Singapore School Mathematics Operating Manual · Chapter 34
Mathematics constantly compresses information.
A mean compresses an entire data set into one number. A ratio compresses two raw quantities into a relationship. A graph compresses a table into visual structure. A formula compresses repeated calculations into one rule. A percentage compresses a part-whole relationship into a common scale. A factorised expression can compress a long polynomial into visible root structure.
Compression is useful because it reduces complexity. But every compression has a cost: some information disappears.
This chapter develops a practical question for school Mathematics: what does this summary preserve, and what can no longer be recovered from it?
1. A summary is a smaller representation of a larger object
The data set 10,20,30 has mean 20.
The one number 20 is a compressed representation of three observations.
It preserves total divided by count, but not the individual values.
2. Compression can preserve exactly what a task needs
If the only target is total score from mean and count, the mean is sufficient together with the count.
For five values with mean 12, total is 60.
The original five values are unnecessary for that target.
3. A summary can be sufficient for one question and insufficient for another
Mean 50 may be enough to compare average level.
It is not enough to compare spread.
Data sets 50,50,50 and 0,50,100 have the same mean but very different variability.
4. Information loss means different originals share the same summary
Whenever two distinct states map to the same compressed value, inversion is ambiguous.
This links directly to Observability and Identifiability.
5. The arithmetic mean compresses location but not shape
A mean tells us the balancing point of numerical data.
It does not tell us whether the distribution is symmetric, skewed, clustered, bimodal or full of outliers.
Additional summaries are needed for those features.
6. Median compresses order information differently
The median records a central position after sorting.
It is less sensitive to extreme values than the mean.
But it does not tell us the total or exact distances between observations.
7. Range compresses spread very aggressively
Range = maximum − minimum.
It uses only two observations and ignores every value in between.
Two very different data sets can share the same range.
8. Multiple summaries reduce ambiguity
Knowing mean and range reveals more than either alone.
Knowing mean, median, quartiles and sample size reveals still more.
Yet even several summaries may not reconstruct the original data set uniquely.
9. A histogram is a compressed data representation
Raw values are grouped into bins.
Exact within-bin positions disappear.
The histogram preserves frequency by interval while discarding fine resolution.
10. Wider bins increase compression
Using 10-unit classes instead of 1-unit classes reduces detail.
The representation becomes simpler but may hide clusters or gaps.
Compression strength and resolution move in opposite directions.
11. Tables can compress repeated verbal information
A table places categories and quantities into a structured layout.
It reduces prose while preserving selected relationships.
The table still reflects a choice about which variables were kept.
12. Graphs compress many values into shape
A line graph can reveal trend, turning points and relative change faster than a long table.
But reading exact values may become less precise, especially at coarse scale.
Representation trades one kind of access for another.
13. A formula compresses infinitely many evaluations
y=2x+3 replaces a potentially infinite table of input-output pairs.
The formula preserves the exact relationship under the model.
This is highly efficient lossless compression for that mathematical object.
14. Not all mathematical compression is lossy
12/18 and 2/3 represent the same rational number exactly.
Simplifying the fraction removes redundant common scale without losing value.
Factorising x²−9 into (x−3)(x+3) also preserves the polynomial exactly.
Some transformations compress representation while preserving all mathematically relevant information.
15. Canonical forms can remove redundancy without losing meaning
Reducing a ratio to lowest terms gives one simpler representative of the same proportional relationship.
This is the role developed in Equivalence and Canonical Forms.
16. Rounding is deliberately lossy
7.241 and 7.249 can both round to 7.2 at one decimal place.
The compressed display is easier to use but does not identify the exact original.
The inverse of rounding is an interval.
17. Significant figures are a controlled compression rule
Reporting 12,347 as 12,300 to three significant figures discards fine detail while preserving approximate scale.
The correct number of significant figures should match the precision needed and justified.
18. Percentages normalise and compress
18 out of 24 becomes 75%.
The percentage preserves proportional performance while hiding the original denominator unless it is reported separately.
18/24 and 75/100 look equivalent proportionally but came from different raw counts.
19. Rates compress two quantities into one relationship
60 km/h combines distance and time.
