Displaying One-Variable Data

A good graph lets you see the shape of a data set at a glance: where the values cluster, how spread out they are, and whether anything is unusual. But the right graph depends on the type of data, and a badly made graph (by accident or on purpose) can tell a story the numbers don’t support. This page covers both: choosing a graph, and catching a misleading one.

Key ideas

Match the graph to the data

Start by asking what kind of data you have (see types of data).

Data type	Example	Good graphs
Categorical	favourite sport, type of pet	bar graph, circle graph
Discrete numerical	number of siblings	bar graph, stem-and-leaf plot, boxplot
Continuous numerical	heights, times, masses	histogram, stem-and-leaf plot, boxplot

Bar graphs

A bar graph shows a count (or percentage) for each category. The bars are separate, with gaps between them, because the categories are separate. The order of the categories usually doesn’t matter, so you can sort the bars from tallest to shortest to make comparisons easier.

Circle graphs

A circle graph (pie chart) shows how a whole is divided into parts. Each sector’s angle is its share of $360^\circ$:

$$\text{sector angle} = \frac{\text{frequency}}{\text{total}} \times 360^\circ$$

Use a circle graph only when the parts add up to one whole (every person is counted exactly once). Circle graphs are good for showing shares, but bar graphs are better for comparing categories that are close in size.

Histograms

A histogram shows continuous data grouped into intervals (also called classes or bins).

• The intervals should have equal widths and must not overlap. Use ”$20$ to under $30$”, so a value of exactly $30$ goes in only one interval.
• The bars touch, with no gaps, because the number line is continuous: one interval ends exactly where the next begins.
• The height of each bar is the frequency (or relative frequency) of that interval.

About $5$ to $10$ intervals usually works well. Too few hides the shape; too many makes it bumpy.

Stem-and-leaf plots

A stem-and-leaf plot splits each value into a stem (the leading digits) and a leaf (the last digit). It looks like a sideways histogram but keeps every individual value, so you can still find the median and quartiles. It works best for small data sets (up to about $50$ values). Always include a key, like ”$2 \mid 1$ means $21$ minutes”.

Boxplots

A boxplot shows the five-number summary and any outliers (see quartiles and percentiles). Boxplots hide the individual values, but they’re excellent for comparing two or more data sets side by side.

How graphs can mislead

Watch for these tricks:

• Truncated axis: the vertical axis doesn’t start at $0$, so small differences look huge. Bar graphs should always start at $0$, because we compare bars by their lengths.
• Uneven intervals: in a histogram, a wider interval collects more data and makes its bar look more important than it is.
• 3-D effects: a tilted 3-D circle graph makes the sectors at the front look bigger than equal sectors at the back.
• Misleading pictographs: if a picture is scaled up in both height and width to show “twice as much”, its area grows four times, so it looks four times as big.
• Missing labels or scales: a graph with no units or no numbers on the axis can’t be checked at all.

On the SAT

Desmos can’t read a graph for you, so on the SAT this topic is mostly careful reading. Questions show a histogram, dot plot, or bar graph and ask for a mean, median, or count, so read the scale and the bin labels first. For the median, add up the frequencies to get $n$ and find the value in position $\dfrac{n + 1}{2}$ (counting from the smallest); for the mean, multiply each value by its frequency, add, and divide by $n$, using Desmos for the arithmetic if you like. See using Desmos on the SAT.

Worked examples

Example 1: Choosing a graph

Which type of graph would you use for each data set?

• (a) The number of students in each club at a school.
• (b) How a town’s budget is split among roads, parks, police, and other services.
• (c) The masses of $150$ newborn babies.
• (d) The quiz marks of a class of $24$, when you want to see every mark.

Solution.

(a) A bar graph: the clubs are categories.

(b) A circle graph: the parts make up a whole budget.

(c) A histogram: mass is continuous, and $150$ values is too many to list.

(d) A stem-and-leaf plot: it shows the shape and keeps every individual mark.

Example 2: Sector angles for a circle graph

A survey of $120$ students asked for their favourite cafeteria lunch: pizza $42$, sushi $18$, sandwiches $30$, salad $12$, other $18$. Find the sector angle for each.

