63, \ 64, \ 64, \ 67, \ 68, \ 70, \ 71, \ 72, \ 72, \ 73, \ 75, \ 78, \ 81, \ 84, \ 93

Make a stemplot.

The tens digits $5$ through $9$ become the stems and the ones digits become the leaves.

Stem	Leaves
5	6 8
6	1 3 4 4 7 8
7	0 1 2 2 3 5 8
8	1 4
9	3

Key: $6 \mid 1$ means $61$ beats per minute.

Check: $2 + 6 + 7 + 2 + 1 = 18$ leaves, one for each student. Turned on its side, the stemplot looks like a histogram: the values pile up in the $60$ s and $70$ s, with a single student at $93$ .

A stemplot keeps every data value (like a dot plot) while also showing the overall shape (like a histogram). It works best for small to moderate data sets, roughly $15$ to $50$ values.

Two useful variations:

Split stems. If most of the data land on just two or three stems, list each stem twice: leaves $0$ – $4$ go on the first copy and leaves $5$ – $9$ on the second. This spreads the picture out so the shape is easier to see.
Back-to-back stemplots. To compare two groups, write the stems down the middle, one group's leaves to the right and the other group's leaves to the left. You'll use these in the lesson on comparing distributions.

Histograms

For large data sets, the standard display is a histogram. Divide the range of the data into classes of equal width, called bins, and draw a bar over each bin. The height of the bar is the bin's frequency (a frequency histogram) or relative frequency (a relative frequency histogram). Unlike a bar graph, the bars touch, because the bins together cover a continuous stretch of the number line.

By convention, a value that falls exactly on the boundary between two bins goes in the bin on the right. A bin labeled $20$ to $30$ holds values from $20$ up to, but not including, $30$ .

Worked example: From table to histogram

A store tracked how many minutes each of $40$ customers spent shopping.

Time (minutes)	$0$ – $10$	$10$ – $20$	$20$ – $30$	$30$ – $40$	$40$ – $50$	$50$ – $60$
Frequency	$2$	$7$	$12$	$9$	$6$	$4$

Time spent shopping by 40 customers (minutes)

What is the relative frequency of the $20$ to $30$ minute bin?
What percent of the customers shopped for less than $20$ minutes?

Solutions.

$\dfrac{12}{40} = 0.30$ . In a relative frequency histogram, this bar would have height $0.30$ .
The first two bins hold $2 + 7 = 9$ customers, and $\dfrac{9}{40} = 0.225$ , so $22.5\%$ .

A relative frequency histogram has exactly the same shape as the frequency histogram of the same data; only the numbers on the vertical axis change. Relative frequency histograms are the better choice for comparing groups of different sizes.

Choosing the bin width

The bin width changes the picture. With too few wide bins, everything lands in one or two bars and the shape disappears. With too many narrow bins, the histogram becomes a jagged row of short bars with gaps that are just noise. Aim for a width that shows the overall pattern; for most data sets that means somewhere around $5$ to $15$ bins. Technology will pick a width for you, but you should try a few to make sure the shape you describe isn't an accident of the bins.

What each display keeps

Display	Shows every value?	Best for
Dot plot	Yes	Small data sets
Stemplot	Yes	Small to moderate data sets
Histogram	No (only counts per bin)	Large data sets

Common mistake

A histogram is not a bar graph. Bar graphs display categorical data: the bars are separated and can be put in any order. Histograms display quantitative data: the bars touch and follow the number line in order. Also, from a histogram you can't recover the individual data values, so you can't compute the exact mean or median, only locate the bin that contains the median.

Practice

Practice 1

The dot plot shows the number of siblings of $25$ students. What percent of the students have more than $2$ siblings?

Number of siblings