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Statistics guide, Unit 1, step 4 of 22Previous stepNext step

Step 4: Describing Data with Graphs

By the EconTutorials team. 10-minute read with 6 practice questions.
SyllabusAP Stats 1.3–1.6
In 30 seconds
  • Every graph starts as a frequency table: frequency (count), relative frequency (proportion) and cumulative frequency (running total).
  • Categorical data gets bar charts, with gaps; quantitative data gets dotplots, stem-and-leaf plots, histograms (bars touching) and boxplots.
  • Describe any distribution by shape, centre, spread, outliers. Skew is named after the direction of the long tail.
  • Graphs can mislead through truncated axes, area tricks, cherry-picked windows and dual axes. Always read the axes before reacting to the shape.
Video lesson coming soonA short animated lesson for this step will appear here.

Frequency tables: the table behind every graph

Imagine you ask 15 households how many cars they own and get: 1, 2, 2, 0, 3, 1, 2, 4, 1, 2, 0, 1, 3, 2, 1. That list is hard to read. Counting how often each value appears makes it clear at a glance.

Key termFrequency distribution: a table recording how often each value, or group of values, occurs.

Three columns matter:

  • Frequency: the raw count.
  • Relative frequency: the count as a proportion or percentage of the total. This lets you compare groups of different sizes fairly.
  • Cumulative frequency: the running total, which answers “how many are at or below this value?”
Worked example

Build the frequency table for the 15 households above.

Step 1Count: 0 cars \( \to 2 \); 1 car \( \to 5 \); 2 cars \( \to 5 \); 3 cars \( \to 2 \); 4 cars \( \to 1 \). Check: \( 2+5+5+2+1 = 15 \).
Step 2Relative frequency (divide by 15): \( 0.13,\ 0.33,\ 0.33,\ 0.13,\ 0.07 \)
Step 3Cumulative frequency: \( 2,\ 7,\ 12,\ 14,\ 15 \)

One or two cars covers \( 10 \) of \( 15 \) households (two-thirds), and the cumulative column shows 12 of 15 own two cars or fewer. The final cumulative value always equals the total.

When you group values into ranges (40–49, 50–59 and so on), each range is a class. Good classes have equal widths, don’t overlap and cover every value; aim for 5 to 15.

Graphs for categorical data

A bar chart draws one bar per category. Categories have no number scale, so the bars have gaps and can go in any order; sorted tallest to shortest, it becomes a Pareto chart.

A pie chart shows the same counts as slices. Your eye judges lengths far more accurately than angles, so a bar chart is usually safer; keep pies for two or three categories.

To compare groups, use a segmented bar chart in percentages (every bar reaches 100%) or a grouped bar chart.

Two-way tables

When two categorical variables are recorded for the same people, the counts form a two-way table. Here 300 people are classified by age and preferred device:

Age groupPhoneLaptopRow total
18–256238100
26–404852100
41–603070100
Column total140160300

The marginal distribution comes from the totals: overall, \( 140 \div 300 \approx 47\% \) prefer a phone. A conditional distribution looks at one row or column only: among 18–25s, 62% prefer a phone, against only 30% of 41–60s. Laptop preference rises with age (38%, 52%, 70%), so device preference is associated with age.

Graphs for quantitative data

For small datasets, two displays keep every value visible. A dotplot stacks one dot per observation above a number line; it shows clusters, gaps and outliers quickly. A stem-and-leaf plot splits each value into a stem (leading digits) and a leaf (final digit). For the quiz scores 52, 55, 58, 61, 63, 63, 67, 68, 71, 72, 74, 74, 79, 83, 88:

StemLeaves
52 5 8
61 3 3 7 8
71 2 4 4 9
83 8

Turn it on its side and it has the shape of a histogram, but you can still read every exact score.

Histograms, polygons and ogives

For larger datasets, group values into bins and draw a histogram.

Key termHistogram: a graph of a quantitative variable in which bars for equal-width bins sit on a real number line, so the bars touch and cannot be reordered.

Two practical points. If bins have unequal widths, plot density = frequency ÷ width so that area, not height, shows frequency. And the number of bins matters: too few can hide a second peak, too many turn the shape into noise.

A frequency polygon joins the tops of the bins with lines. An ogive plots cumulative frequency, rising from 0 to the total; read percentiles from it.

