6.1 - Tables, pictograms, bar charts and pie charts

6.1 - Tables, pictograms, bar charts and pie charts

Turn a collection of answers into a clear table or chart. You will construct and read pictograms, bar charts and pie charts, and decide what each display can tell you.

Two-way tables

Can a larger pie-chart sector represent fewer pupils? We will return to that puzzle after organising a survey of 4040 pupils.

You already know

You can add and subtract counts, find a fraction of a total and measure angles with a protractor. A full turn is 360∘360^\circ.

Ask each pupil for one main way of travelling to school. This wording matters: counting every way they sometimes travel could count one pupil several times. Make categories that cover every answer without overlap, then tally each answer once. A frequency is a count.

Two-way table

A table that sorts the same observations by two characteristics, one in its rows and one in its columns. An interior cell meets both labels; the grand total counts all observations once.

Rows might show year group and columns might show travel choice. The totals also give a frequency table ready for a chart. Choose the totals for the group you want: the Year 77 row describes Year 77, while the bottom row describes everyone.

Worked example 1: complete a travel table

Question. Find the two missing entries.

Year groupBusWalkBikeCarTotal
Year 7995544222020
Year 8??44222020
Total1616121288444040

Plan. Subtract the known entry from each column total.

Year 8 Bus=16−9=7complete Bus columnYear 8 Walk=12−5=7complete Walk column\begin{aligned} \text{Year 8 Bus}&=16-9=7 &&\text{complete Bus column}\\ \text{Year 8 Walk}&=12-5=7 &&\text{complete Walk column} \end{aligned}

Why subtract? Each column total includes both year groups, so removing one leaves the other.

Answer. Both missing entries are 77 pupils.

Check. The Year 88 row totals 7+7+4+2=207+7+4+2=20, and the bottom row totals 4040.

Change only the Year 77 Bus entry to 1111, with its row total now missing. Keep all column totals and the Year 88 Bike and Car entries unchanged. This tests whether you follow the labels rather than reuse the earlier answer.

Worked example 2: change one survey entry

Question. With that change, find Year 88 Bus and both year-group totals.

Plan. Complete the Bus column, then add across each row.

Year 8 Bus=16−11=5fixed column totalYear 7 total=11+5+4+2=22add its categoriesYear 8 total=5+7+4+2=18add its categories\begin{aligned} \text{Year 8 Bus}&=16-11=5 &&\text{fixed column total}\\ \text{Year 7 total}&=11+5+4+2=22 &&\text{add its categories}\\ \text{Year 8 total}&=5+7+4+2=18 &&\text{add its categories} \end{aligned}

Why do the row totals change? The same 4040 pupils are now distributed differently between the two rows.

Answer. Year 88 Bus is 55; the row totals are 2222 and 1818.

Check. 22+18=4022+18=40, matching the unchanged grand total.

Use either a row or a column to find a missing cell, then use the other direction as a check. The counts must agree in both directions.

Watch out

The line grand total=40+40=80\text{grand total}=40+40=80 double counts everyone. Row totals and column totals describe the same pupils. Add either set once to get 4040.

Now try a different table. Its row and column labels supply the meaning of each subtraction.

Pictograms and their keys

A pictogram replaces counts with repeated equal-size symbols, so its key is part of the mathematics.

Pictogram

A display in which one full symbol represents a stated number of observations. A half symbol represents half that number.

Tabulate the frequencies first. Choose a key that gives manageable rows and easy fractions of symbols. Label the rows, align the symbols and keep their sizes equal. Show half of the same symbol for a half, rather than a smaller complete symbol whose meaning is unclear.

Worked example 3: read the key

Question. Read the library loans and find their total and the difference between Tuesday and Wednesday. One full circle represents 66 books.

[DIAGRAM: asset_slug: m086_pictogram; caption: Library loans with one full circle representing six books and a half circle representing three; description: Monday has two circles for 12 books, Tuesday three for 18 and Wednesday one and a half for 9; file: m086_pictogram.png; recommended_method: matplotlib]
Library loans with one full circle representing six books and a half circle representing three

Plan. Convert each row to books before adding or subtracting.

Monday=2×6=12two full symbolsTuesday=3×6=18three full symbolsWednesday=1.5×6=9include the halftotal=12+18+9=39add all three daysdifference=18−9=9compare the days\begin{aligned} \text{Monday}&=2\times6=12 &&\text{two full symbols}\\ \text{Tuesday}&=3\times6=18 &&\text{three full symbols}\\ \text{Wednesday}&=1.5\times6=9 &&\text{include the half}\\ \text{total}&=12+18+9=39 &&\text{add all three days}\\ \text{difference}&=18-9=9 &&\text{compare the days} \end{aligned}

Why use the key first? The symbols count groups of books, not individual books.

Answer. There are 3939 loans altogether; Tuesday has 99 more than Wednesday.

Check. The 6.56.5 symbols together represent 6.5×6=396.5\times6=39 books.

Keep the library frequencies fixed but change only the key: one full circle now represents 33 books. More symbols are needed for the same observations.

Worked example 4: construct rows with a new key

Question. Redraw Monday 1212, Tuesday 1818 and Wednesday 99 loans, using one full circle for 33 books.

Plan. Divide each frequency by 33, then draw that many equal-size circles.

12÷3=4Monday: four full18÷3=6Tuesday: six full9÷3=3Wednesday: three full\begin{aligned} 12\div3&=4 &&\text{Monday: four full}\\ 18\div3&=6 &&\text{Tuesday: six full}\\ 9\div3&=3 &&\text{Wednesday: three full} \end{aligned}

Why divide? Each new symbol stands for a group of 33 books.

Answer. Draw the three labelled rows with 4,6,34,6,3 equal-size circles and state the key of 33 books per circle.

Check. Multiplying by 33 recovers 12,18,912,18,9; changing the key has not changed the loans.

With the earlier key of 66, Thursday 1515, Friday 2424 and Saturday 99 loans would require 15/6=2.515/6=2.5, 24/6=424/6=4 and 9/6=1.59/6=1.5 symbols. A half circle represents 33 books exactly.

A pictogram suits small categorical counts. A bar chart is often clearer if the key would require awkward fractions.

Watch out

The line 1.5 symbols=1.5 books1.5\text{ symbols}=1.5\text{ books} ignores the key. With this key, 1.5×6=91.5\times6=9 books.

Read the whole key in a paper question; its value can change from one pictogram to another.

Bar charts and categorical comparisons

A bar chart compares separate categories by the heights of equal-width bars.

Continue this lesson.

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