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Play: Counting, sorting and charts

Statistics and data · step 1 of 5

A question to start

Your class wants to choose a class pet. Someone shouts "Everybody wants a dog!" Someone else says "No, most people want a cat!" They cannot both be right. How could you find out what the class really thinks, so that nobody can argue?

You could ask everybody, write down every answer, and count. That is exactly what people who work with data do, whether they are asking a class about pets or counting dust storms on Mars. The answers you collect are called data, and the math of collecting, sorting and understanding data is called statistics.

The idea

Step 1: ask a clear question

A survey is a set of questions you ask lots of people. A good survey question has clear answers to choose from. "Which pet do you like best: dog, cat, fish, rabbit, or no pet?" is a good question. "Are pets nice?" is not, because everyone will answer in a different way and you will not be able to count the answers.

Step 2: keep a tally

As people answer, you make a mark for each one. These marks are called tally marks. To make counting easy, you draw four lines and then cross them with the fifth, so every bundle is 5.

= 5 and 2 more = 7

Tally marks are fast to write while people are talking, and bundles of 5 are fast to count afterwards: 5, 10, 15, and then the single lines.

Step 3: make a table

When the survey is finished, count each tally and write the totals in a table. Here are the answers from a class of 20 children.

Pet Tally How many
Dog five and two 7
Cat five 5
Fish three 3
No pet four 4
Rabbit one 1

A good habit: add up the last column. 7 and 5 make 12, then 15, then 19, then 20. There are 20 children in the class, so nobody was missed and nobody was counted twice.

Step 4: draw a bar chart

A table is useful, but a picture lets you see the answer at a glance. In a bar chart, each answer gets a bar, and the taller the bar, the more people chose it.

013 57 DogCatFish No petRabbit

Every bar starts at zero, at the bottom line. The faint lines across the chart mark 1, 3, 5 and 7, and the numbers on the left tell you which is which. So the "No pet" bar, which stops halfway between 3 and 5, stands for 4.

Step 5: answer the question

Now nobody needs to argue. Dog has the tallest bar, so dog is the most popular answer. But look again: 7 children chose a dog, and 13 children did not. So "Everybody wants a dog" was wrong, and "most people want a cat" was wrong too. The honest sentence is: dog was chosen more often than any other pet, by 7 of the 20 children.

The answer that turns up most often has a name. It is the mode. In this survey the mode is "dog".

What is typical?

Another question statisticians love is "what is a typical one?" Try this with your friends. Ask everyone to stand in a line from shortest to tallest. Now find the person in the very middle of the line. That person's height is a good answer to "how tall is a typical person here?", because half the line is shorter and half is taller. This middle value is called the median. You will meet it again in the next step.

Worked example

The question: A teacher asks 12 children how many brothers and sisters they have. The answers are:

1, 0, 2, 1, 1, 3, 0, 1, 2, 1, 0, 1

Which answer is most common, and how many children have at least two brothers and sisters?

  1. Sort the answers into groups. Go through the list once, making a tally for each answer.
    • 0: three marks
    • 1: six marks
    • 2: two marks
    • 3: one mark
  2. Check the total. 3 and 6 make 9, and 2 make 11, and 1 makes 12. That matches the 12 children.
  3. Find the most common answer. The biggest group is "1", with 6 children. So the mode is 1 brother or sister.
  4. Answer the second question. "At least two" means two or more, so we add the "2" group and the "3" group: 2 and 1 make 3 children.

Notice how much easier the questions became once the answers were sorted. A jumble of numbers tells you very little; a sorted table tells you a story.

Try it

  1. Count these tally marks: three full bundles of five and then four single lines.
  2. In a survey about favourite colours, red got 6 votes, blue got 9, green got 4 and yellow got 6. Which colour is the mode? How many people were asked?
  3. Using the colour survey, how many more people chose blue than green?
  4. Five children line up from shortest to tallest. Which child in the line has the median height: the 1st, 2nd, 3rd, 4th or 5th?
  5. Sam draws a bar chart of the pet survey, but the bars start halfway up the page instead of at zero. Why might the chart trick people?
Answers
    1. Three bundles of five make 15, and four more make 19.
  2. Blue, with 9 votes. People asked: 6 and 9 make 15, and 4 make 19, and 6 make 25 people.
  3. 5 more. Blue had 9 and green had 4, and from 4 up to 9 is 5.
  4. The 3rd child. Two children are shorter (1st and 2nd) and two are taller (4th and 5th), so the 3rd is right in the middle.
  5. If the bars do not start at zero, a bar that is only a little bigger can look many times bigger. The rabbit bar might even vanish, as if nobody chose it. Bars must start at zero so that their heights are fair to compare.

Common mistake

Thinking the biggest group is "most people". The tallest bar only tells you which answer was chosen more often than any other. In the pet survey, dog was the most popular pet, yet most children (13 of 20) did not choose a dog. Always compare the biggest group with the total before you say "most".

Where it leads