SPI Mathematics · Data Interpretation

Pictographs, Bar Graphs and Pie Charts

A pictograph counts with symbols, a bar graph compares with bar heights, and a pie chart shares out 360° in proportion to the parts of a whole.

All Data Interpretation notes

Why this matters

Six questions in the bank, all EASY. Two read a pictograph's key, one asks which axis of a bar graph carries the categories, and three work with a pie chart: a share from the chart, a central angle from a percentage, and an amount from an angle.

Concept 1 of 3: Reading a pictograph

A pictograph has a key, such as one symbol for 10 students. Every row is just the key multiplied by the number of symbols, and a part symbol counts for the same part of the key.

Definition

Value $=$ number of symbols $\times$ value of one symbol.
A half symbol stands for half the key, a quarter symbol for a quarter.
If the question gives the value of a part symbol, find the value of one full symbol first.

Pictograph value

\text{value} = \text{number of symbols} \times \text{value of one symbol}

Worked example

In a pictograph one symbol stands for 8 cars. How many cars do 3 and a quarter symbols stand for?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Data Interpretation · Pictographs, Bar Graphs and Pie Charts

If half a picture of an apple represents 5 fruits, then 2 full pictures and one half represent:

Using the part symbol's value as the key

If half a symbol is 20 units, a full symbol is 40 units. Multiplying the number of full symbols by 20 gives only half the answer.

Counting a half symbol as a full one

3 and a half symbols with a key of 10 stand for 35, not 40. Each part symbol adds only its part of the key.

Drill 1 more on reading a pictograph

Concept 2 of 3: The parts of a bar graph

A bar graph puts each item being compared on one axis and a number scale on the other, so the taller bar is the bigger value. Read a bar by lining its top up with the scale.

Definition

Three facts settle most bar graph questions:

In the usual (vertical) bar graph the categories go on the x-axis and the values on the y-axis.
The bars have equal widths and equal gaps. Bars that touch belong to a histogram.
A scale such as 1 cm for 5 students turns a bar's height into a value.

Part of a bar graph	What it shows
x-axis (horizontal)	The categories being compared, such as years, subjects or cities
y-axis (vertical)	The values, marked on an even scale
Height of a bar	The value of that category
Gaps between bars	Equal gaps and equal widths, because the categories are separate
Scale	How much one unit of height stands for, such as 1 cm for 5 students
Horizontal bar graph	The bars lie sideways: categories on the y-axis, values on the x-axis Only the sideways graph swaps the axes. The usual bar graph has its categories on the x-axis.

Value of a bar = its height × what one unit of height stands for.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Data Interpretation · Pictographs, Bar Graphs and Pie Charts

Which axis is generally used to represent categories in a bar graph?

Mixing up a bar graph and a histogram

A bar graph compares separate items, so its bars have gaps. A histogram shows class intervals that run on from each other, so its bars touch.

Reading the height without the scale

A bar 4 cm tall on a scale of 1 cm for 25 units shows 100 units, not 4. Multiply the height by the scale.

Concept 3 of 3: Pie chart shares, angles and amounts

A pie chart is a whole circle, 360°, cut into slices in proportion to the parts. A part that is a quarter of the whole gets a quarter of the circle, 90°. The same share works in both directions: from a part to its angle, and from an angle back to the part.

Definition

Central angle $= \dfrac{\text{part}}{\text{whole}} \times 360^\circ$ .
From a percentage: angle $= \text{percentage} \times 3.6^\circ$ , since $1\% \text{ of } 360^\circ = 3.6^\circ$ .
From an angle back to the amount: amount $= \dfrac{\text{angle}}{360^\circ} \times \text{whole}$ .
The percentages of all the sectors add to 100%, and their angles add to 360°. A missing share is 100% minus the others.

Pie chart

\text{central angle} = \frac{\text{part}}{\text{whole}} \times 360^\circ \qquad \text{amount} = \frac{\text{angle}}{360^\circ} \times \text{whole}

Worked example

In a class of 40 students, 15 like cricket best. Find the central angle of the cricket sector in a pie chart.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Data Interpretation · Pictographs, Bar Graphs and Pie Charts

Monthly salary of a person is ₹15000. The central angle of the sector representing his expenses on food and house rent on a pie chart is

60^\circ

. The amount he spends on food and house rent, is:

Taking a percentage as the angle

A sector showing 30% of the whole has a central angle of

30 \times 3.6^\circ = 108^\circ

, not 30°. Percentages are out of 100, angles out of 360.

Dividing an angle by 100

To turn a central angle into an amount, divide the angle by 360, not by 100. A 45° sector of ₹8000 is

\frac{45}{360} \times 8000 = 1000

, that is ₹1000.

Forgetting the share that is not printed

When one sector is marked with a question mark, its share is 100% minus all the others. Find that share first, then take it of the total.

Drill 2 more on pie chart shares, angles and amounts

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Reading a pictograph
Pictograph value
$\text{value} = \text{number of symbols} \times \text{value of one symbol}$
Pie chart shares, angles and amounts
Pie chart
$\text{central angle} = \frac{\text{part}}{\text{whole}} \times 360^\circ \qquad \text{amount} = \frac{\text{angle}}{360^\circ} \times \text{whole}$

Reference tables (1)

The parts of a bar graph6 rows

Part of a bar graph	What it shows
x-axis (horizontal)	The categories being compared, such as years, subjects or cities
y-axis (vertical)	The values, marked on an even scale
Height of a bar	The value of that category
Gaps between bars	Equal gaps and equal widths, because the categories are separate
Scale	How much one unit of height stands for, such as 1 cm for 5 students
Horizontal bar graph	The bars lie sideways: categories on the y-axis, values on the x-axis Only the sideways graph swaps the axes. The usual bar graph has its categories on the x-axis.

Value of a bar = its height × what one unit of height stands for.

Watch out for (7)

Using the part symbol's value as the key
→ Reading a pictograph
Counting a half symbol as a full one
→ Reading a pictograph
Mixing up a bar graph and a histogram
→ The parts of a bar graph
Reading the height without the scale
→ The parts of a bar graph
Taking a percentage as the angle
→ Pie chart shares, angles and amounts
Dividing an angle by 100
→ Pie chart shares, angles and amounts
Forgetting the share that is not printed
→ Pie chart shares, angles and amounts

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