Two districts can have success rates of 90% and 50%. Their average rate is 70% if each district counts once. Yet if the first district contains 100 cases and the second contains two, the pooled rate is 91 successes out of 102 cases, about 89.2%. Neither calculation is an arithmetic mistake. They answer different questions because they use different weights.
This guide brings that distinction into zonal statistics. A zone may be a district, catchment, habitat polygon or administrative area. A local ratio has a numerator and denominator; combining zones requires a decision about what should count equally. The examples are invented and designed for transparent checking. They are not official indicators or software instructions for a particular dataset.
This specialist guide belongs under The Gold Standard of Zonal Statistics. It connects to The Gold Standard of Education: Learning That Transfers into Life and the verified How X Works hub.
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Keep numerators, denominators and weights visible, then choose whether zones, cases, people, area or exposure should count equally.
ROUTE 1 · CHAPTERS 1–12
Define the ratio account
Build the central two-zone example and distinguish zone, case, area and cell weights.
ROUTE 2 · CHAPTERS 13–24
Control aggregation
Check stability, hierarchy, spatial support, time and the changed-context example.
ROUTE 3 · CHAPTERS 25–35
Audit software and evidence
Validate formulas, misconceptions, targets and reporting choices.
ROUTE 4 · CHAPTERS 36–47
Extend the weighting model
Handle survey weights, standardisation, uncertainty, hierarchy and confidentiality.
ROUTE 5 · CHAPTERS 48–57
Transfer and continue
Test synthetic data, revisions, benchmarks, sensitivity and verified routes.
Chapter contents
Define the ratio account · Chapters 1–12
1. Define the numerator and denominator
2. Work the central two-zone example
3. Ask what should count equally
4. Derive pooled rates as denominator-weighted means
5. Keep counts when publishing percentages
6. Avoid averaging rounded rates
7. Distinguish people weighting from area weighting
8. Distinguish cell weighting from zone weighting
9. Keep the spatial support compatible
10. Handle NoData separately from zero
Control aggregation · Chapters 13–24
13. Check for Simpson’s paradox
14. Preserve time exposure in rates
15. Work an area-versus-population example
16. Keep ecological fallacy limits visible
17. Audit boundary changes over time
18. Treat fractional overlap carefully
19. Choose the map legend for the denominator
20. Build a four-column audit table
21. Work a changed-context example
Audit software and evidence · Chapters 25–35
25. Separate a policy target from an empirical average
26. Compare median zone performance
27. Use benchmark attainment counts
28. Test sensitivity to weighting choices
29. Build a reproducible formula record
30. Use population-at-risk denominators
31. Distinguish prevalence and incidence
32. Handle age-standardised rates
33. Separate averages of percentages from percentage-point change
Extend the weighting model · Chapters 36–47
36. Keep confidential suppression honest
37. Transfer the method beyond percentages
38. Plan a reader-facing summary
39. Distinguish macro and micro averages
40. Choose weights before viewing preferred outcomes
41. Normalise weights and check their sum
43. Handle survey weights separately
44. Standardise when populations differ
45. Preserve uncertainty through aggregation
Transfer and continue · Chapters 48–57
48. Avoid double counting across time windows
49. Treat moving populations carefully
50. Compare rates on common definitions
51. Separate target weights from convenience weights
52. Test aggregation with a synthetic dataset
53. Maintain revision provenance
54. Use inequalities as a quick reasonableness check
55. Communicate disagreement between summaries
A rate or proportion is not one free-standing number. It is a relationship between a numerator and denominator under a rule. “90% access” might mean 90 households with access out of 100 surveyed households, or 90 square kilometres served out of 100. Those denominators represent different things.
Write the pair beside every zonal rate. Include unit, population, time and missing-data rule. When only a percentage survives, later analysts cannot safely reconstruct the pooled result. Metadata should preserve the counts that make the ratio interpretable.
