A rise in the average sale price does not necessarily mean that comparable homes became more expensive. The homes sold may have changed. A period dominated by smaller homes and a period dominated by larger homes can produce very different averages even when neither kind changes price.
The short answer is to separate three questions: what did the transacted homes cost, how did a defined comparable set change, and how well does that set represent the wider housing stock? Matching the same properties helps with the second question. It does not automatically answer the third or remove renovations, transaction differences and missing observations.
This lesson continues The Gold Standard of Urban Economics within Learning That Transfers into Life and How X Works. All homes, prices and transactions below are invented. Prices use abstract credits, not local currency. The exercises explain measurement; they do not forecast prices, value an actual property or recommend a purchase.
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ROUTE 2 · CHAPTERS 5–8
Match and aggregate
Compare explicit weighting rules without confusing their questions.
ROUTE 3 · CHAPTERS 9–12
Preserve evidence and scope
Keep identity, intervals and coverage with the calculation.
Chapter contents
Define the comparison · Chapters 1–4
1. Name the population behind the headline
2. Separate a transaction flow from the housing stock
Match and aggregate · Chapters 5–8
5. Hold a stated reference mix fixed
6. Follow the same property across two transactions
Preserve evidence and scope · Chapters 9–12
9. Understand a geometric summary without confusing methods
10. Preserve transaction dates before annualising
Audit the interpretation · Chapters 13–16
13. Avoid turning a sample result into a causal story
14. Check prices, areas and the bundle being sold
Practise and continue · Chapters 17–20
17. Diagnose the first mistaken comparison
18. Try a changed-context price comparison
Suppose a headline says that a district’s average home price increased. Begin by asking which homes supplied the numbers. Were they homes sold during the period, all homes in the district, listings, valuations or a repeated set of properties? These are different populations and records.
A sale-price mean is a legitimate description of the transactions included in it. The problem begins when that description becomes a claim about every home without accounting for the selection of transactions. Homes that did not sell contribute no observed sale price to that particular mean.
Write a measurement sentence before calculating: this is the arithmetic mean price of these completed transactions in this area during this period. Add whether the data include particular property types and whether the price describes the whole property or a stated unit of area.
That sentence makes the denominator visible. It also prevents a change in the number or composition of sales from disappearing behind the familiar word average. A larger sample can still contain a different mix.
The Urban Economics root explains why location, housing and evidence belong together. Here the question is narrower: which part of an observed price comparison comes from the items being compared? Start there before discussing why a market changed or what might happen next.
CHAPTER 2 OF 20 · Define the comparison
2. Separate a transaction flow from the housing stock
The homes that sell in a quarter form a flow of transactions. The homes that exist on a particular date form a stock. The two sets overlap, but they are not identical. A quiet housing period does not mean most homes ceased to exist; it means fewer entered the observed transaction flow.
Imagine a district with many long-held family homes and a smaller group of frequently traded apartments. A transaction dataset may contain relatively many apartment sales. Its mean describes that observed flow more directly than the entire stock.
This selection can change between periods. If larger homes trade more frequently later, the mean sale price can rise even without a change in either type’s price. Conversely, a burst of lower-priced transactions can reduce the mean while comparable homes hold their prices.
Neither pattern proves manipulation or makes transaction statistics useless. It identifies the question that the statistic answers. A useful report can describe the transaction market while clearly distinguishing it from a claim about all properties.
For a classroom check, draw ten home cards and mark which ones sold. Calculate a transaction mean using only the marked cards. Then change the marked subset without changing any home prices. The exercise shows how a flow summary can move when the underlying stock’s stipulated prices do not.
CHAPTER 3 OF 20 · Define the comparison
3. Work a price change caused entirely by the sales mix
Create two fictional types. A Compact home sells for 100 credits and a Large home for 300 credits. In the first period, eight Compact and two Large homes sell. The total transaction value is 8 × 100 + 2 × 300 = 1,400 credits across ten sales.
The mean is therefore 140 credits. In the second period, two Compact and eight Large homes sell at exactly the same type prices. Total value is 2 × 100 + 8 × 300 = 2,600 credits, and the mean is 260 credits.
The mean increased by 120 credits, or approximately 85.71% relative to 140. Yet neither stipulated type price changed. The entire difference comes from the transaction mix in this example.
| Fictional period | Compact sales at 100 | Large sales at 300 | Mean sale price |
|---|---|---|---|
| First | 8 | 2 | 140 credits |
| Second | 2 | 8 | 260 credits |
The correct conclusion is not that prices never rose. It is that this particular mean-price comparison cannot establish comparable-home appreciation. We supplied a complete counterexample to that inference.