Worked example

Twenty resting pulse rates (beats per minute): 58, 62, 64, 65, 67, 68, 69, 70, 71, 72, 72, 73, 74, 75, 76, 77, 78, 82, 85, 90. Group them in classes of width 8 starting at 56 and find the modal class.

Step 1Classes: 56–63, 64–71, 72–79, 80–87, 88–95.
Step 2Tally: \( 2,\ 7,\ 8,\ 2,\ 1 \). Check: \( 2+7+8+2+1 = 20 \).
Step 3The tallest class, the modal class, is 72–79 with 8 values.

A histogram of these bars would be roughly symmetric, with a slight tail to the right towards the single value of 90.

Describing shape, and the boxplot

Whenever you look at a graph of a quantitative variable, describe four things in this order: shape, centre, spread, outliers. Shape comes first because it decides which average and which measure of spread to trust (Step 4).

  • Modality counts the peaks: unimodal (one), bimodal (two), multimodal (several) or uniform (flat). Two peaks often mean two groups are mixed, such as ages at a playground with peaks near 5 (children) and 35 (parents).
  • Symmetry and skew: symmetric means the two halves roughly mirror each other. A long tail to the right is right-skewed (incomes, house prices); a long tail to the left is left-skewed (exam scores when most students do well).
  • Outliers are points far from the rest. They may be errors or the most interesting cases in the data.
Common mistakeNaming the skew after the hump. A distribution with its peak on the left and a long tail stretching right is right-skewed, even though most of the data sits on the left. Follow the tail, not the crowd.

A boxplot is built from the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3) and maximum. The box runs from Q1 to Q3 (the middle 50%), a line marks the median, and whiskers reach the most extreme values that are not outliers. Boxplots side by side compare groups at a glance.

Worked example

Find the five-number summary of these 11 ordered values and describe the shape: 12, 15, 18, 20, 22, 24, 27, 30, 33, 38, 45.

Step 1\( n = 11 \), so the median is the 6th value: \( 24 \).
Step 2Lower half 12, 15, 18, 20, 22 gives \( Q_1 = 18 \); upper half 27, 30, 33, 38, 45 gives \( Q_3 = 33 \).
Step 3Summary: \( 12,\ 18,\ 24,\ 33,\ 45 \). IQR \( = 33 – 18 = 15 \).
Step 4Right whisker \( 45 – 33 = 12 \) is longer than left whisker \( 18 – 12 = 6 \), and the upper box \( 33 – 24 = 9 \) exceeds the lower box \( 24 – 18 = 6 \).

The upper half is more stretched out, so the distribution is right-skewed.

Time series and choosing the right graph

When you measure something repeatedly, plot it against time: a time-series plot. Look for a trend (long-run rise or fall), seasonality (a regular repeating cycle) and irregular jumps. A histogram throws away the order, so it cannot show growth.

For example, quarterly sales (in thousands of dollars) of 40, 55, 60, 80 in Year 1 and 46, 62, 68, 88 in Year 2 show an upward trend (every quarter is higher than a year earlier) and seasonality (a Q4 peak each year, falling back in Q1).

You have…Best display
One categorical variableBar chart or Pareto chart; pie for 2–3 categories
One quantitative variable, small datasetDotplot or stem-and-leaf
One quantitative variable, large datasetHistogram, frequency polygon; ogive for percentiles
A quantitative variable over timeTime-series plot
Quantitative variable across groupsSide-by-side boxplots
Categorical variable across groupsSegmented or grouped bar chart

How graphs mislead

Four tricks explain most misleading graphs:

  1. Truncated axis: starting the vertical axis above zero turns a small change into a dramatic one.
  2. Area or volume trick: scaling a picture in both width and height to show a one-dimensional number. Doubling a value then looks about four times bigger, because \( 2 \times 2 = 4 \).
  3. Cherry-picked window: zooming in on a flattering stretch of time and hiding the wider trend.
  4. Dual axes: two series on two different vertical scales, which can be set so the lines appear to move together.
Worked example

A bar chart’s y-axis runs from 90 to 100. January’s sales are 94 and February’s are 98, and February’s bar looks twice as tall. What was the real change?

Step 1Real percentage change: \( \dfrac{98 – 94}{94} \times 100 = \dfrac{4}{94} \times 100 \approx 4.3\% \)
Step 2Drawn heights above 90: \( 94 – 90 = 4 \) and \( 98 – 90 = 8 \)
Step 3Ratio of drawn heights: \( 8 \div 4 = 2 \), so the bar looks twice as tall.