Zone A records 90 successful cases out of 100, so its local rate is 90%. Zone B records one successful case out of two, so its local rate is 50%. The unweighted mean of the two local rates is (90% + 50%)/2 = 70%.
The pooled rate is (90 + 1)/(100 + 2) = 91/102 ≈ 89.22%. The difference is large because equal-zone averaging gives tiny Zone B the same influence as Zone A, while pooling weights each zone by its denominator. Name the intended estimand before choosing between them.
If the question is “What is the average rate of a zone in this set?”, equal-zone weighting can be appropriate. Each zone is one observational unit. If the question is “What proportion of all cases succeeded?”, the pooled rate is appropriate because each case contributes equally.
These are not rival ways to conceal the truth. They are different summaries. Publish both when both perspectives matter. Avoid labelling one simply “the overall average” without its weighting rule.
CHAPTER 4 OF 57 · Define the ratio account
4. Derive pooled rates as denominator-weighted means
The pooled rate equals a weighted mean of local rates where each weight is the zone denominator. For Zone A and B, (0.90×100 + 0.50×2)/(100+2) = 91/102. This identity makes the weighting visible.
It also provides an audit check. If supplied numerators, denominators and local rates do not reconcile within rounding tolerance, investigate the data. Possible causes include excluded cases, inconsistent time windows or percentages rounded before aggregation.
CHAPTER 5 OF 57 · Define the ratio account
5. Keep counts when publishing percentages
A table containing only 90% and 50% cannot recover 89.22% unless denominators are known. Many combinations of counts yield the same percentages. Ten of twenty and one of two both give 50%, but contribute differently to pooled totals.
Store numerator, denominator, rate and validity flags. Use the rate for display and the counts for recombination. When confidentiality prevents small counts being released, document that pooled reconstruction is limited rather than encouraging users to average displayed percentages.
Suppose a dashboard displays 33% and 67%, while the underlying fractions are one-third and two-thirds. Averaging displayed values gives 50%, which happens to match here. In other cases, early rounding accumulates error, especially with many zones and small differences.
Aggregate from underlying counts or higher-precision values, then round the final display. State rounding rules. A displayed percentage should not become the authoritative source when auditable components remain available.
CHAPTER 7 OF 57 · Define the ratio account
7. Distinguish people weighting from area weighting
A population-weighted rate asks about people or cases. An area-weighted rate asks about land area. A large sparsely populated zone can dominate an area-weighted summary but contribute little to a resident-weighted summary.
Name the unit represented by each weight. Do not use polygon area as a proxy for population merely because GIS software can calculate it. Join an appropriate denominator dataset and examine its date, coverage and spatial compatibility.
CHAPTER 8 OF 57 · Define the ratio account
8. Distinguish cell weighting from zone weighting
Raster zonal statistics often summarise cells within each zone. A later average across zone means introduces another level. An equal-zone average gives every zone one vote, regardless of its valid-cell count. A valid-cell-weighted average gives zones with more included cells more influence.
Neither is automatically the physical area average. Cell areas can differ in latitude-longitude grids, and boundary overlaps may be fractional. Track the hierarchy: observation or cell, zone, then multi-zone summary.
Zone rates may refer to different boundaries, years or inclusion rules. Pooling them assumes the numerators and denominators partition or validly cover the intended total. Overlapping zones can double-count cases; gaps can omit them.
Check whether zones are mutually exclusive and collectively cover the target. If they overlap by design, do not sum totals without a rule assigning shared cases. A visually complete map can hide duplicate records beneath boundaries.
Zero successes out of ten valid cases is a 0% rate. No valid denominator is undefined, not 0%. Treating an unavailable zone as zero lowers an average and invents evidence.
Create distinct states for zero, missing and not applicable. If a zone has numerator zero and denominator zero, its rate is undefined. Decide whether it is excluded from an equal-zone mean, represented separately or handled by a model. Report the number of zones and cases excluded.
