Lesson 14 of 36 · Data insight
Bar charts, line charts and histograms
Read values, changes and trends off bar charts, line charts and histograms, and avoid the slips that steep lines and cut-off axes invite.
About 20 minutes
In this lesson
- Turn a bar or line chart into the few numbers a question needs, using the axis scale.
- Tell the largest absolute change (the steepest segment) from the largest percent change.
- Read a histogram as counts in intervals, and say what it cannot tell you.
The idea
A chart is a table drawn as shapes
Bocconi lists tables, histograms and graphs among the displays that data insight questions are built on. Every chart encodes numbers, and the work is to decode the few you need, so start the way you start a table: title, units, what each bar or point stands for, and the scale of each axis. In this course charts are described in words or given as the table behind them; on screen you will be reading the same numbers off gridlines.
Reading values off the scale
Find the axis unit first: "thousands of visitors", "billions of euros", "%". Then read each value from the gridlines, estimating between them when you must. If a question's options are far apart, a rough reading is enough; if they are close, read carefully and check against a gridline. When an exact value is printed on a bar, use it rather than your estimate.
Bar charts: grouped and stacked
- Grouped bars stand side by side, one per series, for each category (exports and imports for each year). Compare bars within a group to compare series; compare the same series across groups to see change.
- Stacked bars pile the series on top of each other. The top of the stack is the total; the height of one segment is its top minus its bottom, not the level where it ends. Only the bottom segment can be read directly from the axis.
Line charts: level, change and slope
A line chart shows a quantity over time. Three readings matter:
- Level: the height of a point.
- Absolute change: the difference in height between two points. The steepest segment is the largest absolute change between neighbouring points.
- Percent change: the absolute change divided by the starting level.
The steepest segment is not always the largest percent change. A rise of 15 from 45 is a third; a rise of 20 from 70 is under 30%. When the question says "percent", "fastest growth" or "relative", compute or compare the fractions; when it says "by how much", subtract.
A cut-off axis exaggerates differences
If the vertical axis starts at 36% instead of 0%, a bar for 42% is six units tall and a bar for 38% is two, so the first looks three times the second. The real ratio is , about 1.1. Never compare bar heights by eye when the axis does not start at zero; read the values and compare those.
Two vertical axes
Some charts plot two series against different axes, left and right. Read each series against its own axis. Where two lines cross on such a chart means nothing about the two quantities being equal; it is an accident of the two scales.
Histograms: counts in intervals
A histogram groups values into intervals (classes) and shows how many fall in each. Its bars touch because the intervals do. What it can tell you:
- how many values fall in an interval or a run of intervals;
- the total count (add all bars);
- which interval holds the median: find where the middle position falls in the running total;
- the modal class, the interval with the tallest bar (when the classes have equal widths).
What it cannot tell you is anything about single values inside an interval. If 26 people commute 20 to 30 minutes, you do not know how many commute more than 25 minutes, nor the single most common commute time. A statement that needs this is not determined by the chart. Check the boundary rule too: a class "20–30" usually means at least 20 and under 30.
Working without a calculator
Write the values you read onto your sheet as a small table. Most chart questions then become the table questions of the previous lesson: a difference, a ratio against a benchmark, or a count of categories that meet a condition.
Worked examples
Example 1: Steepest segment or fastest growth?
A line chart shows a museum's monthly visitors, in thousands, from January to June. The points are: January 40, February 45, March 60, April 55, May 70, June 90. Between which two consecutive months was the increase largest in absolute terms, and between which was it largest in percent terms?
Write the month-to-month changes: , , , , . The largest absolute rise, the steepest segment, is May to June, thousand.
For percent change divide each rise by its starting level: February to March ; April to May ; May to June ; January to February .
Confirm the top two without decimals: compare with by cross-multiplying, , so is larger.
Answer
The largest absolute rise is May to June (20 thousand visitors); the largest percent rise is February to March (one third). Answering May to June to the percent question is the tempting slip: it is the steepest segment, but it starts from a higher level.
Example 2: What a histogram does and does not say
A histogram shows the commute times of 80 employees in 10-minute classes (each includes its lower end and excludes its upper end): 0–10 minutes, 6 people; 10–20, 18; 20–30, 26; 30–40, 16; 40–50, 10; 50–60, 4. What percent commute for at least 30 minutes? In which class is the median? How many commute for more than 35 minutes?
Check the total: , as stated.
"At least 30 minutes" is the classes 30–40, 40–50 and 50–60: people, and .
The median of 80 values sits between the 40th and 41st in order. Running totals: 6, then 24, then 50. Both the 40th and 41st fall in the 20–30 class.
"More than 35 minutes" splits the 30–40 class. The chart says 16 people are somewhere from 30 to under 40, but not how many are above 35, so the count is between and and cannot be fixed.
Answer
37.5% commute for at least 30 minutes; the median is in the 20–30 class; the number above 35 minutes cannot be determined from the histogram. Answering 22 by taking half of the 30–40 bar assumes the 16 people are spread evenly, which the chart does not say.
Example 3: A bar three times as tall
A bar chart shows the market shares of two firms: P has 42% and Q has 38%. The vertical axis starts at 36%. A reader says "P's bar is three times Q's, so P's share is three times Q's". Is that right?
