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Counting what we sorted

Statistics and data · level 1, lesson 2 of 5

A question to start

Look at this fruit. Are there more bananas or more grapes?

bananas grapes

Many people say "more bananas!" The bananas are big and they are spread all across the page. The grapes are small and squeezed into a corner.

Now count. Bananas: 1, 2, 3. Grapes: 1, 2, 3, 4, 5, 6.

There are more grapes. Six grapes, and only three bananas. Our eyes were tricked, because big things and spread-out things look like more. Counting does not get tricked. That is why people who work with data always count.

The idea

Count each group

In the last lesson we sorted ten buttons by colour. Now we count each group. Touch each button once and say one number for each button. The last number you say is how many are in the group.

  • Orange: 1, 2, 3, 4, 5.
  • Blue: 1, 2, 3.
  • Green: 1, 2.

Write the number under each group, so you do not have to count again. These numbers are called counts. A count tells you how many things are in a group.

Which group has the most?

Once every group has a number, you can compare them. The group with the biggest number has the most. The group with the smallest number has the fewest.

  • 5 is bigger than 3, and bigger than 2. So orange has the most.
  • 2 is smaller than 3, and smaller than 5. So green has the fewest.

"Most" and "fewest" are two of the first questions anyone asks about data. Which fruit do most children like? Which day had the fewest sunny hours? You answer them by sorting and then counting.

Comparing by lining up

There is a second way to compare two groups, and it works even before you can count very far. Line them up, one next to one.

Put the orange buttons in a row. Put the blue buttons in a row just under them, one blue under each orange. Start both rows at the same place, and space them out the same.

orange blue 2 left over

Each blue button has an orange partner, shown by the dashed lines. But two orange buttons have no partner. They are left over. So there are more orange buttons, and the leftovers tell you how many more: 2 more.

This is a very old and very clever idea. You do not need to know any numbers to see which row is longer, as long as the rows start at the same place and the things are spaced the same. In the next lesson you will see that a picture graph is exactly this: rows of things, lined up so they can be compared.

The same

Sometimes two groups have the same count. Then neither one has more. When we sorted the buttons by holes, we got 5 with two holes and 5 with four holes. If you line them up, every button has a partner and none are left over. We say the groups are the same size, or equal.

A group with nothing in it

Suppose your rule was colour, and someone asks "How many purple buttons?" There are none. The count for purple is zero, which we write 0. Zero is a real answer. It tells you something: no purple buttons at all. When you sort, a group can be empty, and its count is 0.

Count them all: the total

Add up all the counts and you get the total: the number of things altogether.

  • Colour: 5 and 3 make 8, and 2 more make 10.

Now count the whole pile, all at once, without sorting: 1, 2, 3, ..., 10. Also 10.

The two answers must be the same, because sorting does not make buttons appear or disappear. This is your total check. If the two numbers are different, something went wrong: a button was counted twice, or a button was missed. Then you count again, slowly. Checking the total is one of the most useful habits in all of statistics.

Counting with a list

When the groups get bigger, it helps to write the counts in a little list, like this:

Colour How many
orange 5
blue 3
green 2
purple 0
all together 10

This is a table. Each line is called a row. One row for each group, and a last row for the total. A table keeps your counts safe, so you can look at them again tomorrow.

Worked examples

Example 1. A box of crayons. A box holds these crayons: red, blue, red, yellow, blue, red, green, red, blue. How many of each colour? Which colour is there most of, and which fewest?

  1. Sort them: put each colour in its own pile.
  2. Count each pile. Red: 1, 2, 3, 4. Blue: 1, 2, 3. Yellow: 1. Green: 1.
  3. Most: red, with 4.
  4. Fewest: yellow and green are both fewest, with 1 each. When two groups tie, say both.
  5. Total check: 4 and 3 make 7, and 1 more is 8, and 1 more is 9. Count the crayons in the list: there are 9. Good, none missed.

Example 2. Lining up cars and trucks. Leo has a row of toy cars and a row of toy trucks. He lines them up, one truck under each car.

cars trucks 3 left over
  1. Each car has a truck partner, but some trucks have no car partner.
  2. So there are more trucks than cars.
  3. Count the trucks with no partner: 3 are left over. So there are 3 more trucks than cars.
  4. Check by counting: cars: 4. Trucks: 7. From 4, count up to 7: 5, 6, 7. That is 3 more. Both ways agree.

Example 3. The total check finds a mistake. Sara sorted 12 shells into three groups. She wrote: white 5, pink 4, brown 4.

  1. Add up her counts: 5 and 4 make 9, and 4 more make 13.
  2. But there were only 12 shells! 13 is one too many.
  3. So one group has a mistake. Sara counts each group again, slowly, touching each shell. She finds that pink has only 3. She had counted one pink shell twice.
  4. New counts: white 5, pink 3, brown 4. Total: 5 and 3 make 8, and 4 more make 12. Now it matches.

Notice that the total check does not tell you which group is wrong. It only tells you that something is wrong. That is still very useful: it tells you to look again.

Try it

  1. Count the groups: a bag holds 3 red marbles, 6 blue marbles and 4 yellow marbles. Which colour is there most of? Which colour is there fewest of?
  2. In the bag from question 1, how many marbles are there altogether?
  3. A row of 6 apples is lined up above a row of 4 pears, one pear under each apple. How many apples have no partner? Are there more apples or more pears, and how many more?
  4. You sort your toys into "has wheels" and "no wheels". You count 0 toys with wheels. What does that tell you?
  5. Ana sorted 10 leaves: green 4, yellow 3, red 4. Do the total check. What do you find?
  6. A group of children sorted themselves by the month they were born. January had 2 children, February had 2, and March had 2. Which month had the most?
  7. Here are some animals at a farm: cow, hen, hen, pig, cow, hen, duck, hen, pig. Sort them, count each group, and make a table with a total row.
Answers
  1. Blue has the most (6). Red has the fewest (3).
  2. 3 and 6 make 9, and 4 more make 13 marbles.
  3. 2 apples have no partner. There are more apples, 2 more. Check: from 4 count up to 6: 5, 6. That is 2.
  4. You have no toys with wheels. Every toy is in the "no wheels" group. A count of 0 is still an answer.
  5. 4 and 3 make 7, and 4 more make 11. But there were only 10 leaves. One leaf was counted twice (or put in two groups). Ana needs to count again.
  6. None of them. All three months have the same count, 2. They are equal, so no month has the most.

  7. Cow: 2. Hen: 4. Pig: 2. Duck: 1.

    Animal How many
    cow 2
    hen 4
    pig 2
    duck 1
    all together 9

    Check: there are 9 animals in the list, and 2 and 4 and 2 and 1 make 9. Hens are the most, ducks the fewest.

Common mistake

Deciding "more" by how big the pile looks. Three big bananas spread out can look like more than six small grapes squeezed together. Size and spreading out trick the eye. To compare fairly, count, or line the things up one next to one, starting at the same place.

Forgetting the total check. If your counts add up to more or less than the number of things you started with, something went wrong. Always add the counts and compare.

Where it leads