KS3 · Maths

Constructing and interpreting bar charts and pictograms

Two pictograms can look identical yet show completely different numbers. The secret is the key: that small box most people skip.

Maths · Pictograms

One paw print stands for 4

Paw prints: 10/20. Paw prints drawn 2½. Key: 1 paw print = 4. Frequency = icons × key 10

Paw prints drawn2½Key: 1 paw print =4Frequency = icons × key10

Cover the key and you could still say which row is longest, but not how many. With the key, every part of a paw print turns into a number.

Watch out: The key never moves. You change the number of paw prints, and the frequency follows.

From a messy list to a bar chart

Problem

Fifteen pupils each name the pet they own: dog, cat, dog, fish, cat, dog, rabbit, cat, dog, rabbit, fish, dog, hamster, cat, rabbit. Turn the list into a bar chart.

Maths · Check the chart

One of Priya’s steps makes her pictogram impossible

Priya is drawing a pictogram of the dogs she saw in the park. Her dog row has 2.5 icons. Read her plan: which line goes wrong?

Priya’s plan — which line goes wrong?

I’ll use a dog’s-head icon, because it matches my data.
For my key I’ll say 1 icon = 5 dogs.
So my dog row of 2.5 icons shows 2.5 × 5 = 12.5 dogs.
I’ll label the row “12.5 dogs”.

Chart reader

Reading bar charts: one marked value is enough

View: Scale from one value. Showing 1 layer: Vertical bars

View

tap a view ↓

Only 48 is marked, and it sits on the 4th grid line. So each line is worth 48 ÷ 4 = 12, and the lines read 12, 24, 36, 48, 60, 72. Flapjack reaches the 5th line, so 60: the most common. Scone reaches the 2nd line, so 24: the least common.

Switch the view to read four different bar charts.

Maths · Before you draw

Does the order of the bars matter?

Sort each data set: does it need its bars in a relevant order, or can they go in any order?

Still to sort

Needs a relevant order (0)

The data has a natural order the bars should follow.

Order doesn’t matter (0)

No relevant order, so the bars can go in any order.

6 of 6 still to sort.

Maths · Why the gaps?

Your friend asks why you left gaps between the bars

You are drawing a bar chart of the types of farm animal on a farm, and your friend points at the gaps between your bars.

Which idea is closest to what you think right now?

The gaps are just decoration, to make the chart look neater.The gaps don’t matter. As long as the heights are right, the bars can be any width.There shouldn’t be gaps. The bars should touch, like one solid shape.Each bar is its own category and nothing exists between them, so the bars are separate, equally wide and equally spaced.

How sure are you?

I'm sureI think soI'm guessing

The gaps are just decoration, to make the chart look neater.

This is a common misconception

They have a job. The data has nothing in between the categories: you have 7 shirts or 8 shirts, never 7.2, and nothing exists between goat and horse. So every category gets its own separate bar.

The gaps don’t matter. As long as the heights are right, the bars can be any width.

This is a common misconception

The heights carry the frequencies, yes. But the bars are the same width, with equal gaps between them, so that they read as separate categories.

There shouldn’t be gaps. The bars should touch, like one solid shape.

This is a common misconception

Touching bars would suggest something sits between one category and the next. Between goat and horse there is nothing, so the bars stand apart.

Each bar is its own category and nothing exists between them, so the bars are separate, equally wide and equally spaced.

This matches the evidence

That’s the idea. Separate categories get separate bars, the same width, with equal gaps.

Gaps between the bars say “nothing exists in between”. Same-width bars with equal gaps show separate categories.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Reading them, drawing them, and spotting when they mislead

