Pie charts, bar charts and histograms · Statistics: data & averages

Paper 2 often asks you to draw a chart, and Paper 1 asks you to read one. There are three to know, and the most common mistake is drawing the wrong one.

Pie chart, bar chart, histogramSwitch chart

Each slice's angle is its share of 360°: frequency ÷ total × 360°. Maize: 12 ÷ 30 × 360° = 144°. Draw the angles with a protractor, and label the angles in degrees, not percentages.

Chart	Use it for	What it looks like
Pie chart	parts of a whole (shares of money, of a population)	a circle cut into sectors
Bar chart	separate categories, or separate whole-number values (scores on a die)	bars of equal width with gaps
Histogram	grouped data on a continuous scale (marks 0–9, 10–19, …)	bars touching, standing on the class boundaries

Pie charts

The whole circle is $360^\circ$ , so each item gets its share of $360^\circ$ :

\text{angle} = \frac{\text{frequency}}{\text{total}} \times 360^\circ

A sector's angleangle = frequency ÷ total × 360°

With percentages, $1\% = 3.6^\circ$ , so multiply each percentage by $3.6$ .

Worked example · WAEC 2022

WAEC 2022 · Paper 2 · Q8 (a, b)

Item	Food and drinks	Fuel	Rent	Building project	Education	Savings
Percentage (%)	35	7.5	10	15	17.5

The table shows the monthly expenditure (in percentage) of Mr. Okafor's salary.

Calculate the percentage of Mr. Okafor's salary that was put into savings.

Illustrate the information on a pie chart.

The missing percentage
- They add up to ${100\%}$ .
- The others total ${35 + 7.5 + 10 + 15 + 17.5 = 85}$ .
- So savings are ${100 - 85 = 15\%}$ .
Think first. The percentages must add up to what?
The angles
Multiply each by $3.6$ : food $35 \times 3.6 = 126^\circ$ , fuel $27^\circ$ , rent $36^\circ$ , building $54^\circ$ , education $63^\circ$ , savings $54^\circ$ .
Check: $126 + 27 + 36 + 54 + 63 + 54 = 360$ ✓.
Think first. What angle is 35%?
Draw it
Draw a circle with compasses and one radius as a starting line. Measure each angle with a protractor, going round in order. Write each item’s name and angle in its sector.

Reading a pie chart is the same formula backwards: $\text{frequency} = \frac{\text{angle}}{360^\circ} \times \text{total}$ .

More: pie charts

Bar charts

One bar for each category or value, all the same width, with equal gaps.
The height of each bar is the frequency. Label the axes and choose a scale that uses most of the graph paper.

Histograms

For grouped data, each bar covers its whole class, so the bars stand on the class boundaries and touch:

classes 10–19, 20–29, … have boundaries 9.5, 19.5, 29.5, …;
classes 1.0–1.9, 2.0–2.9, … have boundaries 0.95, 1.95, 2.95, ….

When all the classes are the same width, the height of each bar is its frequency.

The mode from a histogram

The mode is in the tallest bar. To estimate it, draw two lines:

from the top-left corner of the tallest bar to the top-left corner of the bar after it;
from the top-right corner of the tallest bar to the top-right corner of the bar before it.

Where they cross, read down to the axis. Switch to the histogram on the board above to see the two lines.

Reading the modeThe crossed lines lean towards the taller neighbour

Worked example · WAEC 2018

WAEC 2018 · Paper 2 · Q3

Price (₦1,000,000)	1.0–1.9	2.0–2.9	3.0–3.9	4.0–4.9	5.0–5.9
Number of vehicles	23	48	107	90	32

The frequency distribution shows the range of prices of a brand of car sold by a dealer and the corresponding quantity demanded.

Represent the information in a histogram.

Use the histogram to determine the most preferred selling price for the brand of car (in millions of naira).

The boundaries
- Halfway between 1.9 and 2.0 is 1.95.
- So the boundaries are 0.95, 1.95, 2.95, 3.95, 4.95, 5.95. Each bar is 1 unit wide.
Think first. The classes are 1.0–1.9, 2.0–2.9, …. Where do the bars start and end?
Draw the bars
Heights $23, 48, 107, 90, 32$ , touching, on those boundaries. Label the axes “Price (₦ million)” and “Number of vehicles”.
The most preferred price
- The price bought most often is the mode.
- The tallest bar is 2.95–3.95. Draw the crossed lines on it and read down: about 3.7, so about ₦3,700,000.
As a check by formula, put the values into the mode formula (the class width is 1):
$\begin{aligned} & 2.95 + \frac{107 - 48}{(107 - 48) + (107 - 90)} \\ &= 2.95 + \frac{59}{76} \approx 3.73 \end{aligned}$
Think first. 'Most preferred' means which average?

Your turn

WAEC 2018 · Paper 2 · Q5

Statistics: data & averages

Score	1	2	3	4	5	6
Frequency	2	5	13	11	9	10

The table shows the distribution of scores obtained when a fair die was rolled 50 times.

(a)
Draw a bar chart for the distribution.
Model answer
Put the scores on the horizontal axis and the frequency on the vertical axis, with equal-width bars and equal gaps between them. The bar heights are $2, 5, 13, 11, 9, 10$ ; the tallest bar is at score 3.
(b)
Calculate the mean score of the distribution.

Worked solution (try it first)

(a)

The scores 1 to 6 are separate values, so draw a bar chart: six bars of equal width with equal gaps between them, of heights $2, 5, 13, 11, 9, 10$ .
Label the axes "Score" and "Frequency", and state the scale.

(b)

$\sum fx = 1(2) + 2(5) + 3(13) + 4(11) + 5(9) + 6(10)$
$= 2 + 10 + 39 + 44 + 45 + 60$
$= 200$ and $\sum f = 50$ , so the mean score is $\frac{200}{50} = 4$ .

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JAMB 2012 · UTME · Q41

Statistics: data & averages

The grades of 36 students in a class test are shown in the pie chart (Pass $120^\circ$ , Credit $80^\circ$ , Very good $90^\circ$ ). How many students have Excellent?

Worked solution (try it first)

The angles add up to $360^\circ$ , so Excellent is $360^\circ - (120^\circ + 80^\circ + 90^\circ) = 70^\circ$ .
Its share of the 36 students: $\frac{70}{360} \times 36 = 7$ , option D.

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