CHAPTER 11 OF 57 · Define the ratio account
11. Do not divide by a tiny denominator without context
Rates from tiny denominators are unstable. Zone B’s one success out of two gives 50%, but one changed case moves it by 50 percentage points. A zone with 500 cases changes far less from one record.
Show denominators and uncertainty. Consider suppressing or aggregating small cells under the applicable governance rules. Do not hide a small denominator inside a bright colour scale that makes the estimate look as stable as a large-zone rate.
CHAPTER 12 OF 57 · Define the ratio account
12. Separate descriptive weighting from precision weighting
Denominator weighting answers a population aggregation question. Statistical precision weighting may use inverse variances in a model or meta-analysis. These weights serve different purposes and need not be equal.
Do not describe an inverse-variance estimate as the pooled case rate unless the formula actually produces that quantity. Likewise, a large denominator is not the only determinant of precision when sampling designs, clustering and measurement error differ.
An aggregate comparison can reverse the direction seen within subgroups when subgroup composition differs. Suppose two regions have different mixtures of easy and difficult cases. A pooled rate can favour one region even if the other has higher rates within each case type.
Stratify by relevant, pre-specified groups and inspect counts. Do not search indefinitely for a subgroup that reverses the result. Explain which comparison answers the decision question and why composition matters. Aggregation is a model of what can be combined.
Some rates use person-time, vehicle-kilometres or hours of operation rather than simple case counts. Adding numerators across zones is valid only when denominators share compatible units and definitions. An annual rate and a monthly rate cannot be pooled by adding their displayed percentages.
Convert to a common exposure basis where justified, retain the period and account for partial coverage. A zone observed for twice as long may deserve twice the exposure weight for a pooled event rate, but only if the process and reporting definitions are compatible.
Zone C covers 90 square kilometres with 100 residents and has 80 residents served. Zone D covers 10 square kilometres with 900 residents and has 630 served. The resident rates are 80% and 70%.
The equal-zone mean is 75%. The pooled resident rate is (80+630)/(100+900)=71%. An area-weighted mean of the zone rates is (0.80×90+0.70×10)/100=79%. Each result is traceable, but only the 71% figure answers “What share of all residents is served?”
A zone rate describes an aggregate. It does not state the outcome for every person or place inside the zone. Two districts with the same rate can have different within-zone patterns and inequalities.
Use finer data or distributional measures when the decision concerns subgroups or local hotspots. Do not assign the zone mean to individuals as if it were an observed personal value. The root guide’s ecological-fallacy warning applies directly to ratio maps.
If zone boundaries change, a rate difference may reflect reassignment rather than changed outcomes. Preserve boundary versions and consider a common geography, areal interpolation or case-level reaggregation where lawful and justified.
Do not compare an old Zone A with a new Zone A merely because the name is unchanged. Map the actual spatial support. If reconstruction is impossible, qualify the trend instead of drawing a seamless line.
Raster cells and administrative cases do not always fit zones neatly. Fractional area weighting may be appropriate for continuous fields, but population or event counts can be distributed unevenly within a cell. Multiplying a cell count by overlap fraction assumes uniform density unless ancillary data support a better allocation.
State the allocation model. The specialist guide Bounding a Subzone Mean When Only Coarse Cell Averages Remain shows how coarse averages can support bounds rather than a falsely exact subzone value.
CHAPTER 19 OF 57 · Control aggregation
19. Choose the map legend for the denominator
A choropleth of rates should display normalised values, while a map of counts answers a different question. Yet a rate map without denominator information can make tiny and large zones look equally certain.
Use labels, tooltips, companion maps or uncertainty layers to show denominators. Consider whether equal intervals, quantiles or policy thresholds fit the decision. A colour class is a display choice, not a statistical conclusion.