What you need to know

Frequency just means how many: how many times something happens, or how many people or things have a property.
A pictogram shows frequency with pictures (icons), and it must come with a key saying what each icon is worth.
No key, no numbers: you can still compare rows (most, least, double) but you can’t read a frequency.
Have a goYour mate squints at a pictogram with no key: “Dogs have more icons, so there are 12 dogs.” What can they actually say?
Only that dogs beat the other rows. The “12” needs the key.
Without the key you can compare categories, but you can’t read a frequency.
To read a pictogram, multiply the number of icons by the key. Part of an icon is part of the key’s value.
You can work out a missing key by dividing the frequency by the number of icons that show it.
Have a goA row of 4 icons shows 24 books. What is each icon worth?
6 books each (24 ÷ 4).
The key is found by dividing the frequency by the number of icons; multiplying would take you the wrong way.
A good key gives whole-number frequencies and suits the size of the data.
Have a goA row has 3.5 icons. Which is the better key: 1 icon = 1 pet, or 1 icon = 2 pets?
1 icon = 2 pets, which gives 3.5 × 2 = 7.
A key of 1 gives 3.5 pets, and a key must give whole-number frequencies.
Keep icons simple enough to split, the same size throughout, and evenly spaced.
A bar chart shows frequency as bar height, read against the frequency axis, which goes up in equal jumps.
Bars get equal widths and equal gaps, because nothing exists between categories: you have 7 shirts or 8, never 7.2.
Pictograms show context more easily; bar charts suit non-rounded frequencies and larger data sets. Which is better depends on the job.

The big picture

Pictograms and bar charts both show frequency visually. A pictogram needs a key, and frequency = icons × key. A bar chart needs a frequency scale that goes up in equal jumps, with separate bars of equal width and equal gaps. Build them carefully and they tell the truth.

Key points

1Frequency = icons × key, and half an icon is half the key.

2No key, no numbers: a pictogram without its key only lets you compare categories.

3On a bar chart, read bar height against a scale that goes up in equal jumps. One marked grid value fixes the rest.

4Bars are the same width with equal gaps. Put them in order only when the data has a relevant order.

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Worked example

Problem

A pictogram of books borrowed shows 3 icons for Monday, which stands for 150 books. Tuesday’s row shows 2.5 icons. How many books were borrowed on Tuesday?

⚠ Watch out

Treating the number of icons as the frequency: counting 4½ paw prints and writing 4½ instead of multiplying by the key. The bar chart twin is reading which grid line a bar reaches (the 4th) instead of the value on that line (48).

🧠

Memory hook

Icons × key = frequency. Bar height + scale = frequency.

✓

Check yourself

Without looking back: why can’t you read a frequency from a pictogram with no key, and what could you still say about it?

Flashcards

(15)

What is frequency?

The number of times something happens, or the number of people or things with a particular property.

What does a pictogram’s key tell you?

What each icon is worth. Without it you can compare categories but you can’t read frequencies.

How do you read a frequency from a pictogram?

Number of icons × the key. Part of an icon is part of the key’s value.

Key: 1 icon = 8 people. A row has 1.5 icons. How many people?

1.5 × 8 = 12 people.

How do you find an unknown key?

Divide a frequency by the number of icons that show it. Example: 90 ÷ 3 = 30 per icon.

What two things should a key do?

Give whole-number frequencies, and be sensible for the size of the data.

What makes a good pictogram icon?

Simple enough to divide without interpretation, the same size throughout, and evenly spaced.

How do you tally in groups of five?

Four strokes, then a bar across them for the fifth. Count up the groups to get each frequency.

How do you read a frequency from a bar chart?

Take the bar’s height and read it against the frequency scale. You need both the bar and the scale.

A chart’s 3rd grid line is marked 30. What does each grid line go up in?

30 ÷ 3 = 10, so the lines read 10, 20, 30, 40 and so on.

What goes on the axes of a bar chart?

A scale going up in equal jumps with the label “frequency”, and the other axis labelled with the context.

Why do bar charts have gaps between the bars?

Nothing exists between the categories (7 shirts or 8, never 7.2), so the bars are separate, equally wide, with equal gaps.

When should bars go in a particular order?

When the data has a relevant order, like shoe sizes. For pets or movie genres the order doesn’t matter.

Comparison bar chart or stacked bar chart?

Comparison: groups of bars inside each category. Stacked: the group bars piled up so the height shows the total. Both need a key.

Pictogram or bar chart?

Pictograms show context and look appealing. Bar charts suit non-rounded frequencies and larger data sets. It depends on the job.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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One of Priya’s steps makes her pictogram impossible

Does the order of the bars matter?

Your friend asks why you left gaps between the bars

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