Secondary Mathematics · Worked Repair Guide 36
A statistical diagram is a representation, not the data itself. It can reveal structure quickly, but the design choices—axis scale, class grouping, angle, area, ordering and labels—can also exaggerate or hide differences. Reading a graph therefore requires both calculation and judgement.
This guide develops one central habit: reconstruct the quantity behind the picture before drawing a conclusion. A bar height represents a value on a scale. A pie sector represents a fraction of a whole. A stem-and-leaf display preserves individual observations. A truncated axis can change visual drama without changing the numerical difference.
All data below are invented teaching material. Use Sampling, Scatter Plots, Correlation and Lines of Best Fit for bivariate data, Histograms, Frequency Density and Grouped Data for continuous grouped distributions, and Averages, Spread and Data Interpretation for numerical summaries.
1. Choose a display to match the data type
Categories such as transport mode can be shown in a bar chart. Numerical observations such as scores can be shown in a stem-and-leaf plot. Parts of one total can be shown in a pie chart. Values changing through ordered time points can be shown in a line graph.
No display is automatically best for every question.
Entry check: a pie chart is poor for showing exact changes through 20 time points because sectors encode shares of one whole rather than temporal sequence.
Start by naming what each observation represents.
2. Bar charts compare categories by height
Suppose category counts are A=12, B=18, C=9 and D=15.
A bar chart uses a common numerical vertical scale, with separated bars because the categories are distinct.
The difference B−C is9 observations.
The visual comparison should agree with that numerical scale rather than with the physical centimetres of the printed bar.
3. A truncated vertical axis can exaggerate differences
Suppose two values are98 and102.
On an axis from0 to110, the bars appear similar. On an axis from95 to103, the visual difference appears dramatic.
The numerical difference remains4.
A truncated axis is not automatically dishonest, but it must be clearly labelled and interpreted carefully. It changes visual emphasis.
4. Unequal picture sizes can mislead through area
If a pictogram doubles both the width and height of an icon to represent “twice as much”, its area becomes four times as large.
Viewers often respond to area rather than one dimension.
For proportional pictograms, the encoding rule must be explicit: number of equal icons is safer than scaling icons in two dimensions without explanation.
Representation should preserve the intended ratio.
5. Pie-chart angles are proportions of 360°
If a category contains15 of60 observations, its fraction is15/60=1/4.
Pie-sector angle=1/4×360°=90°.
The angle represents the category’s share of the complete sample.
All sector angles should sum to360°, subject only to small rounding adjustments if angles are rounded.
6. Recover a frequency from a pie sector
An invented pie chart represents72 observations. One sector is100°.
Frequency=100/360×72=20.
The sector angle is a fraction of the full turn, just as the category frequency is a fraction of the total sample.
If the result is not an integer in a count context, check whether the angle was rounded or whether the chart data were approximate.
7. Compare pie charts only with sample size in mind
A 90° sector is25% of a pie.
In a sample of40, it represents10 observations. In a sample of200, it represents50.
Equal sector angles mean equal proportions, not equal frequencies unless the totals are also equal.
This is a denominator-control problem disguised as a diagram comparison.
8. Stem-and-leaf plots preserve individual values
Consider scores 42,45,47,51,51,56,63.
A stem-and-leaf display can be written:
4 | 2 5 7
5 | 1 1 6
6 | 3
A key such as “4|2 means42” is essential because the same digits could represent4.2,42 or420 under different conventions.
9. Leaves should be ordered within each stem
For data 34,31,37,32, the stem row should normally read3 | 1 2 4 7.
Ordering makes the display immediately useful for median, quartile and range calculations.
It also exposes duplicates clearly.
An unordered leaf row still contains the raw values but sacrifices much of the display’s analytical purpose.
10. Read median and range directly from a stem-and-leaf plot
For 42,45,47,51,51,56,63, the median is the fourth value51.
Range=63−42=21.
Because the raw observations are retained, these summaries are exact rather than grouped-data estimates.
This distinguishes stem-and-leaf plots from histograms, which compress values into intervals.
11. Back-to-back stem-and-leaf plots support comparison
A back-to-back display places two data sets on opposite sides of common stems.
It can reveal differences in centre, spread and clustering while retaining individual observations.
Read the key carefully because leaves on the left side are often written in descending visual order away from the stem to maintain numerical ordering.
Do not compare only the longest row; consider the full distribution.
12. Line graphs emphasise ordered progression
Suppose an invented measurement is12,15,14,19 at times1,2,3,4.
Connecting consecutive points emphasises change through the ordered time sequence.
The segment between two recorded points does not automatically prove the variable changed linearly between observations; that depends on context.
A line joining observations can be a visual guide rather than a continuous physical model.
13. Dual-axis graphs demand extra caution
A graph with two different vertical axes can make two unrelated series appear to track closely if the scales are chosen conveniently.
Before inferring association, read which series belongs to which axis and compare numerical changes rather than visual alignment alone.
Dual axes are sometimes useful, but they increase the burden on the reader.
One visual coincidence is not evidence of causation.
14. Three-dimensional decoration can distort comparison
A tilted 3D pie chart can make front sectors appear larger than equal-angle sectors at the back because of perspective.