For every zone, keep numerator, denominator, local rate and weight. Add validity, time and boundary version when needed. Compute the equal-zone mean from valid local rates and the pooled rate from sums. Reconcile the denominator-weighted identity.
Then write one sentence for each result. “The typical zone rate under equal-zone weighting is…” and “Across all recorded cases, the pooled rate is…”. If they differ substantially, explain the composition rather than selecting the more attractive number.
Three zones report 8/10, 45/50 and 1/10 successes. Their local rates are 80%, 90% and 10%. The equal-zone mean is 60%. The pooled rate is 54/70≈77.14%. The first asks about a zone selected with equal probability; the second asks about a case selected from all 70.
Now change only the third denominator from 10 to 100 while keeping one success. Its local rate becomes 1%; the equal-zone mean becomes 57%, and the pooled rate becomes 54/160=33.75%. Denominator composition drives the pooled result, which is why totals must remain visible.
GIS and statistical packages may offer mean, weighted mean, sum and ratio operations with different expectations. Some tools average an input rate raster cell by cell; others compute ratios from separate numerator and denominator fields. These procedures are not interchangeable.
Read the version-matched documentation and validate a small hand-worked case. Confirm how NoData, zero denominators, duplicate zones and weights are handled. A plausible output column does not prove the intended estimand was calculated.
The average of rates is not generally the rate of totals. Pooling is not always superior; it answers a different question. Equal-zone weighting is not neutral—it gives each zone equal influence. Area weighting does not automatically represent people. A displayed 0% is not a substitute for missing data.
At the same time, different summaries need not be contradictory. They can reveal scale and distribution. The gold standard names the unit, denominator, weight and interpretation so readers can understand why the numbers differ.
CHAPTER 25 OF 57 · Audit software and evidence
25. Separate a policy target from an empirical average
A target such as “every zone should exceed 80%” is a normative threshold, while a mean or pooled rate describes observed data. A pooled 90% can coexist with several zones below target if a large high-performing zone dominates. Conversely, most zones can meet the target while a populous low-performing zone keeps the pooled rate low.
Report target attainment, equal-zone distribution and pooled population outcome separately. Name which one controls the decision. Do not use a strong pooled result to erase local failure or use one small zone to describe every resident.
Where resources are limited, the measures support different priorities: a minimum standard may direct help to the lowest zone, while total benefit may favour a large denominator. Make that value judgment explicit instead of hiding it inside a weighting formula.
The median of local rates describes the middle zone after ordering. It is insensitive to denominator size and less affected by an extreme zone rate than the mean. It answers neither the typical case nor the pooled population share.
Publish the median when distribution across zones matters, alongside quartiles and counts. Do not substitute it for a weighted rate. A dashboard can show median, range and pooled total without forcing one number to carry every interpretation.
Policy may ask how many zones meet a threshold rather than their average rate. Count eligible zones above the benchmark and show denominator sizes. Ten small zones meeting a standard do not imply most residents are covered.
Pair zone attainment with population coverage when both matter. Thresholds create discontinuities, so show values near the line and measurement uncertainty.
CHAPTER 28 OF 57 · Audit software and evidence
28. Test sensitivity to weighting choices
Calculate equal-zone, denominator-weighted and policy-relevant alternatives on the same validated table. Explain the unit each one represents. Large divergence signals heterogeneous sizes or correlation between size and rate.
Sensitivity analysis is not a contest to choose the preferred answer. It reveals which conclusions depend on weighting and which persist. Predefine the primary estimand, then use alternatives as checks.
CHAPTER 29 OF 57 · Audit software and evidence
29. Build a reproducible formula record
Store the exact expression, field names, filters, missing-data handling and software version. For a pooled rate, record sum(numerator)/sum(denominator), not the ambiguous word average. Include grouping and boundary keys.
Recompute selected outputs manually after every pipeline change. A reproducible formula record turns a zonal statistic into inspectable evidence rather than a transient dashboard value.