Solution. Multiply each share of the total by $360^\circ$:

Lunch	Frequency	Sector angle
Pizza	$42$	$\dfrac{42}{120} \times 360^\circ = 126^\circ$
Sushi	$18$	$\dfrac{18}{120} \times 360^\circ = 54^\circ$
Sandwiches	$30$	$\dfrac{30}{120} \times 360^\circ = 90^\circ$
Salad	$12$	$\dfrac{12}{120} \times 360^\circ = 36^\circ$
Other	$18$	$\dfrac{18}{120} \times 360^\circ = 54^\circ$

Check: $126 + 54 + 90 + 36 + 54 = 360$. ✓

Example 3: Stem-and-leaf plot and histogram

Twenty people timed how long (in minutes) it took them to finish the same crossword:

$$12,\ 15,\ 18,\ 21,\ 22,\ 24,\ 25,\ 27,\ 28,\ 29,\ 31,\ 32,\ 33,\ 34,\ 36,\ 38,\ 41,\ 44,\ 47,\ 53$$

Make a stem-and-leaf plot and a histogram, and describe the shape.

Solution. Use the tens digit as the stem and the ones digit as the leaf:

Stem	Leaves
$1$	$2\ 5\ 8$
$2$	$1\ 2\ 4\ 5\ 7\ 8\ 9$
$3$	$1\ 2\ 3\ 4\ 6\ 8$
$4$	$1\ 4\ 7$
$5$	$3$

Key: $2 \mid 1$ means $21$ minutes.

From the plot, the 10th and 11th values are $29$ and $31$, so the median is $30$ minutes.

For the histogram, use intervals of width $10$: $10$ to under $20$, $20$ to under $30$, and so on. The frequencies are the number of leaves on each stem: $3, 7, 6, 3, 1$.

The crossword times as a histogram. The bars touch because time is continuous.

Most people took between $20$ and $40$ minutes. The data trail off more slowly on the right (up to $53$ minutes) than on the left, so the distribution is slightly skewed right.

Example 4: A misleading bar graph

A school’s newsletter shows the recycling collected each month: $412$ kg in September, $418$ kg in October, $425$ kg in November, and $431$ kg in December. Compare the two graphs below.

The same data with a truncated axis (left) and a zero baseline (right).

Solution. On the left, the axis starts at $400$, so the bars show only the part above $400$. September’s bar is $412 - 400 = 12$ units tall and December’s is $431 - 400 = 31$ units, so December looks about $\dfrac{31}{12} \approx 2.6$ times as big.

In fact, recycling went up by $431 - 412 = 19$ kg, which is

$$\frac{19}{412} \approx 0.046 = 4.6\%$$

That’s a real but small improvement, which the honest graph on the right shows correctly.

Common mistakes

Leaving gaps between histogram bars (or none between bar graph bars). Gaps mean separate categories (bar graph). No gaps mean a continuous number line (histogram).

Using a circle graph when the parts aren’t one whole. If people could pick more than one answer, the percentages add to more than $100\%$, and a circle graph makes no sense. Use a bar graph.

Using unequal intervals in a histogram. A wider interval catches more values just because it’s wider. Keep the widths equal.

Overlapping intervals. ”$10$–$20$” and ”$20$–$30$” both seem to include $20$. Write ”$10$ to under $20$” so every value belongs to exactly one interval.

Forgetting the key on a stem-and-leaf plot. Without it, "$2 \mid 1$" could mean $21$, $2.1$, or $210$.

Trusting a graph without reading the axis. Before you react to a dramatic graph, check where the vertical axis starts and whether the intervals are equal.

Practice

1. (Warm-up) Which type of graph would you use for each data set?

• (a) Students’ favourite sports.
• (b) The heights of $200$ Grade 12 students.
• (c) The percentage of a family’s monthly spending on rent, food, transportation, and savings.

Solution

(a) Bar graph (categorical data). (b) Histogram (continuous data, many values). (c) Circle graph (parts of one whole).