A 4.3% rise has been made to look like a doubling.

Exam tipWhen asked to “describe the distribution”, always mention all four features (shape, centre, spread, outliers) with numbers from the graph, and name the skew after the tail. When asked to criticise a graph, name the trick, say what it exaggerates and give the fix, such as “start the y-axis at zero”.

Practise: 6 questions

Try each one on paper first, then check the worked answer.

EasyQ1. Twenty commuters travel by car (9), bus (5), train (4) and bicycle (2). Find the relative frequency of each as a percentage, and the cumulative frequencies in this order. Show worked answer
Divide each by 20 and multiply by 100:
Car \( 9 \div 20 = 45\% \); bus \( 5 \div 20 = 25\% \); train \( 4 \div 20 = 20\% \); bicycle \( 2 \div 20 = 10\% \). Total 100%.
Cumulative: \( 9,\ 14,\ 18,\ 20 \).
EasyQ2. A distribution has its peak on the left and a long tail stretching to the right. Name the skew, and give a real example of data with this shape. Show worked answer
Right-skewed, because skew is named after the direction of the long tail. Examples: incomes or house prices, where most values are modest but a few are very large.
MediumQ3. Fifty exam scores fall in classes 40–49: 4, 50–59: 9, 60–69: 15, 70–79: 13, 80–89: 7, 90–99: 2. Find the relative frequency of the 60–69 class and the cumulative frequency up to 69. Show worked answer
Relative frequency: \( 15 \div 50 = 0.30 \), or 30%.
Cumulative up to 69: \( 4 + 9 + 15 = 28 \).
So 28 of the 50 students scored 69 or below.
MediumQ4. A five-number summary is: min 41, Q1 58, median 66, Q3 72, max 95. Find the range and IQR, and comment on the shape. Show worked answer
Range \( = 95 – 41 = 54 \).
IQR \( = 72 – 58 = 14 \).
Right whisker \( 95 – 72 = 23 \) is much longer than left whisker \( 58 – 41 = 17 \), so the distribution is right-skewed.
MediumQ5. Using the age and device table above, what percentage of the 41–60 group prefer a laptop, and what percentage of all 300 people do? What does the comparison suggest? Show worked answer
Conditional: \( 70 \div 100 = 70\% \) of 41–60s prefer a laptop.
Marginal: \( 160 \div 300 \approx 53\% \) of everyone prefer a laptop.
The older group prefers laptops more than people overall, and the conditional percentages rise with age (38%, 52%, 70%), so device preference is associated with age.
ExamQ6. A company’s profit chart has a y-axis running from 9.8 million to 10.2 million dollars. A rival infographic shows a doubling of sales by drawing an icon twice as tall and twice as wide. Explain how each graphic misleads and how to fix it. [4 marks] Show worked answer
Mark scheme (1 mark each):
1. The profit chart uses a truncated axis: starting at 9.8 million makes tiny changes look dramatic.
2. Fix: start the y-axis at zero, or clearly mark a broken axis.
3. The icon is scaled in both dimensions, so its area grows by \( 2 \times 2 = 4 \) times, overstating a 2-times change.
4. Fix: use a plain bar chart where only height changes, so a doubling looks like a doubling.

Check yourself

1. Why do the bars of a histogram touch, while bar-chart bars have gaps?

Histograms show quantitative data on a continuous scale; bar charts show categories with no scale.

2. Which display best compares centre, spread and outliers across four groups at once?

Boxplots on one shared axis line up medians, box lengths and outliers for easy comparison.

3. Ice-cream sales rise every summer and fall every winter. This pattern is called…

A regular repeating cycle over time is seasonality, visible only on a time-series plot.

Your score: 0 out of 3

The short version

  • Frequency, relative frequency and cumulative frequency tables feed every graph.
  • Categorical: bar, Pareto, segmented bar. Quantitative: dotplot, stem-and-leaf, histogram, ogive, boxplot. Over time: time-series plot.
  • Describe shape, centre, spread and outliers, and name skew after the tail.
  • Read the axes first: truncated axes, area tricks, cherry-picked windows and dual axes all distort.
  • The full chapter in Statistics Made Simple has 52 practice questions with a complete answer key.

Further reading on EconTutorials

Older in-depth articles on this topic from our blog:

Want more questions like these, with worked answers?
The practice book has hundreds more, graded from easy to exam level.
Next: step 5 of 22Describing Data with Numbers