Counting what matters: meeting data

Suppose your teacher says, "This Friday we will have a class party. Which snack should we order?" Thirty hands shoot up and thirty voices shout thirty answers. Samosa! Idli! Fruit! Samosa again! How do you decide?

You could guess. You could order whatever your best friend likes. Or you could ask everyone, write the answers down, count them, and let the numbers decide. The moment you do that, you are doing data handling, one of the most useful kinds of mathematics there is.

In this lesson you will learn to ask good questions, collect answers, organise them neatly, draw them as pictures, and finally squeeze a whole list of numbers into one number that describes it.

Data is a collection of facts, such as numbers, words or measurements, gathered to answer a question. One fact on its own ("Riya likes idli") is a single piece of information. Put thirty of them together ("what does each child in Class 6B like?") and you have data.

Data is everywhere in India once you start looking:

• The scoreboard at a cricket match lists runs, balls, fours and sixes for every batter.
• The weather report on the news gives the highest and lowest temperatures in Delhi, Mumbai, Chennai and Kolkata, and tells you how many millimetres of rain fell.
• Your report card holds your marks in every subject.
• A kirana shop owner notes how many packets of milk, bread and biscuits are sold each day, so she knows how much to buy tomorrow.
• The electricity bill at home shows how many units your family used this month and in past months.
• Every ten years or so, the Census of India counts every single person in the country: about 121 crore people in 2011.

Every good piece of data handling starts with a question, not with numbers. "Which snack does our class like most?" "How much rain fell in Mumbai in July?" "How tall are children in Class 5?" The question tells you what to collect, from whom, and how to write it down. A vague question gives messy, useless data. A clear question gives data you can actually use.

There are three everyday ways to collect data.

1. Ask people (a survey). You might ask each classmate one question out loud, or hand out a questionnaire, a sheet with a few questions and boxes to tick.
2. Watch and count (an observation). Stand at the school gate for ten minutes and count how many children arrive by bus, cycle, auto-rickshaw or on foot. Nobody needs to answer anything; you just record what you see.
3. Measure (an experiment or measurement). Measure the heights of your friends with a tape, or put a jar outside to measure how much rain falls each day.

Often you will use data somebody else has collected, such as rainfall figures from the India Meteorological Department, or census numbers. That is still data; you just did not collect it yourself.

Here are the answers from Class 6B's snack survey, written down in the order the children answered:

Fruit, Samosa, Idli, Biscuits, Samosa, Biscuits, Idli, Samosa, Idli, Samosa, Biscuits, Fruit, Fruit, Idli, Fruit, Fruit, Poha, Samosa, Poha, Fruit, Poha, Idli, Samosa, Poha, Samosa, Samosa, Poha, Idli, Samosa, Idli.

This is called raw data: data exactly as it was collected, before anyone has sorted or counted it. Quick: which snack won? It is hard to tell. Your eyes jump around and you lose your place. We need a better way to count.

The oldest trick for counting is the tally mark. Go through the list once, from start to finish. For each answer, draw one short stroke | next to that snack's name. When you reach the fifth stroke, draw it across the other four, making a little gate: 卌 (written in books as four upright strokes crossed by a fifth). Gates make counting quick, because you can count in fives: 卌 卌 || is 5 + 5 + 2 = 12.

Why go through the list only once, crossing off each answer as you tally it? Because then you never miss one or count one twice. Shepherds, shopkeepers and cricket scorers have used tallies like this for thousands of years.

A pictograph shows data using small pictures or symbols. Each symbol stands for a certain number of things, and a key tells you how many. Without the key, a pictograph is useless: one mango symbol could mean 1 mango or 100.

Mr Rao runs a fruit stall in Ratnagiri. He counted the mangoes he sold each day for a week. Here is his data as a pictograph, where ● = 10 mangoes and ◐ = 5 mangoes (half a symbol means half the key).

Drawing lots of little pictures takes time, and big numbers need too many symbols. A bar graph does the same job more neatly. Each category gets a bar, and the height (or length) of the bar shows the number.