Ask the learner to explain where the extra 1,200 credits of transaction value came from. It came from selling more high-priced homes and fewer low-priced ones, not from increasing either supplied price. The mechanism matters more than the large percentage.
CHAPTER 4 OF 20 · Define the comparison
4. Check whether the median solves the mix problem
A median resists the influence of unusually extreme values, but it can still change with composition. Sort the ten sale prices in the first period: eight are 100 and two are 300. The middle pair are both 100, so the median is 100 credits.
In the second period, two prices are 100 and eight are 300. The middle pair are both 300, giving a median of 300 credits. The median increased by 200% despite unchanged type prices.
The median is not wrong. It accurately describes the middle of each supplied transaction distribution. It simply does not hold the composition constant. Replacing the word mean with median does not automatically convert a changing transaction sample into a comparable-property measure.
This example also separates robustness from comparability. Robustness concerns how a statistic responds to particular observations. Comparability concerns whether the two sets represent the same intended object or population under compatible conditions.
A useful report can show both mean and median, accompanied by counts and composition. That helps readers understand the distribution. Neither number should be given a broader interpretation than its construction supports.
For independent practice, reverse the transaction mix while keeping the two prices unchanged. Both statistics fall. Explain why that is evidence about the observed mix rather than proof that every home lost value.
One transparent classroom comparison fixes category weights. Suppose the reference mix is half Compact and half Large. In both periods of the original example, the standardised mean is 0.5 × 100 + 0.5 × 300 = 200 credits.
That unchanged result answers a different question from the transaction means. It asks what the average would be under the supplied common category mix. It does not describe the actual mean amount paid across either period’s transactions.
Now change the second-period type prices to 110 credits for Compact and 315 for Large. With the same half-and-half reference mix, the standardised mean becomes 212.5 credits. Its increase is 6.25% relative to 200.
The result depends on the reference weights. A reference containing more Compact homes would place more weight on their 10% increase; a reference containing more Large homes would place more weight on their 5% increase. State the target population or analytical purpose behind the weights.
This method also assumes that the categories are sufficiently comparable internally. If later Compact sales contain better-located or renovated homes, fixing only the Compact/Large proportions may leave composition differences within each category.
Standardisation makes one adjustment explicit. It does not eliminate every difference by naming two groups. A useful analysis states what was held constant and which characteristics remain uncontrolled.
CHAPTER 6 OF 20 · Match and aggregate
6. Follow the same property across two transactions
Matching repeated transactions changes the comparison unit from a broad category to a property identity. For each matched home, retain its earlier and later price, transaction dates and evidence that the records describe the same property unit.
Our next dataset contains four invented homes. A and B are Compact; C and D are Large. Unlike the first example, these are individual repeated-property records rather than anonymous type totals.
| Home | Earlier price | Later price | Later divided by earlier |
|---|---|---|---|
| A | 100 | 110 | 1.10 |
| B | 120 | 132 | 1.10 |
| C | 300 | 285 | 0.95 |
| D | 200 | 190 | 0.95 |
The first two matched prices increased by 10%; the last two decreased by 5%. Those statements follow directly from the supplied pairs. They do not yet establish a unique combined index because we have not chosen how to aggregate the changes.
Matching helps prevent one home’s price level from being compared with another home’s different level as though it were a change through time. But a shared identifier is only the beginning. Changes in the property, transaction conditions or data quality can still affect interpretation.
Preserve the pair before summarising it. The four individual ratios reveal differences that a single headline number would hide.
CHAPTER 7 OF 20 · Match and aggregate
7. Define an equal-home average of percentage changes
One possible teaching summary gives each of the four matched homes equal weight in its percentage change. The arithmetic mean ratio is (1.10 + 1.10 + 0.95 + 0.95)/4 = 1.025. Its associated mean percentage change is 2.5%.
This summary treats a one-percentage-point change for A as equally important to the mean as a one-percentage-point change for C. It does not weight the homes by their earlier prices or by floor area.
That is a legitimate stated choice for an equal-home question. It should not be presented as the only possible housing-price measure. Nor should it be labelled an official repeat-sales index merely because it uses matched properties.
Notice the role of the denominator inside each ratio. A’s increase of ten credits is 10% of 100. C’s decrease of fifteen credits is 5% of 300. Adding raw price changes first produces a different weighting from averaging these individual percentage changes.
For a useful check, duplicate every home record once. The equal-home average remains the same because both numerator and count double. Duplicate only A and B, and the result changes because those increases now receive greater representation.
