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Pictographs with a key

Statistics and data · level 2, lesson 4 of 6

A question to start

Four grades in a school counted the books they read in one month, 160 books in all. You want to make a picture graph, with one sticker for each book read. But Grade 3 alone read 60 books. That is 60 stickers in one row! The row would run right off the page.

What if one sticker could stand for more than one thing? One sticker for every 10 books. Then Grade 3 needs only 6 stickers. That small change, a picture that stands for a group, is the idea of this lesson. It is how newspapers and posters show big numbers with a few pictures.

The idea

A picture for a group

A pictograph is a picture graph where each picture can stand for more than one thing. How many? That is written in a little box called the key (some people call it the legend).

Here is a pictograph of favourite pizza toppings in a class of 26 children. Read the key first!

Favourite pizza topping cheese pepperoni veggie mushroom = 2 children

The key says: one whole circle stands for 2 children.

Reading a pictograph

To read a row, count its pictures and use the key.

  • Cheese: 4 whole circles. Each is 2 children. Count by 2s: 2, 4, 6, 8. 8 children.
  • Pepperoni: 5 whole circles and a half circle. Count by 2s for the whole ones: 2, 4, 6, 8, 10. A half circle is half of 2, which is 1. So 10 and 1 make 11 children.
  • Veggie: 2 whole circles and a half: 2, 4, then 1 more. 5 children.
  • Mushroom: 1 whole circle: 2 children.

Check the total: 8 and 11 make 19, and 5 make 24, and 2 make 26. That matches the 26 children.

The key is everything. If you forgot to read it and counted each circle as 1 child, you would say cheese has 4, and you would be wrong. Always read the key first.

Why half a picture means half the key

If one circle is 2 children, then cutting the circle in half cuts the 2 children in half: 1 child. In the same way:

  • If one picture is 10, half a picture is 5.
  • If one picture is 4, half a picture is 2.
  • If one picture is 2, half a picture is 1.

Half a picture always means half of the key.

Making a pictograph

To make a pictograph from a table:

  1. Choose a key. Look at the numbers. Choose a key that makes the rows a good length (not too long, not too short) and that works with the numbers.
  2. Work out each row. How many groups of the key fit into each number? Use whole pictures for the whole groups, and a half picture if there is half a group left.
  3. Draw the pictures, all the same size, lined up from the same starting line, with labels, a title and the key.

For the pizza data with a key of 2:

Topping Children Groups of 2 Pictures
cheese 8 4 4 whole
pepperoni 11 5 and 1 left 5 whole and a half
veggie 5 2 and 1 left 2 whole and a half
mushroom 2 1 1 whole

Choosing a good key

Not every key works with every set of numbers.

With a key of 2, you can show any whole number: even numbers with whole pictures, odd numbers with one half picture at the end.

With a key of 10, half a picture is 5. So you can show 10, 20, 30, ... with whole pictures and 5, 15, 25, 35, ... with a half at the end. But how would you show 23? It is 2 whole pictures and 3 more, and 3 is not half of 10. You would need a strange little piece of a picture, about three tenths of a circle, which is very hard to draw and very hard to read.

So choose a key that fits your numbers:

  • Numbers that are all multiples of 5 or 10 (like 30, 45, 60, 25): a key of 10 works well.
  • Small numbers, some odd and some even: a key of 2 works well.
  • A key of 5: whole pictures show 5, 10, 15, ...; but half a picture would be 2 and a half, which is a strange amount of children. So a key of 5 works best when every number is a multiple of 5.

Also choose a key that keeps the rows a sensible length. With the school books data, a key of 1 gives a row of 60 pictures, much too long. A key of 10 gives a row of 6. Just right.

Pictograph or bar graph?

A pictograph is friendly and fun to look at, and a key lets it show big numbers. But reading half pictures is not very exact, and some numbers do not fit the key. A bar graph with a scale can show any number. Use a pictograph to make a point quickly and clearly; use a bar graph when you need exact values.

Worked examples

Example 1. Reading with a key of 10. Here is the school's book reading pictograph.

