Math Core

A long list of numbers is hard to make sense of. A graph lets you see the data all at once: where most of the values are, how spread out they are, and whether any values stand out. Two of the most useful graphs for numerical data are dot plots and histograms.

Dot plots

A dot plot (also called a line plot) shows each data value as a mark above a number line. If a value appears more than once, the marks stack up.

Worked example: Reading a dot plot

Thirteen students were asked, "How many books did you read over winter break?" The dot plot shows their answers. Each ✕ is one student.

Books read over winter break

How many students read exactly 3 books? There are 3 marks above 3, so 3 students.
How many students read fewer than 2 books? Count the marks above 0 and 1: $1 + 2 = 3$ students.
What was the most common answer? The tallest stack is above 2, so 2 books.

Look at the overall picture, too. Most students read between 1 and 4 books. There is a gap from 5 to 7 where no one answered, and one student read 8 books. A value that is far away from the rest of the data is called an outlier.

Making a dot plot

To make a dot plot, draw a number line that covers the smallest to the largest value. Then put one mark above the number line for each data value.

Worked example: Goals per game

A soccer team recorded the number of goals it scored in each of 12 games:

$2, \ 0, \ 3, \ 1, \ 2, \ 2, \ 4, \ 1, \ 2, \ 3, \ 0, \ 2$

The values go from 0 to 4. Count how many times each value appears:

Goals	0	1	2	3	4
Number of games	2	2	5	2	1

Goals scored per game

Check: $2 + 2 + 5 + 2 + 1 = 12$ marks for 12 games. ✓

Histograms

When the data has lots of different values, a dot plot can get too wide. A histogram groups the data into intervals of equal width (such as 40–49, 50–59, 60–69) and shows how many values fall in each interval. The number of values in an interval is called its frequency.

Definition

Histogram

A histogram is a bar graph for numerical data. Each bar covers an interval of values, and the height of the bar is the frequency: the number of data values in that interval. The bars touch because the intervals are next to each other on the number line.

Worked example: Making a histogram

A basketball team scored these points in 16 games:

$42, \ 55, \ 61, \ 48, \ 57, \ 63, \ 50, \ 45, \ 59, \ 66, \ 52, \ 58, \ 71, \ 54, \ 49, \ 60$

Use intervals of 10 points. Tally each score into its interval:

Points	Scores in the interval	Frequency
40–49	42, 48, 45, 49	4
50–59	55, 57, 50, 59, 52, 58, 54	7
60–69	61, 63, 66, 60	4
70–79	71	1

Check: $4 + 7 + 4 + 1 = 16$ games. ✓

Now draw one bar per interval. The number above each bar is its frequency. A score of 50 goes in the 50–59 bar, not the 40–49 bar.

Points scored in 16 games

The team most often scored in the 50s. Only one game had 70 or more points.

Common mistake

A histogram tells you how many values are in each interval, not the exact values. From the histogram above, you know 7 games had between 50 and 59 points, but you can't tell what any single score was. If a question asks for an exact value, you need the original data or a dot plot.

Describing the shape of data

When you look at a dot plot or histogram, describe what you see:

Describing a distribution

Peak: Where is the tallest stack or bar? That's where the data clusters.
Symmetric or skewed: If the left and right sides look roughly like mirror images, the data is symmetric. If one side stretches out farther, like a tail, the data is skewed toward that side.
Gaps, clusters and outliers: Are there empty spaces? Groups of values bunched together? Values far away from the rest?

Worked example: Describing a histogram

The table shows the homework time of 24 students on one night.

Minutes	0–9	10–19	20–29	30–39	40–49
Frequency	3	6	8	5	2

How many students spent at least 30 minutes on homework?
Describe the shape.

Solutions.

"At least 30" means 30 or more: the 30–39 and 40–49 intervals. $5 + 2 = 7$ students.
The peak is the 20–29 interval. The frequencies rise to the peak and then fall off on both sides, so the data is roughly symmetric, with most students spending between 10 and 39 minutes.

Tip

Always add up the frequencies (or count the marks) and check that you get the number of data values. It catches tally mistakes right away.

Practice

This dot plot is used in the first three problems. It shows how many pets each student in a class has.

Number of pets

Practice 1

Look at the pets dot plot above. How many students are in the class?