Likewise, 3D bars introduce depth that does not encode an additional quantity.
When exact comparison matters, flat two-dimensional displays are often easier to read correctly.
Decorative depth should not be mistaken for data.
15. Percentage change and percentage-point change are different
If a category share rises from20% to30%, the percentage-point increase is10 points.
The relative percentage increase is(30−20)/20×100%=50%.
A chart headline saying “up 50%” and one saying “up10 percentage points” can both describe the same change, but they answer different comparison questions.
Read the denominator before reacting to the size of the stated increase.
16. Missing categories can change the apparent story
Suppose a report displays only the three largest categories and omits several smaller ones.
The shown bars may be accurate but the display no longer represents the complete distribution.
Check whether the graph includes all relevant categories, an “other” category, or a note explaining exclusions.
A correct local picture can still support an overbroad conclusion if the displayed subset is mistaken for the whole.
17. Absolute change and relative change can rank cases differently
Case A rises from100 to120: absolute increase20, relative increase20%.
Case B rises from10 to15: absolute increase5, relative increase50%.
A chart of absolute values may make A look more changed; a chart of percentage changes makes B larger.
Neither is automatically wrong. The chosen comparison must match the question being asked.
18. Capstone: audit a misleading display
An invented bar chart compares values98,100 and102 using a vertical axis from97 to103 and labels the third bar “massive increase”.
The numerical increase from98 to102 is4 units, about4.08% relative to98.
The truncated axis magnifies the visual difference because only a six-unit window is shown.
A responsible description would state the actual values and percentage change, then mention that the axis is truncated.
The graph may still be useful for showing small differences, but the dramatic visual impression should not substitute for numerical interpretation.
19. Independent practice
- Name a suitable basic display for counts across distinct categories.
- Values are98 and102. Find their absolute difference.
- Explain how an axis from95 to103 changes visual emphasis compared with an axis from0 to110.
- A category contains15 of60 observations. Find its pie angle.
- A100° sector represents a sample of72. Find the frequency.
- A90° sector appears in samples of40 and200. Find the two frequencies.
- Write a stem-and-leaf plot for42,45,47,51,51,56,63 using tens as stems.
- State a suitable key for Question7.
- Find the median of the data in Question7.
- Find the range of the data in Question7.
- Why should leaves normally be ordered?
- Explain one advantage of stem-and-leaf over a histogram.
- A series changes12,15,14,19 at four ordered times. What does connecting points emphasise?
- Why can a dual-axis graph be misleading?
- Why can 3D pie charts distort visual comparison?
- A share rises20% to30%. Find the percentage-point increase.
- Find the relative percentage increase for Question16.
- Case A rises100→120 and B rises10→15. Which has larger absolute increase?
- Which case in Question18 has larger relative increase?
- Explain why omitting several categories can make a display incomplete even if every shown bar is accurate.
20. Worked answers
1. Bar chart.
2. 4.
3. The truncated axis makes the same four-unit difference occupy a much larger fraction of the visible scale.
4. 90°.
5. 20.
6. 10 and50. Same proportion, different sample sizes.
7. 4|2 5 7; 5|1 1 6; 6|3.
8. For example, 4|2 means42.
9. 51.
10. 21.
11. Ordering makes rank, median, range and duplicates easier to read correctly.
12. It preserves individual observations rather than only grouped frequencies.
13. Change through the ordered sequence of times.
14. Separate vertical scales can be chosen so unrelated series appear visually aligned; each series must be read against its own axis.
15. Perspective changes apparent sector size even when angles are equal.
16. 10 percentage points.
17. 50%.
18. Case A. Increase20 versus5.
19. Case B. 50% versus20%.
20. The display represents only a selected subset, so conclusions about the full distribution require information about the omitted categories.
21. Diagnose diagram errors by reconstructing the encoded quantity
Common failures include comparing pie sectors without sample totals, reading a truncated axis as though it began at zero, scaling icons in two dimensions for a one-dimensional quantity, omitting a stem-and-leaf key, or interpreting decorative 3D perspective as data.
A useful correction note says “sector angle=share×360°”, “read the actual scale”, “equal proportions can hide unequal counts”, or “representation must preserve the intended ratio”.
Then redraw the same data using a different valid display. If the conclusion changes merely because the picture changed, inspect whether the original interpretation depended on presentation rather than data.
22. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Averages, Spread and Data Interpretation for summaries, Histograms, Frequency Density and Grouped Data for continuous grouped displays, and Sampling, Scatter Plots, Correlation and Lines of Best Fit for bivariate evidence.
Within Batch09, continue to Matrices, Data, Scalar Operations and Matrix Multiplication, Sine Rule, Cosine Rule and Non-Right Triangles, or Power Functions, Exponential Graphs and Tangent Gradients.
23. Sources and scope
The statistical tables, chart values, stem-and-leaf data and practice questions are original teaching material. The interpretive guidance separates numerical evidence from design choices in the display.
For current Singapore Secondary curriculum documents and assessment scope, consult the relevant MOE/SEAB syllabus for the learner’s subject level and year. Match stem-and-leaf, pie-chart and misleading-display depth to the actual course being studied.