CHAPTER 30 OF 57 · Audit software and evidence
30. Use population-at-risk denominators
A numerator should be divided by the population that could experience the event. Total residents may be unsuitable if only a subgroup is eligible. Different eligibility rules across zones make rates incomparable.
Define start and end dates, inclusion criteria and source. If eligibility is estimated, preserve its uncertainty. A polished map cannot compensate for an incorrect population at risk.
CHAPTER 31 OF 57 · Audit software and evidence
31. Distinguish prevalence and incidence
Prevalence describes existing cases at a time or during a period, while incidence concerns new events among those at risk. Both may be shown as percentages or rates, but their denominators and interpretations differ.
Do not pool prevalence and incidence columns. Label the measure, time reference and repeat-event rule. A zone can have high prevalence and falling incidence, so one cannot substitute for the other.
Age-standardised rates apply weights from a reference population to age-specific rates. They support comparison when age structures differ. They are not literal proportions of current residents.
Use the same reference population and age bands across zones. Publish the crude rate alongside the standardised rate when actual burden matters. Do not recover event counts by multiplying a standardised rate by the local population.
CHAPTER 33 OF 57 · Audit software and evidence
33. Separate averages of percentages from percentage-point change
If Zone A rises from 40% to 50%, the change is ten percentage points and a 25% relative increase. Zone B moving from 80% to 90% also gains ten points but only 12.5% relatively.
Choose and label the change measure before averaging across zones. Percent and percentage point are not interchangeable. Weighting changes the summary of change just as it changes the level.
CHAPTER 34 OF 57 · Audit software and evidence
34. Use covariance to explain the gap
The difference between equal-zone and denominator-weighted means reflects the relationship between local rates and denominator sizes. When larger zones tend to have higher rates, the pooled rate exceeds the equal-zone mean; when they tend to have lower rates, it falls below.
Plot rate against denominator and inspect influential points. This descriptive check explains composition without implying causality. It can guide questions about service design, urbanisation or data quality.
CHAPTER 35 OF 57 · Audit software and evidence
35. Guard against denominator mismatch after joins
GIS joins can attach a numerator from one year to a population denominator from another, or duplicate a denominator across several records. The resulting rate may look plausible while being wrong.
Validate join cardinality, dates and identifiers. Sum joined denominators and compare with the source. A one-to-many join should be intentional and accompanied by an allocation rule.
CHAPTER 36 OF 57 · Extend the weighting model
36. Keep confidential suppression honest
Small counts may be suppressed, rounded or perturbed to protect privacy. A displayed dash is not zero. Pooled totals reconstructed from published cells may not equal official totals because disclosure control alters components.
Follow the provider’s instructions and use authorised aggregate totals where available. Explain why arithmetic may not reconcile. Do not reverse-engineer protected small values.
CHAPTER 37 OF 57 · Extend the weighting model
37. Transfer the method beyond percentages
The same issue appears with speeds, prices, densities and error rates. Average speed across trips differs depending on whether trips, distance or time receive equal weight. Mean price per unit differs from total price divided by total units when item quantities vary.
For every ratio, ask what its denominator represents and which unit should count equally. This transfer check turns one GIS formula into a general quantitative habit.
Lead with the decision question, then give the chosen summary and denominator. Add the alternative weighting when it changes interpretation materially. Show a small table of zone counts and avoid a map-only result.
End with coverage, missing zones, boundary date and revision status. The reader should be able to reconstruct why a number exists and what it does not describe.
CHAPTER 39 OF 57 · Extend the weighting model
39. Distinguish macro and micro averages
An equal-zone mean is often called a macro average because it calculates each zone’s metric and gives zones equal weight. A pooled rate is often called a micro average because it combines case-level numerators and denominators. The names help only when definitions accompany them.
Macro averages highlight performance across reporting units and can keep small zones visible. Micro averages describe the total recorded population and can be dominated by large zones. Publishing both can reveal whether performance differs systematically with zone size.