Books read in one month Grade 1 Grade 2 Grade 3 Grade 4 = 10 books
  1. Read the key: one circle is 10 books. Half a circle is 5 books.
  2. Grade 1: 3 circles. 10, 20, 30: 30 books.
  3. Grade 2: 4 circles and a half. 10, 20, 30, 40, and 5 more: 45 books.
  4. Grade 3: 6 circles: 60 books.
  5. Grade 4: 2 circles and a half: 20 and 5, 25 books.
  6. Altogether: 30 and 45 make 75, and 60 make 135, and 25 make 160 books.
  7. How many more books did Grade 3 read than Grade 1? From 30 up to 60 is 30 more. On the graph, that is 3 more circles, and 3 circles are 30 books. You can answer "how many more" by counting the extra pictures and using the key.

Example 2. Making a pictograph. Make a pictograph of this table, with a key of 2.

Fruit Children
apple 6
banana 9
grapes 4
  1. Apple: 6 is three groups of 2: 3 circles.
  2. Banana: 9 is four groups of 2 (8) and 1 left: 4 circles and a half.
  3. Grapes: 4 is two groups of 2: 2 circles.
  4. Draw them in rows, with labels, a title, and the key "one circle = 2 children".

Example 3. Choosing the key. A survey of 200 people asked about their favourite season: winter 40, spring 50, summer 80, fall 30. Choose a key.

  1. With a key of 1, summer would need 80 pictures. Too many.
  2. With a key of 2, summer needs 40 pictures. Still too many.
  3. With a key of 10: winter 4, spring 5, summer 8, fall 3. Every number is a multiple of 10, so all whole pictures. The longest row has 8 pictures. That fits nicely.
  4. (A key of 20 would also work: winter 2, spring 2 and a half, summer 4, fall 1 and a half. But the halves make it harder to read, so 10 is better.)

Try it

  1. In the pizza pictograph, how many more children chose pepperoni than mushroom?
  2. A pictograph has the key "one star = 5 votes". A row has 4 stars. How many votes?
  3. With the key "one circle = 4 children", what does half a circle stand for? What do 3 circles and a half stand for?
  4. Make the rows for a pictograph with a key of 2: red 10, blue 7, green 3.
  5. A pictograph about pets uses "one paw = 10 pets". The cat row has 3 paws and a half. How many cats?
  6. Why would "one picture = 5 children" be a bad key for the numbers 7, 12, 9?
  7. Choose a good key for 20, 35, 50, 15, and say how many pictures each row needs.
  8. Jamal reads a pictograph without looking at the key. He says "the dog row has 6 pictures, so 6 people chose dog". The key says one picture is 3 people. What is the right answer?
Answers
  1. Pepperoni 11, mushroom 2: from 2 up to 11 is 9 more.
  2. 5, 10, 15, 20: 20 votes.
  3. Half a circle is half of 4: 2 children. 3 circles and a half: 4, 8, 12, and 2 more: 14 children.
  4. Red: 10 is five 2s, 5 circles. Blue: 7 is three 2s and 1 left, 3 circles and a half. Green: 3 is one 2 and 1 left, 1 circle and a half.
  5. 10, 20, 30, and half of 10 is 5: 35 cats.
  6. None of them is a multiple of 5, or half of one. 7 is one 5 and 2 more; 12 is two 5s and 2 more; 9 is one 5 and 4 more. You would need awkward bits of pictures that nobody could read. A key of 1 (or 2, with halves) would be better.
  7. A key of 10 works: 20 is 2, 35 is 3 and a half, 50 is 5, 15 is 1 and a half. (A key of 5 also works, with whole pictures only: 4, 7, 10, 3.)
  8. Each picture is 3 people: 3, 6, 9, 12, 15, 18. 18 people chose dog.

Common mistake

Forgetting the key. Counting the pictures is only half the job. Each picture stands for the number in the key, so always read the key first and count by that number.

Reading half a picture as 1. Half a picture is half of the key. With a key of 10, half a picture is 5, not 1.

Where it leads