Data in the wild: pie charts, tricks, chance and projects

Data handling does not stop at the classroom. Newspapers print pie charts of the government budget; advertisers show bar graphs of how much whiter their toothpaste makes your teeth; weather apps say "70% chance of rain"; the Census of India counts over a hundred crore people. This layer takes your skills out into that world. You will learn a new kind of graph, learn to catch graphs that lie, learn the language of chance, and try some real projects and hard puzzles.

One note before you start: the family budget, the toothpaste brands, the coin experiment and the Pune electricity bills in this lesson are invented examples. The census figures are real, and the chapter says where they come from.

A pie chart (or circle graph) shows how a whole is split into parts. The circle is the whole; each slice (called a sector) is one part, and its size shows what fraction of the whole that part is. Pie charts answer questions like what share of the family budget goes on food?

The key fact: a full turn round the centre of a circle is 360° (see /topics/angles). So a part that is a quarter of the whole gets a quarter of 360°, which is 90°, a right angle. In general:

central angle of a sector = (part ÷ whole) × 360°.

A graph can be made from perfectly true numbers and still give a false impression. Sometimes this happens by accident, sometimes on purpose. Here are the most common tricks.

1. The broken (truncated) axis. Toothpaste brand A is preferred by 42% of people in a survey, brand B by 45%. Draw the bars from 0 and they look almost the same, which is honest. Now start the axis at 40%: bar A is 2 units tall and bar B is 5 units. Brand B suddenly looks two and a half times as popular, when the real difference is 3 percentage points.

2. Pictures scaled by area. A company's sales doubled, so the designer draws a coin twice as wide and twice as tall. But the bigger coin covers 2 × 2 = 4 times the area, and your eye reads area. Double the height and width of a 3D object and its volume looks 2 × 2 × 2 = 8 times bigger.

3. Cherry-picked time windows. Showing a share price only for the one month it rose, or rainfall only for the two driest years, can tell any story you like.

4. Uneven scales. Gaps on the axis labelled 0, 10, 20, 50, 100 at equal spacing squash the big values.

5. Missing context. "Crime reports doubled!" might mean 2 cases became 4, or that a new online reporting system made reporting easier. A total without the population behind it can mislead too: a big state will almost always have more of everything than a small one.

Data describes what has happened. Probability describes what might happen. The two are linked: we use data to estimate chances, and chances to predict data.

We describe chance with words:

• Impossible: it cannot happen. Rolling a 7 on an ordinary die.
• Unlikely: it could happen but probably will not. Snow in Chennai.
• Even chance: as likely as not. A fair coin landing heads.
• Likely: it will probably happen. Rain in Mumbai on a July afternoon.
• Certain: it must happen. The sun rising tomorrow morning.

Mathematicians put these on a probability scale from 0 (impossible) to 1 (certain), with ½ in the middle for an even chance.

When all outcomes are equally likely, the probability of an event is

P(event) = (number of outcomes where it happens) ÷ (total number of outcomes).

For a die, P(even number) = 3 ÷ 6 = ½, and P(a 5 or a 6) = 2 ÷ 6 = ⅓.

But probability does not promise exact results. Picture a Class 8 experiment in which 10 groups each toss a coin 20 times. Their numbers of heads might come out like this: 9, 12, 10, 11, 8, 10, 13, 9, 10, 11. (These figures are invented, but they are the kind of spread you really get.) Only three groups got exactly 10 heads. Altogether there were 103 heads in 200 tosses, a relative frequency of 103 ÷ 200 = 0.515, close to ½. The more trials you do, the closer the relative frequency usually gets to the probability.

Your home's electricity bill is a ready-made data set (see /topics/electricity). Most bills print the units (kilowatt-hours) used in recent months. Here is an invented but realistic year for a family in Pune (units per month, January to December): 180, 170, 190, 260, 340, 380, 300, 270, 250, 220, 190, 180.

The summaries: total 2,930 units; mean 2,930 ÷ 12 ≈ 244.2 units; median (the 6th and 7th ordered values are 220 and 250) 235 units; modes 180 and 190; range 380 − 170 = 210 units. The peak in May and June is the cooler, fans and fridge working hardest in the heat before the monsoon.

Once a decade, India attempts something extraordinary: to count every person in the country on the same day, and record their age, sex, education, work, language, religion, housing and more. The Census of India is one of the largest data collection exercises on Earth, done by lakhs of enumerators, usually school teachers, going house to house.

The census answers questions no other data can: How many children need schools in each district? Where are hospitals most needed? How many people speak each language? It is also the frame from which smaller sample surveys are chosen.

The last full count was in 2011. The next one, Census 2027, is India's 16th and its first digital one: houses are listed between April and September 2026, people are counted in February 2027, and the reference moment — the instant everyone is counted as of — is midnight on 1 March 2027.

Every time you use a phone app, play an online game, or even walk past a CCTV camera, data about you may be collected. Data about people is powerful, and power needs rules.

• Consent: people should know what data is being collected about them and agree to it. In a class survey, tell people what the survey is for, and let them say no.
• Only what you need: a survey about favourite snacks does not need names, phone numbers or home addresses.
• Anonymity: report results for the group ("9 children chose samosa"), not for individuals ("Riya chose idli"), unless they agree.
• Keep it safe: personal data such as marks, health information or identity numbers must be stored securely and shared only with the right people.
• Fairness: data can reflect unfairness. If a survey only asks people with smartphones, it leaves out those without them, and decisions based on it may ignore their needs.

India's Digital Personal Data Protection Act, 2023 sets rules for how organisations may collect and use people's personal data, including extra care for children's data.

These problems need more than a formula. Most can be cracked with one key idea: switch from averages to totals. Try each one before reading the solution.