CHAPTER 40 OF 57 · Extend the weighting model
40. Choose weights before viewing preferred outcomes
If analysts choose equal, population or area weights after seeing results, the summary can become outcome-driven. Define the estimand and weighting plan in advance where possible.
Document alternatives used for sensitivity analysis. A policy may legitimately care about both residents and places, but those perspectives should appear as named summaries rather than one selected after comparison. Governance is part of statistical quality.
CHAPTER 41 OF 57 · Extend the weighting model
41. Normalise weights and check their sum
Weighted means can use raw weights and divide by their sum, or normalised weights that sum to one. Errors arise when percentages are used as if they were proportions or when omitted zones leave weights summing to less than expected.
For the 90% and 50% example, denominator weights are 100 and 2. Normalised weights are 100/102 and 2/102. Multiplying rates by these and adding yields 89.22%. Displaying the check makes implementation errors visible.
A local rate may already incorporate population exposure. Multiplying it by both denominator and a second population field can double-weight large zones. Understand how each published statistic was constructed.
Rebuild from numerators and denominators where possible. If only modelled rates exist, follow the model documentation. A weight should represent the target aggregation, not a vague preference for “more important” zones.
Survey data may attach a sampling weight to each respondent so estimates represent a population. The zonal numerator and denominator should then be weighted totals, not raw respondent counts. Pooling raw counts across differently sampled zones can be biased.
Respect strata, clusters and replicate weights when estimating uncertainty. GIS boundaries do not simplify a complex survey design. Consult the survey documentation and statistical expertise rather than treating a weighted estimate as an ordinary fraction.
CHAPTER 44 OF 57 · Extend the weighting model
44. Standardise when populations differ
Crude rates can differ because zone populations have different age, risk or case mixes. Direct or indirect standardisation may support a fairer comparison under a stated reference distribution.
Standardised rates are modelled summaries and should not be pooled by adding their displayed numerators unless those components exist and the method permits it. Publish crude and standardised results with their purposes. A comparison metric is not necessarily the actual share of all cases.
CHAPTER 45 OF 57 · Extend the weighting model
45. Preserve uncertainty through aggregation
Local rates have sampling or measurement uncertainty. An equal-zone mean’s uncertainty depends on between-zone variation and the sampling process. A pooled rate’s uncertainty depends on the combined design, not simply the average of local confidence intervals.
Use a method appropriate to the data-generating process. Avoid drawing error bars by averaging lower and upper bounds. Keep descriptive arithmetic separate from inferential claims.
A very large denominator can dominate a pooled rate. Recalculate after excluding each zone in turn and report the sensitivity where decisions are affected. This is a diagnostic, not a licence to remove inconvenient data.
Check data quality, coverage and definition for influential zones. A coding error in one large region can shift the total dramatically while leaving an equal-zone mean almost unchanged. Both summaries help detect different problems.
Neighbourhoods may nest within districts and districts within regions. Averaging neighbourhood rates to a district and then averaging districts gives each district equal weight, not each neighbourhood or resident. Multi-stage averaging can silently change the estimand.
Carry sufficient totals upward. If a region-wide case rate is needed, sum compatible numerators and denominators across the leaves. If a typical-district rate is needed, calculate each district consistently and average at that level.
CHAPTER 48 OF 57 · Transfer and continue
48. Avoid double counting across time windows
Monthly numerators may describe unique people within each month. Summing twelve months can count the same person repeatedly. That may be correct for visit rates and wrong for annual unique coverage.
Define whether the unit is events, people or person-periods. An annual rate may require deduplication or exposure accounting. A zonal dashboard should not pool periods simply because columns can be added.
Residents may live in one zone and receive service in another. Workplace populations and daytime exposure differ from residential counts. Choosing the residential denominator can misrepresent a transport hub or business district.
