Reading and interpreting graphs of ungrouped data – Interpretation of Statistical Graphs (Senior 3 Mathematics) | Ganzaa

Interpretation of Statistical Graphs

Lessons

Senior 3 · Mathematics

Interpretation of Statistical Graphs

3 lessons

Lesson 2 of 3

Reading and interpreting graphs of ungrouped data

25 min

Five bars, thirteen million people

The bar chart below shows the population of each province of Rwanda at the 2022 census. Before reading a single bar, read the frame around it: the title says what is counted, the vertical axis says it is counted in millions, and the source line says where the numbers come from. Each labelled gridline is 0.5 million apart, and the faint lines between them divide each gap into 5, so one small division is 0.1 million.

A bar chart answers "how many?" for each category, and lets you compare categories at a glance. It does not answer every question: it says nothing about area. The City of Kigali has the shortest bar, yet it is by far the most crowded province, with 2,401 people per km² against 503 for Rwanda as a whole (NISR, 2022 census).

Definition

The scale of an axis is the value of one division. Find it by dividing the gap between two labelled values by the number of divisions between them: here $\frac{0.5}{5} = 0.1$ million per small division.

To read a bar, follow its top across to the axis, count whole divisions, and estimate any part of a division. A value read from a graph is only as exact as the scale allows, so give it to the nearest division (or half-division) and say "about".

Worked example 1: reading the province bar chart

(a) Read the populations of the Eastern Province and the City of Kigali to the nearest 0.1 million. (b) About how many times as many people live in the Eastern Province as in Kigali? (c) Which two provinces have almost equal populations? (d) Using NISR's exact figures, 1,745,555 for Kigali out of 13,246,394, find the percentage of Rwanda's population that lives in Kigali.

Solution

(a) The Eastern bar ends a little more than half a division above 3.5, so it is about 3.6 million. The Kigali bar ends almost halfway between 1.7 and 1.8, just below the middle, so it is about 1.7 million (the exact figure, 1,745,555, confirms it).

(b) $\frac{3.6}{1.7} \approx 2.1$ : about twice as many. The exact figures give $\frac{3\,563\,145}{1\,745\,555} \approx 2.04$ .

(d) $\frac{1\,745\,555}{13\,246\,394} \times 100 \approx 13.2\%$

Notice the difference between (a) and (d): a value read from a graph is an estimate, and an exact answer needs the table.

Pie charts: reading angles as shares

A pie chart shows how one whole is shared between categories. The whole circle, 360°, stands for the total, so each sector's angle is the same fraction of 360° as the category is of the total. The pie chart below shows how the 72 pupils of one S3 class travel to school.

Important

For a category with frequency $f$ out of a total $N$ , and sector angle $\theta$ :

\theta = \frac{f}{N} \times 360^\circ \qquad \text{and} \qquad f = \frac{\theta}{360^\circ} \times N.

The second formula is the one you use to read a pie chart. If $N$ is not given, find it from any sector whose frequency you know: $N = \frac{360^\circ}{\theta} \times f$ .

Worked example 2: reading the pie chart

(a) How many pupils walk? (b) What fraction of the class comes by moto or by car? (c) A second class's chart was printed without its total, but you are told that 15 pupils come by bus and the bus sector is 54°. How many pupils are in that class?

Solution

(a) $f = \frac{200^\circ}{360^\circ} \times 72 = 40$ pupils.

(b) Moto and car together have $30^\circ + 30^\circ = 60^\circ$ , which is $\frac{60}{360} = \frac{1}{6}$ of the class (12 pupils).

Warning

A bigger slice is not always more people. A pie chart shows shares, not numbers. If 40 of 72 pupils walk in one class (200°) and 45 of 100 walk in another (162°), the second class has the smaller slice but more walkers. Never compare the slices of two pie charts unless you know both totals.

