Sec 2 Statistical Diagrams: Examples & Interactive Guide

Secondary 2 G3 Mathematics · Study guide

Read dot diagrams, stem-and-leaf diagrams and histograms, choose useful representations and spot misleading claims.

Concept → worked examples → practice14 min read · practice linked below

Choose a diagram that answers the question

A statistical diagram organises observations so that we can see patterns, compare values and ask useful questions. A dot diagram or a stem-and-leaf diagram preserves individual values. A grouped histogram shows how values are distributed across intervals, but no longer tells us every exact observation.

Start by identifying the variable, its units and the number of observations. Label axes, state a stem-and-leaf key, and use consistent scales. For calculating and choosing averages, continue to mean, median and mode.

Dot diagrams: one dot for each observation

Place each dot above its value on a number line. Stack repeated values vertically. The horizontal position gives the value; the number of dots in the stack gives its frequency. Do not add the horizontal labels to count the people.

Worked example 1 — count carefully and interpret a condition

Ten students report the number of books they read in a month: $0,1,1,2,2,2,3,3,4,4$ .

Repeated observations remain separate dots.

The total frequency is $1+2+3+2+2=10$ , agreeing with the ten students.
“At least 3 books” includes both 3 and 4. Their frequencies total $2+2=4$ students.
The fraction reading at least 3 books is $\frac4{10}=\frac25$ , or $40\%$ .

Check: six students read fewer than 3; $6+4=10$ . The modal number is 2 books because its stack is tallest. The mode is a value, not the frequency 3.

The same ungrouped data as a histogram: give each integer value a unit-width bar centred on that value. Adjacent bars meet halfway between the integers. Heights are the frequencies 1, 2, 3, 2 and 2. Unlike the dots, the bars do not show individual people, but no distinct book counts have been merged into a wider class.

The first bar is centred on 0; its left edge is a drawing boundary, not a claim that negative book counts occurred.

Stem-and-leaf diagrams: retain the actual values

A stem contains the leading digit or digits and a leaf contains the final digit. Always give a key: $2\mid3$ could mean 23 marks, 2.3 seconds or something else. Sort the leaves within each stem and preserve repeated values and empty stems.

Worked example 2 — build and read a stem-and-leaf diagram

Ten test marks are $12,14,14,17,21,23,23,26,28,31$ . Use tens as stems and units as leaves.

Test marks; key: 2 | 3 means 23 marks
Stem	Leaves in increasing order
1	2 4 4 7
2	1 3 3 6 8
3	1

There are $4+5+1=10$ leaves, so every observation is included.
With ten marks, the median lies halfway between the fifth and sixth values. The first four are in stem 1, so these are 21 and 23: $\text{median}=\frac{21+23}{2}=22\text{ marks}.$
To count marks above 25, use 26, 28 and 31: there are 3 students. Reading only stem 2 would miss the 31.

Check: reconstruct the ordered list from the diagram. There need not be a student scoring the median 22. In a back-to-back diagram, both groups share stems and a key; on the left, read the units digit next to its stem and place smaller leaves nearest the stem.

Histograms: continuous intervals share boundaries

A histogram has a numerical horizontal axis divided into adjoining intervals. Its bars touch because the intervals meet. For equal-width intervals, frequency can be shown directly as bar height. A bar chart usually compares separate categories, so gaps between its bars are appropriate.

Worked example 3 — interpret grouped data without inventing exact values

The journey times of 20 students are grouped below. The class notation $10\le t<20$ includes 10 minutes but excludes 20 minutes.

Journey time t, in minutes
Time interval	Frequency
$0\le t<10$	4
$10\le t<20$	8
$20\le t<30$	6
$30\le t<40$	2

Every interval has width 10 minutes, so comparing heights compares frequencies fairly.

Check the total: $4+8+6+2=20$ students.
Journeys taking at least 20 minutes belong to the last two intervals: $6+2=8$ students, or $\frac8{20}\times100\%=40\%$ .
The interval with the highest frequency is $10\le t<20$ . This is the modal class; it does not establish an exact modal time of 15 minutes.

What cannot be recovered: the number taking between 12 and 18 minutes is unknown, because all eight observations are grouped together. Do not assume they are evenly spread. With unequal-width classes, frequency heights alone are misleading: bar area must represent frequency, which calls for a frequency-density scale.

Read the scale before trusting the impression

Worked example 4 — evaluate a misleading claim

A display compares two classes with mean test marks 60 and 66. Its vertical axis starts at 54. The visible bar heights are therefore 6 and 12 units, and a caption claims the second mean is “twice as high”. Is that correct?

Recover the actual values from the axis: 60 and 66, not 6 and 12.
The ratio of the means is $\frac{66}{60}=1.1$ , not 2.
The percentage increase relative to the first mean is $\frac{66-60}{60}\times100\%=10\%.$
A bar chart starting at zero represents the size comparison fairly. If a shortened axis is used to show a small change, it must be clearly marked and the interpretation must use the labelled values.

Answer: the second mean is 10% higher, not twice as high. A difference of 6 marks also does not mean an increase of 6%.

Check the wider claim: means alone cannot show that every student in the second class scored higher. To compare distributions, inspect the underlying marks or suitable diagrams as well.

Explore what grouping hides

The ten marks are 12, 14, 14, 17, 21, 23, 23, 26, 28 and one final mark. Switch diagrams, then change the final mark from 31 to 38. The individual value changes, but its histogram class stays the same.

Each representation uses the same ten observations. In the histogram, all classes have width 10 marks.

There are ten observations. The highest individual mark is 31. A dot diagram preserves each exact mark.

Common mistakes and checking strategies

Dropping repeated data: count dots, leaves or frequencies and compare with the sample size.
Misreading a key: reconstruct two values before answering a stem-and-leaf question.
Double-counting a class boundary: follow the inequality signs. A value of 20 belongs in $20\le t<30$ , not $10\le t<20$ .
Overclaiming from grouped data: say “modal class” and “estimated mean” when exact individual values are unavailable.
Choosing an unsuitable diagram: dots are clear for small sets with few distinct values; stem-and-leaf diagrams retain values in an ordered display; histograms summarise larger numerical sets at the cost of detail.

Practise reading and comparing diagrams

Use the Sec 2 statistics question bank. Read the labels and key first, count the observations, and distinguish what a diagram shows from what it cannot tell you.