The rate does not reveal the actual trip length or duration by itself.
Many journeys share the same average speed.
20. Density is another compressed relation
Density = mass/volume.
Knowing density alone does not identify mass or volume separately.
It preserves their ratio.
21. Probability compresses a random model into event weights
P(A)=0.3 tells us the model weight of event A.
It does not tell us which exact outcome will occur on the next trial.
A probability is a model-level summary, not an event-level prediction.
22. Expected value compresses a distribution
A fair die has expected value 3.5.
This one value summarises the weighted average of all outcomes.
It does not identify the distribution uniquely: other distributions can also have expectation 3.5.
23. A checksum can preserve error-detection information without preserving the original object
A digit-sum modulo 9 can help detect arithmetic errors.
It compresses a number to one remainder class.
Many different integers share the same remainder, so reconstruction is impossible, but the summary is sufficient for a narrow checking task.
24. Parity is extreme compression
Every integer becomes only even or odd.
Almost all numerical detail disappears.
Yet parity can be sufficient to prove impossibility or classify an argument.
This is a powerful lesson: a very small summary can be enough if it preserves exactly the property the target depends on.
25. Residue classes preserve modular information
For divisibility by 3, exact integer magnitude may be irrelevant; remainder modulo 3 can be enough.
State compression removes irrelevant detail while retaining the decision variable.
26. A sufficient summary is target-dependent
There is no universally best summary.
For total, mean plus count may be sufficient.
For maximum, the mean is not.
For parity, exact value is unnecessary.
For exact reconstruction, compressed summaries may be inadequate.
27. Compression should be chosen after the target is understood
Compress too early and you may discard information needed later.
For example, replacing a data set by its mean before an outlier analysis makes the later task impossible.
Keep raw information until you know which features are safe to discard.
28. Premature rounding is premature compression
Replacing exact surds or full calculator values with rounded decimals at an intermediate stage discards information.
If later operations amplify the rounding, the loss matters.
Preserve exact or higher-precision values until the requested output stage where practical.
29. Compression can improve reasoning by exposing structure
Let u=x+1/x in a symmetric algebra problem.
The temporary variable compresses repeated structure into one object.
The reduction is useful because the target depends on that symmetric combination.
30. Compression can improve communication
A carefully chosen graph, ratio or summary table can convey structure more efficiently than raw detail.
Good mathematical communication compresses without hiding information necessary for the reader’s task.
31. Compression can also mislead
Averages can hide inequality, peaks or subgroups.
Percentages can hide small sample sizes.
Indexes can hide baseline values.
Graphs can hide truncated axes.
Whenever a representation is compact, ask what was omitted.
32. A compression audit
Ask:
What original information existed? What summary replaced it? Which properties are preserved exactly? Which are only approximated? Which are lost? Could two different originals share the same summary? Is the summary sufficient for the current target? Might a later subproblem need discarded detail?
33. Independent practice
1. Give two different data sets with mean 10.
2. Why does knowing a mean not determine a range?
3. What information does a histogram lose relative to raw data?
4. Why is rounding lossy?
5. What does simplifying 12/18 to 2/3 preserve?
6. Why can percentage alone hide information?
7. What does parity preserve?
8. Give one example where a very small summary is sufficient for a task.
9. Why should raw information often be retained until the target is clear?
10. What is the main danger of premature rounding?
34. Worked answers
1. 10,10,10 and 0,10,20 both have mean 10.
2. Data with the same mean can have different minimum and maximum values.
3. Exact within-bin values and ordering of individual observations.
4. Several exact values can map to the same rounded display.
5. The exact rational value and proportional relationship.
6. It may hide the original numerator, denominator and sample size.
7. Whether an integer is even or odd.
8. Remainder modulo 9 can be sufficient for one digit-sum consistency check.
9. A later task may depend on information the summary discarded.
10. Lost precision can propagate or be amplified in later calculations.
35. Continue through Batch 09
Use Observability and Identifiability to test whether a compressed summary still reveals the target, Mathematical Provenance and Result Lineage to record what source information a summary came from, and Error Budgets and Cumulative Uncertainty when compression includes approximation.
Return to the BTT Mathematics Hub for Batch 09.