Line graphs: one quantity over time

A line graph plots one quantity against time and joins the points with straight lines. Rwanda has held five censuses; the graph shows the population counted at each, with the years spaced to scale on the horizontal axis. Each small division is 0.5 million on the vertical axis and 5 years on the horizontal axis.

Definition

A time series is a set of values of one quantity recorded at successive times, such as the population at each census.

The average rate of change between two points is the change in the quantity divided by the time between them: $\text{average rate of change} = \frac{\text{change in value}}{\text{change in time}}.$ On the graph it is the steepness (gradient) of the segment joining the two points.

Interpolation is reading a value between two plotted points from the straight segment that joins them. It assumes steady change between the points, so the result is an estimate.

Worked example 3: reading the census line graph

(a) Read the population in 1991 and in 2012. (b) Find the average increase per year from 1991 to 2002 and from 2002 to 2012. (c) Which part of the line is flattest, and what does that mean? (d) Estimate the population in 2017.

Solution

(a) The 1991 point is a little above 7, about 7.2 million; the 2012 point is at about 10.5 million.

(b) In 2002 the point is at about 8.1 million. From 1991 to 2002: $\frac{8.1 - 7.2}{11} \approx 0.08$ million, about 80,000 people per year. From 2002 to 2012: $\frac{10.5 - 8.1}{10} = 0.24$ million, about 240,000 people per year. (NISR's exact counts give 88,273 and 238,742 per year.)

(c) The 1991 to 2002 segment is the flattest: the population grew most slowly in those years. NISR links this to the high death toll of the 1990 war and of the 1994 Genocide against the Tutsi.

(d) 2017 is halfway between the 2012 and 2022 points, so read halfway along that segment: $\frac{10.5 + 13.2}{2} \approx 11.9$ million. There was no census in 2017; this is an estimate that assumes steady growth from 2012 to 2022.

Graphs that mislead

A graph can show true numbers and still give a false impression. Three tricks appear again and again in newspapers, adverts and reports.

1. A truncated axis. Both charts below show the same pass rates, 92% and 96%. On the left the axis starts at 0 and the bars look almost equal, which is the truth. On the right the axis starts at 90, so only the top of each bar is shown, and School B seems to have done three times as well.

2. Unequal intervals on the time axis. Here the census populations are plotted again, but the five years are placed at equal steps, although the gaps are 13, 11, 10 and 10 years. The 13-year rise from 1978 to 1991 is squeezed into the same width as a 10-year rise, so it looks as steep as the 2002 to 2012 rise, when the population really grew more slowly then (about 179,000 a year against 239,000).

3. A 3-D pie chart. Tilting a pie chart squashes the back slices and adds a thick side wall to the front ones. In the 3-D chart below, slice R (25%) sits at the front and covers about one and a half times as much of the picture as slice P (30%), so the eye reads the smaller share as the bigger one.

Worked example 4: measuring the distortion

In the right-hand chart of the truncated pair, the axis runs from 90 to 100. (a) How tall is each bar, in percentage points above the axis? (b) How many times as tall as A's bar is B's bar? (c) How many times as large is B's pass rate as A's really?

Solution

(a) A: $92 - 90 = 2$ ; B: $96 - 90 = 6$ percentage points. (b) $\frac{6}{2} = 3$ times as tall. (c) $\frac{96}{92} \approx 1.04$ : B's pass rate is only about 4% larger. The truncated axis turns a 4% difference into a bar three times as tall.

Warning

Before you believe a graph, check

Does the vertical axis of a bar chart start at 0? If not, the bar heights cannot be compared as ratios. (A line graph may start above 0, but it should show a break in the axis and you must read the numbers, not the heights.)
Are the steps on each axis equal? Check the labels, especially years.
Is it 3-D, or drawn with pictures whose area grows faster than the value?
Is the source given, and do the pie chart's percentages add to 100%?

