Interpreting and questioning graphs

Drawing a graph is half the battle. Reading one , really reading it , is the other half. Every day you will see graphs in newspapers, on TV, in textbooks, and online. A careful reader does not just glance; they question.

Concept

To interpret a graph means to look at it and pull out the information it carries. Three questions are always worth asking:

1. What is being shown? What are the categories, the axes, the scale, the title?
2. What does the graph tell us? Which category is biggest, smallest? Are there trends (rising, falling, steady)?
3. What does the graph not tell us? What is missing? Could the data be biased? Could the scale mislead?

A few common tricks that misleading graphs use:

• Cut axis. If the vertical axis starts at $80$ instead of $0$, small differences look huge. A bar at $90$ looks ten times taller than a bar at $81$, but the real difference is tiny.
• Unequal scale. If the spacing between numbers on an axis is uneven, the graph distorts the data.
• Cherry-picked categories. Choosing only "good" months or only "loud" customers gives a one-sided picture.
• Pictures of different sizes. In a pictograph, if the pictures themselves grow in size (not just in count), the bigger pictures lie about the count.

When you read a chart in a newspaper, do this little checklist:

• Does it start at zero?
• Are the units and the scale clearly written?
• Is the source mentioned (who collected the data)?
• Are the categories fair, or is something missing?

Asking these questions does not mean every chart is wrong. It just means we read graphs the way a careful person reads a newspaper article: alert and curious.

A good graph honestly conveys what the data says. A bad graph hides, exaggerates, or tricks. Mathematics gives you the tools to spot the difference.

Worked examples

Example 1. A bar graph shows daily sales of a shop. The vertical axis starts at $\rupee 90$ and goes to $\rupee 100$. Bar heights: Mon $\rupee 91$, Tue $\rupee 97$, Wed $\rupee 100$. Why is this graph misleading?

• The differences (a few rupees) are tiny compared to the absolute sales.
• Because the axis starts at $90$, the bar for Wednesday looks $10$ times taller than for Monday , visually exaggerating a trivial difference.
• Honest graph: start at $0$.

Example 2. From a bar graph of monthly book sales, with months Jan–Jun shown but Jul–Dec missing, what is one question a careful reader should ask?

• "Why are only the first six months shown?" Maybe the rest of the year had poor sales, deliberately hidden.

Example 3. A pictograph claims to compare ice cream sales. For shop A, $1$ small cone = $50$ ice creams. For shop B, the picture is a big cone that is $4$ times the size of the small one. The label says "Shop B sells more!" Why is this misleading?

• If both cones each count as one unit, the bigger cone is just a fancier picture. But to a casual eye it suggests Shop B sells $4$ times more.
• In pictographs, every picture must be the same size.

Example 4. A graph shows that "people who eat ice cream are more likely to drown". The reader sees a positive bar relationship. What's the explanation?

• Both increase during hot summer months. They are correlated but neither causes the other. Don't confuse correlation with causation.

Try it yourself

1. A bar graph shows test scores: A $80\%$, B $82\%$, C $79\%$. The axis starts at $75\%$. Is this graph trustworthy? Why or why not?
2. A pictograph compares the number of children in $3$ schools. School A has $5$ pictures of a small child; School B has $5$ pictures of a giant child. What's wrong?
3. From a bar graph of rainfall: Jan $0$ mm, Feb $0$ mm, Mar $0$ mm, Apr $5$ mm, May $20$ mm. What's the trend?
4. A bar graph in a magazine claims petrol prices went "shockingly up". The actual numbers: $\rupee 100, \rupee 101, \rupee 101.5$ over three months. Is "shockingly" honest?
5. What three things must always appear on a clean bar graph?
6. If a graph has no scale on its vertical axis, what should a reader assume?
7. Critical thinking: A graph compares school exam results of two schools, only for the top $10$ students in each. What is being hidden?

Activity

Graph hunt. Over a single day, find $3$ different bar graphs or pictographs , in a newspaper, on a TV news channel, or in your textbook. For each, write down: (1) what data is shown, (2) one true thing the graph tells you, (3) one thing the graph does not tell you or could mislead about. Share with a classmate and discuss whether the graphs were fair.