Why a Recycling Bin Is a Statistics Problem
A recycling programme cannot be understood by looking at one colourful bin. We need to know what entered the waste stream, how much was correctly separated, how much recyclable material was missed, how much contamination entered the recycling stream and whether the audit sample represents normal operations.
Singapore’s National Environment Agency describes a waste audit as a structured process of identifying and quantifying the sources, amounts and types of waste generated. Its official 3R Guidebook for Schools says audits help collect baseline data and measure programme effectiveness. NEA’s Food Waste Segregation and Treatment Guidebook identifies recycled tonnage and contamination percentage as monitoring parameters.
This makes recycling an excellent answer to why mathematics is important. Ratios define rates, weighted averages combine unequal sites, sampling theory limits generalisation, confidence intervals express precision and classification tables expose disagreement. Mathematics does not decide policy alone, but it makes evidence clearer and more accountable.
For safety, students should prefer supplied or synthetic data. Do not hand-sort mixed waste containing food residues, sharps, chemicals, batteries, medical material or personal information. A real waste audit needs adult risk controls and site procedures.
Define the System Before Calculating a Rate
The audit population
Is the population one day of bins in a canteen, all school waste for a term, or every recycling station in a building? A sample can support conclusions only about a clearly defined population under a defensible sampling plan.
The material categories
Create a codebook. Categories might include accepted paper, accepted plastic, accepted metal, accepted glass, food residue, liquids, tissues, flexible packaging, electronics and unknown. Local acceptance rules should come from current official guidance, not assumptions based on another country’s recycling symbols.
The unit
Mass is common because different items have very different sizes. Item counts answer another question. Ten light wrappers can outnumber one heavy glass bottle while contributing less mass. Volume can describe bin capacity but depends on compaction. State which denominator is used.
Three Rates That Should Not Be Confused
Contamination rate
For a recycling stream,
Contamination rate = contaminant mass / total sampled recycling-bin mass.
If a sampled recycling bin contains 42 kg of accepted recyclables and 8 kg of contaminants, the contamination rate is 8/(42 + 8) = 16%.
Capture rate
For a recyclable material,
Capture rate = recyclable mass correctly placed in recycling / total recyclable mass found across recycling and general waste.
If 42 kg is correctly captured and 18 kg of recyclable material is found in general waste, capture rate is 42/(42 + 18) = 70%.
Diversion or recycling rate
A broader recycling rate may compare material recycled with total waste generated, depending on the reporting definition. It is not interchangeable with bin contamination or capture rate. Always reproduce the official denominator before comparing figures.
| Metric | Numerator | Denominator | Main question |
|---|---|---|---|
| Contamination | wrong material in recycling | all sampled recycling-bin material | How clean is the stream? |
| Capture | correctly recycled target material | target material in all sampled bins | How much target material was caught? |
| Recycling rate | recycled mass | total waste under the stated definition | How much waste entered recycling? |
Did You Know? A programme can improve capture and worsen contamination at the same time. More people may use recycling bins, but some may place wrong items inside. One percentage cannot describe both outcomes.
Weighted Averages: The Most Common Rate Mistake
Suppose Site A has 10 kg of recycling with 10% contamination, while Site B has 90 kg with 30% contamination. Averaging 10% and 30% gives 20%, but this treats the small and large sites equally.
Contaminant mass is 1 kg at A and 27 kg at B. Combined contamination is 28/(10 + 90) = 28%.
The correct combined rate is a mass-weighted average:
p = (m₁p₁ + m₂p₂)/(m₁ + m₂).
Unweighted averages are appropriate only when equal weighting matches the question. If the question is “average site rate”, 20% may be meaningful; if it is “fraction of all sampled mass contaminated”, 28% is correct. Mathematics forces the question into view.
Sampling Plans
Convenience sampling
Auditing the easiest bin after a clean-up event is convenient but may be biased. A confidence interval cannot repair selection bias. Randomness must enter the selection process, not merely the formula.
Simple random sampling
List all eligible bin-time units and select randomly. This supports familiar statistical inference if units are sufficiently independent and measurement is consistent.
Stratified sampling
Divide the population into meaningful strata such as canteen, classrooms, offices and event spaces, then sample within each. Combine strata with weights based on their share of waste mass or another defined population measure.
