Interpretation of Statistical Graphs

Lessons

Senior 3 · Mathematics

Interpretation of Statistical Graphs

3 lessons

Lesson 1 of 3

Ungrouped and grouped data

25 min

Thirteen million answers

In August 2022, Rwanda's Fifth Population and Housing Census counted 13,246,394 people (National Institute of Statistics of Rwanda, NISR). Behind that one number are millions of answers: every person's age, sex, district, schooling and work. A list of answers that long cannot be read. Statisticians first sort the answers into frequency tables and then draw graphs from the tables. Only then can anyone see a pattern.

In S1 and S2 you built frequency tables and drew bar charts, pie charts and histograms. This unit works from the other side: you are handed a graph that someone else drew, and you must read exact values from it, compare groups, and judge whether it tells the truth. It starts with tables, because every statistical graph is a frequency table turned into a picture, and you cannot read the picture without knowing what kind of table is behind it.

Definition

Ungrouped data are counted value by value: each different value has its own row in the frequency table. They suit a variable with only a few different values, such as the number of people in a household.

Grouped data are collected into classes (class intervals) such as 150–169 g, and only the number of values in each class, its frequency, is kept. They suit a variable with many different values, and every continuous variable (one that is measured, such as mass, time or height).

Example: "households of 4 people: 7" is one row of an ungrouped table; "avocados of 150–169 g: 4" is one row of a grouped table.

Worked example 1: an ungrouped table

The number of people living in each of 30 households in one village was recorded (illustrative data). Find (a) how many households have at most 4 people, (b) how many have more than 5 people, (c) the fraction of households with exactly 4 people, and (d) the total number of people.

People in household, $x$	1	2	3	4	5	6	7	8
Frequency, $f$	2	3	5	7	6	4	2	1
Cumulative frequency	2	5	10	17	23	27	29	30

Solution

(a) "At most 4" means 1, 2, 3 or 4 people. The cumulative frequency under 4 already adds these: 17 households.

(b) "More than 5" means 6, 7 or 8 people: $30 - 23 = 7$ households, since 23 households have at most 5.

(d) Multiply each value by its frequency and add: $\sum fx = 1(2) + 2(3) + 3(5) + 4(7) + 5(6) + 6(4) + 7(2) + 8(1) = 127 \text{ people.}$

The vocabulary of a class

Suppose avocados at a market are weighed and each mass is recorded to the nearest gram. The class 170–189 g then holds every avocado recorded as 170, 171, …, 189 g. But an avocado that really weighs 189.7 g is recorded as 190 g, and one of 169.6 g is recorded as 170 g. So the class 170–189 really covers every mass from 169.5 g up to (but not including) 189.5 g. The diagram shows the difference between the numbers written in the table and the interval the class really covers.

The top row shows the classes as they are written; the bottom row shows them as they really are. On a graph of grouped data you always draw the bottom row, so the bars or points sit at the class boundaries and mid-points.

Definition

For a class written as 170–189 (masses to the nearest gram):

The class limits are the numbers written in the table: lower limit 170, upper limit 189.
The class boundaries are the true ends of the interval, halfway between one class's upper limit and the next class's lower limit: lower boundary 169.5, upper boundary 189.5. In interval notation the class is $169.5 \le m < 189.5$ .
The class width is upper boundary minus lower boundary: $189.5 - 169.5 = 20$ .
The mid-point (class mark) is halfway along the class: $\frac{170 + 189}{2} = 179.5$ . It stands for every value in the class when you plot a frequency polygon or estimate a mean.

Important

For any class,

\text{class width} = \text{upper boundary} - \text{lower boundary}

\text{mid-point} = \frac{\text{lower boundary} + \text{upper boundary}}{2}

The width comes from the boundaries, not the limits: $189 - 170 = 19$ is the wrong width for 170–189. For rounded measurements the limits give the same mid-point, $\frac{170 + 189}{2} = 179.5$ ; for ages last birthday they do not (see the warning below).

Worked example 2: completing a grouped table

The masses of 40 avocados, to the nearest gram, are grouped below (illustrative data). Complete the boundaries, widths and mid-points, and find how many avocados have a true mass under 189.5 g.

Mass (g)	Frequency	Boundaries	Width	Mid-point
150–169	4	149.5 to 169.5	20	159.5
170–189	9	169.5 to 189.5	20	179.5
190–209	14	189.5 to 209.5	20	199.5
210–229	8	209.5 to 229.5	20	219.5
230–249	5	229.5 to 249.5	20	239.5

Solution Each boundary lies halfway across a 1 g gap, for example $\frac{169 + 170}{2} = 169.5$ . Every width is $169.5 - 149.5 = 20$ , so the classes are of equal width. Masses under 189.5 g are the first two classes: $4 + 9 = 13$ avocados. The frequencies add to $4 + 9 + 14 + 8 + 5 = 40$ , which checks the table.

Definition

The frequency density of a class is its frequency divided by its width:

\text{frequency density} = \frac{\text{frequency}}{\text{class width}}.

It measures how crowded the class is: the number of values per one unit of the variable. Example: 12 children in the age class 0–4 (width 5 years) gives $\frac{12}{5} = 2.4$ children per year of age.

Worked example 3: classes of unequal width

A health centre recorded the age, in completed years (age last birthday), of the 60 patients seen one morning (illustrative data).

Age (years)	0–4	5–14	15–24	25–44	45–64	65 and over
Number of patients	12	9	10	16	9	4

(a) Find the boundaries and width of each closed class. (b) Which class has the most patients? (c) Which class is most crowded for each year of age?

