Graphs and the shape of data

Dot plots, stem-and-leaf plots, histograms and frequency polygons, sector graphs and divided bar graphs, line graphs, choosing a graph, describing shape, and spotting misleading graphs.

A good graph lets someone understand the data in five seconds. A bad one (or a sneaky one) leaves them with the wrong idea just as quickly. You need to be able to read graphs, choose them, and catch the ones designed to fool you.

Dot plots and stem-and-leaf plots

For small sets of numerical data, these two keep every individual value while showing the shape.

A dot plot puts a dot above a number line for each value. Repeated values stack up.
A stem-and-leaf plot splits each value into a stem (usually the tens) and a leaf (the units). Leaves are written in order, so the plot is also a sorted list. Always include a key.

Example 1●●

Twenty students recorded how many minutes it takes them to get to school: $8$, $12$, $14$, $15$, $15$, $17$, $18$, $21$, $22$, $22$, $24$, $25$, $27$, $31$, $33$, $36$, $42$, $45$, $51$, $63$. Draw a stem-and-leaf plot, find the median and describe the shape.

Stem	Leaf
0	8
1	2 4 5 5 7 8
2	1 2 2 4 5 7
3	1 3 6
4	2 5
5	1
6	3

Key: $3 \mid 6$ means $36$ minutes

Stems are the tens digits 0 to 6. Leaves are the units, in order.

Median $= \dfrac{22 + 24}{2} = 23$ minutes

20 values, so the median is halfway between the 10th and 11th. Count leaves from the top: 1 + 6 = 7, so the 10th and 11th are the 3rd and 4th leaves on stem 2.

The data is positively skewed

Most values are bunched at the low end, with a few long trips stretching out to the high end.

Histograms and frequency polygons

For larger sets of numerical data, group the values (as in 8C) and draw a histogram: a column graph with no gaps between the columns, because the classes run on continuously from one to the next. A frequency polygon joins the midpoints of the tops of the columns (at the class centres) with straight lines, dropping to zero half a class beyond each end.

Describing shape

Symmetric: the two sides are roughly mirror images, with the peak in the middle. Heights of students are usually close to symmetric.
Positively skewed: the tail stretches out to the right (towards the larger values). Commute times, incomes and house prices are typically positively skewed.
Negatively skewed: the tail stretches out to the left. Marks on an easy test are often negatively skewed.

A useful memory hook: the skew is named after the tail, not the peak.

InteractiveClass width and the shape of a histogram

Try this

For each data set, try every class width. Which width shows the shape most clearly? What goes wrong with the narrowest and the widest?

Describe the shape of each data set in a sentence that mentions the context, for example "Most students get to school quickly, but a few have very long trips."

Graphs for categorical data

Column or bar graphs: one bar per category, with gaps between them.
Sector (pie) graphs: show each category as a share of the whole. The angle of each sector is $\dfrac{f}{\text{total}} \times 360^\circ$.
Divided bar graphs: one long bar split into parts. Each part's length is $\dfrac{f}{\text{total}} \times$ the bar's length.

Example 2●

Of $120$ students, $42$ catch the train, $30$ the bus, $36$ come by car and $12$ walk or cycle. Find the sector angles, and the lengths for a $12$ cm divided bar graph.

Train $126^\circ$, bus $90^\circ$, car $108^\circ$, walk or cycle $36^\circ$

Angle = f ÷ 120 × 360°. Dividing by 120 and multiplying by 360 is the same as multiplying by 3.

$126 + 90 + 108 + 36 = 360^\circ$ ✓

Check the angles make a full circle.

Train $4.2$ cm, bus $3$ cm, car $3.6$ cm, walk or cycle $1.2$ cm

Length = f ÷ 120 × 12 cm, which is f ÷ 10.

Choosing a graph

Data	Good choices
Categorical, comparing categories	column or bar graph
Categorical, parts of a whole	sector graph, divided bar graph
Numerical, small set	dot plot, stem-and-leaf plot
Numerical, large or continuous	histogram, frequency polygon
Change over time	line graph
Comparing spread of groups	parallel box plots (see 8F)

Misleading graphs

Graphs can mislead without a single false number. Check for:

a vertical axis that doesn't start at zero (with no break symbol), which exaggerates differences
uneven scales, or missing labels and units
pictures or 3D shapes scaled in two or three directions, so their area or volume grows far faster than the data
a time period cherry-picked to show a trend that isn't there over a longer period
a sector graph whose percentages don't add to $100\%$

InteractiveHow to make a small change look huge

Try this

Compare how big Term 4 looks with the real ratio in each of the four settings. Which setting misleads the most?

Why does the picture version still mislead when the axis starts at $0$?

Your turn

In a survey of $250$ people, one sector of a pie chart has an angle of $72^\circ$. How many people does it represent?

Show the working

$\dfrac{72}{360} = 0.2$, so the sector is $20\%$ of the people: $0.2 \times 250 = 50$ people.

Flashcards

Sector angle

$\dfrac{f}{\text{total}} \times 360^\circ$

Positively skewed

Tail stretches to the right (high values)

Histogram vs column graph

A histogram has no gaps: its classes are continuous

Practice

$45$ of $180$ people chose soccer. Find the sector angle in degrees.
A $72^\circ$ sector in a survey of $250$ people represents how many people?
A distribution has a long tail to the left. It is:
The best graph to show how a city's average daily temperature changes month by month is a:
Why does a histogram have no gaps between its columns?
In Example 1, what is the range of the commute times?
In Example 1, how many students take $30$ minutes or more?
Exam style: A graph of a company's profits has a vertical axis from $\$95$ million to $\$100$ million. The most likely effect is that:
Exam style: In a $15$ cm divided bar graph of $200$ survey responses, how long is the part for a response chosen by $32$ people? Give your answer in cm.