2. (Warm-up) Explain why the bars of a histogram touch, but the bars of a bar graph don’t.

Solution

A histogram shows intervals on a continuous number line: each interval ends exactly where the next one starts, so the bars touch. A bar graph shows separate categories with nothing “between” them, so the bars are separated by gaps.

3. (Warm-up)

• (a) What sector angle represents $25\%$ of a circle graph?
• (b) What percentage does a sector of $72^\circ$ represent?

Solution

(a) $0.25 \times 360^\circ = 90^\circ$.

(b) $\dfrac{72}{360} = 0.2 = 20\%$.

4. (Core) Eighty students were asked how they usually get to school: bus $28$, walk $20$, car $24$, bike $8$. Find the sector angles for a circle graph.

Solution

Bus: $\dfrac{28}{80} \times 360^\circ = 126^\circ$. Walk: $\dfrac{20}{80} \times 360^\circ = 90^\circ$. Car: $\dfrac{24}{80} \times 360^\circ = 108^\circ$. Bike: $\dfrac{8}{80} \times 360^\circ = 36^\circ$.

Check: $126 + 90 + 108 + 36 = 360$. ✓

5. (Core) Make a stem-and-leaf plot of these test marks, then find the median:

$$56,\ 42,\ 67,\ 45,\ 38,\ 60,\ 53,\ 72,\ 45,\ 51,\ 64,\ 58$$

Solution

Stem	Leaves
$3$	$8$
$4$	$2\ 5\ 5$
$5$	$1\ 3\ 6\ 8$
$6$	$0\ 4\ 7$
$7$	$2$

Key: $4 \mid 2$ means $42$.

There are $12$ marks, so the median is the mean of the 6th and 7th: $\dfrac{53 + 56}{2} = 54.5$.

6. (Core) A company’s sales doubled this year. Its ad shows a picture of a shopping bag for last year and, for this year, the same bag drawn twice as tall and twice as wide. Why is this misleading?

Solution

Doubling both the height and the width multiplies the area of the picture by $2 \times 2 = 4$. Our eyes judge size by area, so the new bag looks four times as big, even though sales only doubled. An honest pictograph would show two bags of the same size, or scale only one dimension.

7. (Core) A histogram of the ages of people at a community event uses these intervals: $0$ to under $10$ (frequency $8$), $10$ to under $20$ (frequency $12$), and $20$ to under $50$ (frequency $15$). The bars are drawn with heights $8$, $12$, and $15$.

• (a) Which age group looks the most common?
• (b) Why is this misleading? How many people are there per $10$ years of age in the last interval?

Solution

(a) The $20$ to under $50$ group has the tallest bar, so it looks the most common.

(b) That interval is $30$ years wide, three times as wide as the others, so it naturally collects more people. Per $10$ years, it has only $\dfrac{15}{3} = 5$ people, compared with $8$ and $12$ in the other intervals. It’s actually the least crowded age range. Equal-width intervals avoid this problem.

8. (Challenge) A bar graph compares two phone plans’ customer ratings: $52$ for Plan A and $56$ for Plan B. The vertical axis starts at $50$.

• (a) How many times as tall does Plan B’s bar look compared with Plan A’s?
• (b) By what percentage is Plan B’s rating actually higher?

Solution

(a) The bars show only the part above $50$: Plan A’s bar is $2$ units tall and Plan B’s is $6$. Plan B’s bar looks $\dfrac{6}{2} = 3$ times as tall.

(b) $\dfrac{56 - 52}{52} = \dfrac{4}{52} \approx 0.077$, so Plan B’s rating is only about $7.7\%$ higher.

9. (Challenge) A survey asked $200$ students “Which streaming services do you use? (Choose all that apply.)” The results were $70\%$, $55\%$, and $35\%$ for three services. A student wants to show this as a circle graph. What’s the problem, and what graph would you suggest instead?

Solution

The percentages add to $70 + 55 + 35 = 160\%$, because many students chose more than one service. The categories don’t split one whole into parts, so a circle graph can’t show them. (The sector angles would add to $576^\circ$.) Use a bar graph, with one bar per service showing the percentage of students who use it.