A bar graph has:

• two axes: a line along the bottom (for the categories) and a line up the side (for the numbers);
• a scale on the number axis, such as 1 small square = 1 child, or 1 square = 10 mangoes;
• bars of equal width with equal gaps between them, so only the height carries meaning;
• a title and labels on both axes, so anyone can tell what it is about.

Here is Class 6C's favourite-sport data drawn sideways as a bar graph made of blocks, where each █ is one child.

Reading a bar graph is quick. The longest bar (cricket, 12) is the most popular; the shortest (kho-kho, 4) the least. You can compare too: cricket is 12 − 7 = 5 more than football, and exactly three times kho-kho, because 4 × 3 = 12. All the bars together add up to 12 + 7 + 5 + 8 + 4 = 36, the whole class.

Tables and graphs show all the data. But often we want one number that describes the whole group. "How tall are children in Class 5?" "About 130 cm." "How much rain does Chennai get in November?" "About 350 mm." That one number is called an average, or a typical value.

There is more than one way to choose it. Let's meet the first: the fair share.

Asha, Bala, Chitra, Dev and Esha have 3, 7, 4, 6 and 5 marbles. It does not seem fair: Bala has more than twice as many as Asha. So they put all the marbles in a heap and share them out equally.

• Heap: 3 + 7 + 4 + 6 + 5 = 25 marbles.
• Shared among 5 children: 25 ÷ 5 = 5 marbles each.

That fair share, 5, is called the mean of the data. Another way to see it: imagine the marbles in five towers of heights 3, 7, 4, 6 and 5. Take marbles off the tall towers and put them on the short ones until every tower is the same height. Bala gives 2, Dev gives 1; Asha gets 2 and Chitra gets 1. Every tower levels out at 5.

Seven children line up for a class photo: their heights are 132, 128, 140, 135, 126, 138, 131 cm. Ask them to line up from shortest to tallest: 126, 128, 131, 132, 135, 138, 140. The child standing exactly in the middle of the line, with 3 children on each side, is 132 cm tall.

That middle value is called the median. To find it: put the data in order, then pick the middle one.

Now the shoe shop. Twelve children in a class wear these shoe sizes: 4, 5, 5, 6, 5, 4, 7, 5, 6, 4, 5, 6. A shopkeeper stocking school shoes does not care about the fair share; nobody can wear size 5.2. She wants to know which size is most common.

Counting: size 4 appears 3 times, size 5 appears 5 times, size 6 appears 3 times and size 7 once. The most common value is size 5. The most common value is called the mode. (Think mode = most.)

Finally, how spread out is the data? In the photo line, the tallest child is 140 cm and the shortest 126 cm. The difference, 140 − 126 = 14 cm, is the range. A small range means the values are bunched close together; a big range means they are spread far apart.

In Delhi an average day in May swings from about 26 °C at dawn to 40 °C in the afternoon: a range of 14 degrees. A rainy day in Mumbai in July swings far less, from about 25 °C to 30 °C: a range of just 5.

You now have four numbers that sum up data. They answer different questions:

• "If we shared it all equally, how much each?" → mean.
• "What is the value in the middle?" → median.
• "What is most popular or most common?" → mode.
• "How far apart are the smallest and biggest?" → range.

For the class-party snacks, the only sensible choice is the mode: samosa, chosen by 9 children. You cannot find the "mean snack"! You cannot add up samosas and idlis. The mode is the only average that works for words as well as numbers.

Data becomes exciting when it describes something real. Mumbai is famous for its monsoon. The table shows how much rain Mumbai gets in an average month, using the India Meteorological Department's long-term averages for 1991–2020 at the Mumbai (Santacruz) station, rounded to the nearest millimetre.

Look at the pattern: almost nothing from November to May, then a flood from June to September. July alone brings about 920 mm, which is more than 90 cm — taller than most classroom desks! The whole year adds up to about 2,502 mm.

Wherever you go now, try spotting data everywhere: the petrol price board, the cricket commentary ("his average this season is 48"), the school attendance register, the calendar on which your grandmother marks the days the milk came. Each one is a list of values waiting for a question.