Measure the bars from the axis start: P's bar shows points, Q's shows points, so P's bar is indeed three times as tall.
The quantities are the full shares, 42% and 38%. Their ratio is , a little over 1.1.
Answer
No. P's share is only about 10% larger than Q's, 4 percentage points more. The bar heights mislead because the axis is cut off at 36%.
Common traps
Steepest means fastest
The steepest segment of a line is the largest change in absolute terms. Percent growth also depends on the starting level, so a gentler segment that starts low can be the faster growth. Read whether the question asks "by how much" or "by what percent".
Comparing bar heights on a cut-off axis
When the axis does not start at zero, height ratios exaggerate differences. Read the values from the scale and compare those, never the lengths of the bars.
Reading a stacked segment from the axis
In a stacked bar only the bottom segment starts at zero. The height of a middle segment is its top minus its bottom; the level where it ends is a running total.
Assuming values inside a histogram class
A histogram class gives a count for the whole interval. Statements about part of an interval, a single most common value, or the exact median need data the chart does not show, so they cannot be decided from it.
Check yourself
Answer each question before you open its answer.
Question 1: Smallest trade deficit
Numerical critical reasoning
A grouped bar chart shows a small country's exports and imports of goods, in billions of euros, for five years. The values behind the bars are:
| Year | Exports | Imports |
|---|---|---|
| 2020 | 24 | 30 |
| 2021 | 30 | 32 |
| 2022 | 36 | 40 |
| 2023 | 33 | 42 |
| 2024 | 40 | 44 |
In which year was the trade deficit (imports minus exports) smallest?
- A.2020
- B.2021
- C.2023
- D.2024
Show the answer
Answer: B. 2021
Working
Deficit each year: 2020 ; 2021 ; 2022 ; 2023 ; 2024 . The smallest is 2 billion, in 2021.
Why the other options are wrong
- A. 2020 has the shortest bars, the lowest exports and imports, but the gap between them is 6 billion, not the smallest.
- C. 2023 has the largest deficit, billion; this reverses the question.
- D. 2024 has the highest exports, but imports are high too, leaving a deficit of 4 billion, more than 2021's.
What carries over
A question about the gap between two series needs the difference in every category, not the size of either bar.
Question 2: Fastest growth in exports
Numerical critical reasoning
Use the exports and imports chart from the previous question.
| Year | Exports | Imports |
|---|---|---|
| 2020 | 24 | 30 |
| 2021 | 30 | 32 |
| 2022 | 36 | 40 |
| 2023 | 33 | 42 |
| 2024 | 40 | 44 |
Between which two consecutive years did exports grow by the largest percentage?
- A.2020 to 2021
- B.2021 to 2022
- C.2022 to 2023
- D.2023 to 2024
Show the answer
Answer: A. 2020 to 2021
Working
Percent rises: 2020 to 2021 ; 2021 to 2022 ; 2022 to 2023 a fall; 2023 to 2024 , and since this is under 25% (about 21%). The largest is 25%, from 2020 to 2021.
Why the other options are wrong
- B. Exports rose by 6 billion, the same as the year before, but from a larger base of 30, so only 20%.
- C. Exports fell from 36 to 33 billion in this period; it is a decrease, not growth.
- D. This is the largest absolute rise, 7 billion, but from 33 that is , which is below a quarter.
What carries over
For percent growth, divide each rise by its own starting level; the biggest jump on the chart is not always the fastest growth.
Question 3: What the age histogram shows
Numerical critical reasoning
A histogram shows the ages of the 120 members of a sports club. Each class includes its lower end and excludes its upper end.
| Age | Members |
|---|---|
| 10–20 | 12 |
| 20–30 | 30 |
| 30–40 | 36 |
| 40–50 | 24 |
| 50–60 | 18 |
Which statement must be true?
- A.Exactly 18 members are 55 or older.
- B.The median age is at least 30 and under 40.
- C.More than half of the members are under 30.
- D.The single most common age is between 30 and 40.
Show the answer
Answer: B. The median age is at least 30 and under 40.
Working
The total is . The median of 120 values lies between the 60th and 61st. Running totals are 12, 42, 78, so both the 60th and 61st members fall in the 30–40 class: the median is at least 30 and under 40. The other statements need individual ages or are false ().
Why the other options are wrong
- A. The 18 members are somewhere from 50 to under 60; the histogram does not say how many of them are 55 or older.
- C. Members under 30 number , which is 35% of 120, not more than half.
- D. 30–40 is the tallest class, but the most common single age could lie in another class; the chart hides individual ages.
What carries over
A histogram fixes counts per class and the class of the median, never single values inside a class.
Key points
Read the axis unit and where the axis starts before comparing anything.
The steepest segment is the largest absolute change; percent change also needs the starting level.
In a stacked bar, a segment is its top minus its bottom.
A histogram gives counts per interval; values inside an interval cannot be determined.
All questions here are original and written for UniExamer; none comes from the test maker.
Written by the UniExamer team with AI assistance. Every worked example and check question was re-solved by an independent checker before publication. No subject expert has reviewed this lesson yet. If you find a mistake, write to support@uniexamer.com.