Solution

(a) "Age last birthday" works differently from a rounded mass. A child is "4" from the 4th birthday until the day before the 5th, so the class 0–4 covers $0 \le \text{age} < 5$ . Its boundaries are 0 and 5 and its width is 5 years. In the same way, 5–14 is $5 \le \text{age} < 15$ (width 10), 15–24 has width 10, 25–44 has width 20 and 45–64 has width 20.

(b) The class 25–44, with 16 patients.

Age	0–4	5–14	15–24	25–44	45–64
Width (years)	5	10	10	20	20
Frequency density	2.4	0.9	1.0	0.8	0.45

The class 0–4 is the most crowded: 2.4 patients for each year of age, three times as many as in 25–44.

The open class "65 and over" has no upper boundary, so its width and density cannot be found until you decide where to close it. Tables of ages often end like this, and the person who draws the graph must state the closing boundary they chose.

Warning

Two boundary traps. (1) Do not take the width from the limits: the class 170–189 has width 20, not 19. (2) Do not treat ages like rounded measurements: for "age last birthday" the class 15–24 has boundaries 15 and 25, not 14.5 and 24.5, because nobody is counted as 25 until the 25th birthday. Its mid-point is therefore $\frac{15 + 25}{2} = 20$ , not 19.5.

Which graph for which data?

Before you read any graph, decide what kind of table is behind it. The kind of data decides which graphs are honest ones to draw.

Kind of data	Example	Graphs that suit it
Categories (qualitative)	province of residence, way of travelling to school	bar chart, pie chart
Discrete values, ungrouped	people per household, goals in a match	bar chart with gaps, or a vertical line graph
Grouped values (usually continuous)	mass, age, time, exam marks	histogram, frequency polygon, cumulative frequency curve
One quantity over time	population at each census, monthly rainfall	line graph
Parts of one whole	how a budget is shared	pie chart, component (stacked) bar chart

The bars of a bar chart have gaps because the categories are separate. The bars of a histogram touch because the classes meet at their boundaries: there is no mass "between" 169.5 g and 169.5 g. Lesson 2 reads graphs of ungrouped data, and lesson 3 reads graphs of grouped data.

Tip

For any class, ask two questions before you calculate: "What is the smallest value that can be recorded in this class, and what is the smallest value that goes into the next class?" The boundary is halfway between them, except for ages last birthday, where it is the next class's lower limit itself.

Corbettmaths: Drawing Histograms. Watch how each class's frequency is divided by its width to get the frequency density when the classes are of unequal width, as in Worked example 3. Lesson 3 reads histograms like these.

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Practice

Say whether each variable is categorical, discrete or continuous, and whether you would group it: (a) the number of goats a family owns; (b) the time taken to run 100 m; (c) a pupil's favourite subject; (d) the mass of a newborn baby.
Marks out of 100 are whole numbers. For the class 35–44, give the class limits, the class boundaries, the width and the mid-point.
The table shows the number of books read in a term by 25 pupils: 0 books: 3 pupils; 1: 6; 2: 8; 3: 5; 4: 3. How many pupils read (a) at most 2 books, (b) at least 3 books? (c) What fraction of the pupils read exactly 1 book?
A pupil groups masses in kilograms as 40–50, 50–60, 60–70. Explain what is wrong, and rewrite the classes for masses recorded to the nearest kilogram.
Lengths of 53 leaves are grouped as $0 \le l < 10$ (15 leaves), $10 \le l < 15$ (20 leaves) and $15 \le l < 30$ (18 leaves), in centimetres. Find the frequency density of each class, and say which class is most crowded.
For ages last birthday, find the boundaries, the width and the mid-point of the class 15–19.
Twenty-five of 80 workers are in the age class 30–49 and 18 are in the class 20–24. Musoni says "the 30–49 group is where workers are most concentrated". Use frequency density to judge his claim.

Show answersHide answers

(a) discrete, few values, so ungrouped; (b) continuous, grouped; (c) categorical, not grouped by size (each subject is its own category); (d) continuous, grouped.
Limits 35 and 44; boundaries 34.5 and 44.5; width $44.5 - 34.5 = 10$ ; mid-point $\frac{35 + 44}{2} = 39.5$ .
(a) $3 + 6 + 8 = 17$ ; (b) $5 + 3 = 8$ ; (c) $\frac{6}{25}$ .
A mass of 50 kg or 60 kg would belong to two classes. Use 40–49, 50–59, 60–69 (boundaries 39.5, 49.5, 59.5, 69.5).
Widths 10, 5 and 15; densities $\frac{15}{10} = 1.5$ , $\frac{20}{5} = 4$ , $\frac{18}{15} = 1.2$ leaves per cm. The class $10 \le l < 15$ is most crowded.
Boundaries 15 and 20; width 5; mid-point $\frac{15 + 20}{2} = 17.5$ .
Widths are 20 and 5 years. Densities: $\frac{25}{20} = 1.25$ and $\frac{18}{5} = 3.6$ workers per year of age. The 20–24 group is almost three times as concentrated, so his claim is wrong: 30–49 has more workers only because it is a wider class.

Sources and credits

Census total: Fifth Rwanda Population and Housing Census 2022, National Institute of Statistics of Rwanda (NISR). Official statistics, cited as a fact.
Video: Drawing Histograms, Corbettmaths, embedded from YouTube.
Written by Ganzaa. The household, avocado, patient and leaf data are illustrative, and the diagram, worked examples and practice items were written by Ganzaa.

1Thirteen million answers

2The vocabulary of a class

3Which graph for which data?

Thirteen million answers

The vocabulary of a class

Which graph for which data?