Cluster and time sampling
Auditing all bins on three selected days is a cluster-like design. Results within a day may be correlated, so treating every item as fully independent exaggerates effective sample size. Weekdays, events and school holidays can differ.
Worked Stratified Example
Imagine three areas:
| Area | Share of normal recycling mass | Sample contamination rate |
|---|---|---|
| Canteen | 50% | 24% |
| Classrooms | 30% | 8% |
| Offices | 20% | 12% |
The weighted estimate is
0.50(0.24) + 0.30(0.08) + 0.20(0.12) = 0.168, or 16.8%.
The unweighted mean is (24 + 8 + 12)/3 = 14.7%. The difference comes from weighting. If the 50%, 30% and 20% shares are themselves uncertain or outdated, run a sensitivity analysis with plausible alternatives.
Confidence Intervals and Their Conditions
If items were independently classified as contaminant or accepted under a random sample, an approximate standard error for a proportion is
SE(p̂) = √[p̂(1 − p̂)/n].
For 40 contaminated items out of 200, p̂ = 0.20 and SE ≈ √(0.20×0.80/200) ≈ 0.0283. A rough 95% interval using p̂ ± 1.96SE is about 14.5% to 25.5%.
But waste audits often use mass, not equal items. A bottle and a paper scrap do not contribute equally by mass. Items in the same bin are clustered. Small samples and rare categories make the simple normal approximation poor. A confidence interval is only as credible as the sampling and measurement model.
For student learning, label the interval “illustrative item-based approximation under independence”. That phrase is not weakness; it is statistical honesty.
Classification Error
Two auditors may disagree whether an item is accepted, contaminated or unknown. Build a confusion matrix against an agreed review classification:
| Auditor label | Reviewed accepted | Reviewed contaminant |
|---|---|---|
| Accepted | 86 | 6 |
| Contaminant | 4 | 24 |
Overall agreement is (86 + 24)/120 = 91.7%. Yet the table also shows six contaminants missed and four accepted items wrongly rejected. These error types have different consequences.
Percent agreement can be inflated when one category dominates. Cohen’s kappa or class-specific precision and recall can add information, but they do not replace a clear codebook and adjudication procedure.
Pre-Post Comparisons
Suppose contamination falls from 24% in a pre-audit to 17% after new signage. The change is −7 percentage points. It is not a 7% relative decrease. Relative decrease is (24 − 17)/24 ≈ 29.2%.
Before claiming causation, check:
- Were the same locations and times sampled?
- Did total mass change?
- Were category definitions identical?
- Was there a school event or holiday?
- Did auditors know which period they were classifying?
- Were bins moved or collection schedules changed?
An improvement after an intervention is encouraging evidence, but without a stronger design it may not prove the signage caused the change.
Measurement Uncertainty in Mass
If each category is weighed, scale resolution, tare, zero drift and repeatability matter. Contamination rate p = C/T depends on contaminant mass C and total mass T. Because C is part of T, they are correlated; blindly combining them as independent can be wrong.
A practical sensitivity check recalculates the rate after adding and subtracting plausible mass errors. If C = 8.0 kg and T = 50.0 kg, p = 16.0%. If uncertain residues could shift classification and mass by ±0.5 kg, evaluate the boundary scenarios rather than hiding the ambiguity in a tiny decimal.
Unknown material should remain visible. Reassigning every unknown to the preferred category biases the result. Report an unknown fraction and, if helpful, a range under “all unknown accepted” and “all unknown contaminant” assumptions.
Common Misconceptions
“A cleaner-looking bin has lower contamination”
Appearance is not a measured mass fraction. Hidden liquid or food residue may matter, while large visible items can distort perception.
“Average the percentages from each bin”
Only if equal bin weighting matches the question. To estimate combined mass contamination, weight by sampled mass.
“A large item count means a precise mass rate”
Items have unequal masses and are clustered within bins. Effective information depends on the sampling unit and design.
“A 95% confidence interval means a 95% chance the true rate is inside”
In the frequentist interpretation, the procedure captures the fixed population parameter in about 95% of repeated samples under its assumptions. The interval does not repair biased selection.
“A post-programme improvement proves causation”
Other changes may explain the difference. Comparable sampling and, where feasible, control groups strengthen causal interpretation.
Guidance for Students and Parents
- Begin with an official local list of accepted materials.
- Define population, time window and sampling unit.
- Use a category codebook with an unknown option.
- Keep mass and item-count analyses separate.