Match denominator location to the process: residence, occurrence, origin, destination or time at risk. State cross-boundary flows and data gaps. A zone is an analytical container, not a guarantee that the relevant population stays inside it.
One district may count completed cases while another counts initiated cases. Their percentages cannot be safely averaged even if both labels say success. Harmonise definitions and reporting cut-offs.
Create a data dictionary with numerator event, denominator eligibility, exclusions, time lag and revision policy. Resolve or flag discrepancies before mapping. Arithmetic cannot repair semantic incompatibility.
CHAPTER 51 OF 57 · Transfer and continue
51. Separate target weights from convenience weights
Valid-cell count is easy to obtain from a raster, so analysts may use it as a weight. It represents the number of included cells, which approximates area only under equal cell areas and consistent coverage. It does not automatically represent people, habitat quality or economic value.
Choose weights from the target quantity. Convenience may be acceptable for an exploratory display, but label it and avoid policy interpretation beyond its support.
CHAPTER 52 OF 57 · Transfer and continue
52. Test aggregation with a synthetic dataset
Before processing thousands of zones, build three synthetic zones with known counts, missing values and one zero denominator. Calculate expected outputs by hand. Run the software and compare every field.
Include unequal denominators so an accidental equal-zone average cannot pass. Add a duplicate identifier and overlapping polygon if relevant. A small adversarial test often detects configuration errors that a realistic map hides.
Numerators and denominators can be corrected after publication. If a large zone revises its cases, every pooled summary depending on it changes. Store record identifiers, version dates and derivation code.
Mark preliminary and final releases. Downstream dashboards need a process to ingest corrections rather than merely append new rows. A rate is reproducible only when its source version is known.
CHAPTER 54 OF 57 · Transfer and continue
54. Use inequalities as a quick reasonableness check
Any positive-weighted mean of local rates should lie between the minimum and maximum local rates. If an output falls outside that range, inspect units, signs, scaling and formula. A pooled proportion from compatible nonnegative counts should also remain between zero and one.
These checks do not prove correctness, but they catch impossible results. Add denominator-sum reconciliation and spot-check selected zones. Automated pipelines need simple invariants alongside visual review.
CHAPTER 55 OF 57 · Transfer and continue
55. Communicate disagreement between summaries
When equal-zone and pooled rates differ, explain the size distribution. “The typical zone is 70%, while 89.2% of all recorded cases succeed because most cases are in the 90% zone” is clearer than presenting one number as a correction of the other.
Invite the reader to choose the perspective relevant to the decision. If resources are allocated per person, pooled or population-weighted results may matter. If every municipality must meet a standard, the zone distribution and lowest zones may matter more than the pooled total.
When are the two averages equal?
They are equal when denominators are equal or when the pattern of rates and denominators happens to produce equality. Do not assume either condition.
Should I always publish pooled totals?
Publish the summary that answers the question, often with the alternative and underlying counts. Confidentiality and disclosure rules may constrain small counts.
Can I weight by population estimates?
Yes when the estimand concerns that population and the estimates match the zones, period and definitions. Record uncertainty and source.
What if a denominator is zero?
The rate is undefined. Handle it explicitly rather than forcing zero or infinity into an ordinary mean.
CHAPTER 56 OF 57 · Transfer and continue
56. Continue through the verified zonal routes
Return to The Gold Standard of Zonal Statistics for zone geometry, raster alignment, NoData, cell area, circular statistics and reproducible GIS summaries. Read Bounding a Subzone Mean When Only Coarse Cell Averages Remain for sharp limits when finer spatial information has been lost.
Statistics Canada, Statistics 101: Proportions, ratios and rates distinguishes these measures and their calculation. IBM SPSS Statistics, Ratio Statistics documents a denominator-weighted relationship between ratios and aggregate numerator/denominator means. The calculations in this guide are original teaching examples. For real work, use the metadata, disclosure rules, sampling design and software documentation attached to the actual dataset.