Tip

Read a graph in the order an examiner marks it: title, axes, scale, then data. Write the scale of one small division in the margin before you read any value; most reading errors in exams come from a misread scale, not a misread bar.

TED-Ed: How to spot a misleading graph, by Lea Gaslowitz. As you watch, name each trick the video shows and say which check in the warning box above would catch it.

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Practice

Questions 1 and 2 use the province bar chart; question 5 uses the census line graph.

Read the population of the Northern Province to the nearest 0.1 million. Which province had the largest population?
Use the bar chart to estimate the combined population of the Southern and Western Provinces. Then find it exactly from 3,002,699 and 2,896,484.
In a pie chart of the favourite sports of 90 pupils, the football sector is 160°. (a) How many pupils chose football? (b) Volleyball was chosen by 36 pupils. What is its angle?
Find, to the nearest degree, the angle of the Eastern Province's sector in a pie chart of the 2022 census (3,563,145 of 13,246,394 people).
Using the exact census counts (1978: 4,831,527; 1991: 7,157,551; 2002: 8,128,553; 2012: 10,515,973; 2022: 13,246,394), find the period with the greatest average increase per year.
A shop's bar chart of monthly sales starts its vertical axis at 400,000 Frw. January's sales were 500,000 Frw, and February's bar is twice as tall as January's. Find February's sales, and say how many times January's they really are.
Choose the most suitable graph, with a reason: (a) the daily maximum temperature in Huye for one week; (b) how a family spends its monthly income; (c) the number of goals scored in each of 30 matches.
A poster shows a 3-D pie chart with no percentages written on it, and the slice at the front is labelled with the poster-maker's own product. Give two reasons why a reader should not trust the size of that slice.

Show answersHide answers

About 2.0 million. The Eastern Province.
About $3.0 + 2.9 = 5.9$ million; exactly $3\,002\,699 + 2\,896\,484 = 5\,899\,183$ .
(a) $\frac{160}{360} \times 90 = 40$ pupils. (b) $\frac{36}{90} \times 360^\circ = 144^\circ$ .
$\frac{3\,563\,145}{13\,246\,394} \times 360^\circ \approx 96.8^\circ$ , so 97°.
Per year: 1978 to 1991, $\frac{2\,326\,024}{13} \approx 178\,925$ ; 1991 to 2002, $\frac{971\,002}{11} \approx 88\,273$ ; 2002 to 2012, $\frac{2\,387\,420}{10} \approx 238\,742$ ; 2012 to 2022, $\frac{2\,730\,421}{10} \approx 273\,042$ . The greatest is 2012 to 2022.
January's bar shows $500\,000 - 400\,000 = 100\,000$ above the axis, so February's shows 200,000 above it: $600\,000$ Frw. That is $\frac{600\,000}{500\,000} = 1.2$ times January's, not 2 times.
(a) A line graph: one quantity changing over time. (b) A pie chart: parts of one whole. (c) A bar chart (or vertical line graph) of ungrouped discrete data: only a few values, such as 0 to 6 goals.
Any two: the front slice gains a side wall, so it covers more of the picture than its share; the tilt squashes the back slices; without percentages the reader cannot check the angles; the maker has a reason to make that slice look big.

Sources and credits

Province populations, population densities and the census totals for 1978, 1991, 2002, 2012 and 2022: Population size and population characteristics and the Fifth Rwanda Population and Housing Census 2022, National Institute of Statistics of Rwanda (NISR). Official statistics, cited as facts; the charts were drawn by Ganzaa.
Video: How to spot a misleading graph, Lea Gaslowitz, TED-Ed, embedded from YouTube.
Written by Ganzaa. The travel, pass-rate, sales and 3-D pie data are illustrative.

1Five bars, thirteen million people

2Pie charts: reading angles as shares

3Line graphs: one quantity over time

4Graphs that mislead

Five bars, thirteen million people

Pie charts: reading angles as shares

Line graphs: one quantity over time

Graphs that mislead