- Calculate contamination, capture and recycling rates separately.
- Weight unequal strata appropriately.
- Preserve raw category totals.
- Report confidence intervals only with their assumptions.
- Use synthetic data if real sorting is unsafe.
Parents can ask, “What was never sampled?” and “What denominator did you use?” These questions cultivate statistical literacy that transfers to health studies, opinion polls and school surveys.
Where This Mathematics Leads
Waste audits connect mathematics with environmental management, logistics, public policy, data science and quality improvement. Students learn that sustainability claims depend on measurement definitions and sampling design, not only good intentions. These skills keep career options open across science, engineering, business and government.
Continue with the eduKate Mathematics Learning Hub, Why Mathematics? | Comparing Percentages Fairly and the related Why Mathematics? | Supermarket Checkouts, Queue Lengths and Little’s Law, which shows another way operational data supports decisions.
Frequently Asked Questions
Should contamination be measured by mass or item count?
Mass usually aligns better with material-flow questions, while item count can reveal frequent mistakes. Report the chosen unit and do not mix the denominators.
Why is an unweighted average of bin percentages often wrong?
Bins can contain very different masses. A small bin should not contribute as much as a large bin when estimating the fraction of total mass.
Can a confidence interval describe an entire school from one convenient bin?
No. The formula addresses random sampling variation under assumptions; it does not remove convenience-sampling bias.
What should happen to unknown items?
Keep an unknown category, review it where safe and authorised, and show how conclusions change under plausible reclassification.
The Bigger Answer to “Why Mathematics?”
Recycling audits show why mathematics matters because every percentage hides a population, denominator and classification rule. Weighted averages prevent small bins from dominating, sampling theory limits overclaiming, and uncertainty keeps apparent improvements in perspective. Mathematics turns a sustainability intention into evidence that can be checked, improved and communicated honestly.
A Practical Mathematics Studio
Prefer supplied or synthetic audit data. Students should not hand-sort mixed waste containing food residues, sharps, chemicals, batteries, medical material or personal information; any real audit needs adult risk controls, gloves and local procedures. Keep raw observations, units, assumptions and limitations beside each result.
Investigation 1: Define the audit population
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 2: Draw a waste-stream map
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 3: Choose a mass-based denominator
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 4: Calculate contamination rate
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 5: Calculate capture rate
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 6: Separate generation from recycling
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 7: Build a category codebook
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 8: Design a random sample
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 9: Design a stratified sample
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 10: Compare weekday and weekend strata
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 11: Calculate a weighted overall rate
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 12: Avoid averaging percentages directly
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 13: Estimate a sample proportion
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 14: Calculate an approximate standard error
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 15: Build a confidence interval
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 16: Inspect small-sample limits
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 17: Measure inter-rater agreement
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 18: Create a confusion matrix
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 19: Separate false contamination labels
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 20: Track missing and unknown items
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 21: Compare pre-post audits
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 22: Use a difference in proportions
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 23: Check seasonality
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 24: Plot category contributions
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 25: Run a sensitivity analysis
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 26: Estimate scale uncertainty
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 27: Document excluded material
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 28: Protect privacy and safety
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 29: Write a policy-evidence boundary
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 30: State a bounded conclusion
Declare the population, time window, site, sampling unit, category definitions and mass denominator. Preserve category totals, calculate the named rate or sampling statistic, and show how excluded or unknown material is handled.
Evidence check: Check that numerator and denominator cover the same sample and that weighted strata are combined by mass, not by an unweighted average of percentages. Inspect selection bias before interpreting a confidence interval. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot estimate a whole school or city from a convenience sample, prove causation or substitute for NEA reporting requirements. Operational conclusions must stay within the sampled places and dates. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Studio Synthesis
Choose four investigations that use different representations: a labelled system diagram, an equation, a raw-data table and a graph. Arrange the report from question to method, calculation, residual or sensitivity check, limitations and conclusion. Another reader should be able to reproduce one result without guessing a unit, category or selection rule.
Change one input or classification rule at a time and recalculate. Identify which conclusion remains stable and which changes direction. Sensitivity analysis reveals the assumptions controlling a result and is usually more valuable than an extra decimal place.
Exchange only the raw data and definitions with a partner. If the calculation reproduces and the same bounded conclusion follows, confidence increases. If it does not, repair the method or labels before treating the output as evidence.