A repeated record is therefore not harmless extra information. Distinguish a genuinely additional transaction pair from a duplicate row before calculating weights.
A different question asks how the total value of this fixed four-home basket changed. Its earlier total is 100 + 120 + 300 + 200 = 720 credits. Its later total is 110 + 132 + 285 + 190 = 717 credits.
The total-value ratio is 717/720, approximately 0.995833. That represents a decrease of about 0.4167%, despite the equal-home mean percentage change being positive 2.5%.
There is no contradiction. The total-value comparison gives larger earlier-price homes more weight. The two declining Large homes accounted for 500 of the original 720 credits, so their declines outweighed the increases in total credits.
Write the identity explicitly: the ratio of basket totals equals the weighted average of individual price ratios, using each home’s earlier price divided by 720 as its weight. Those weights sum to one.
This is a useful way to diagnose apparently conflicting summaries. Ask what was averaged and with which weights before deciding that one calculation must contain an arithmetic error.
Neither summary predicts the value of an actual household’s home. They describe invented records under different aggregation rules. A reader should be able to reconstruct the rule from the report rather than infer it from an attractive label such as overall market performance.
CHAPTER 9 OF 20 · Preserve evidence and scope
9. Understand a geometric summary without confusing methods
A third possible classroom summary is the geometric mean of the four ratios. Here it is (1.10 × 1.10 × 0.95 × 0.95) raised to the one-quarter power, which equals √1.045, approximately 1.022252.
That summary corresponds to an increase of about 2.2252%. It differs from both the arithmetic mean change and the basket-total change. Each has a defined mathematical construction.
Geometric summaries are connected with averaging logarithmic changes. The logarithm of a product becomes a sum, so the log of the geometric mean is the mean of the individual log ratios. Positive prices are required for this construction.
Do not infer that our four-row geometric mean reproduces a real agency’s repeat-sales estimator. Official methods can use regression, transaction timing, selection rules, weights and revisions that this small exercise does not contain.
The FHFA methodology FAQ describes its own weighted repeat-sales approach and why transaction means and medians answer different questions. Our three arithmetic demonstrations are original teaching comparisons, not estimates from FHFA data or implementations of its index.
A sound explanation names both the matched set and the chosen summary. The word matched tells readers which records were compared; it does not by itself specify the averaging rule.
A further check reverses the comparison direction. Home A rises from 100 to 110, an increase of 10% on its earlier base. Going from 110 back to 100 is a decrease of 10/110, approximately 9.09%. The two percentages do not cancel as signed numbers because their denominators differ. Home C falls from 300 to 285 by 5%; returning from 285 to 300 requires an increase of 15/285, approximately 5.26%.
Write earlier and later beside both columns before dividing. Reversing a price ratio gives its reciprocal; reversing a reported percentage does not usually mean changing its sign. That simple check is useful when two reports compare the same dates in opposite directions. It also prevents a learner from treating an aggregation choice as a substitute for defining the direction of each individual change.
CHAPTER 10 OF 20 · Preserve evidence and scope
10. Preserve transaction dates before annualising
A matched price ratio belongs to the interval between its transactions. A 10% increase over one year differs from a 10% increase over five years. Pooling the ratios without accounting for their intervals can answer an unclear question.
In a fictional annualisation exercise, a price doubles over ten years. The constant annual compound factor that reproduces the endpoints is 2 raised to the one-tenth power, approximately 1.071773. That is about 7.1773% per year under a constant-rate representation.
Dividing the total 100% increase by ten gives 10% per year as a simple arithmetic rate on the original base. That does not compound to the same endpoint. The two operations use different accumulation models.
Annualising is a description of endpoints, not evidence that prices rose steadily between them. Many paths can connect the same earlier and later sale prices. Intermediate declines, renovations or transaction events are not recovered from the annualised rate alone.
Before combining records, preserve each pair’s dates and interval definition. Do not silently treat two months, two years and two reporting periods as equivalent.
For actual index methodology, use the responsible publisher’s technical account. The classroom exercise develops interval literacy, leaving the statistical treatment of irregular real transactions within the appropriate method.
CHAPTER 11 OF 20 · Preserve evidence and scope
11. Recognise that the same home can change
Matching a property identity removes some differences between properties, but it does not freeze the property itself. A renovation, extension, subdivision, deterioration or change in associated rights can alter what the later transaction includes.
Consider a fictional home sold for 100 credits and later for 130. One explanation assigns all thirty credits to a price-level change. Another supplied model assigns twenty credits to an added feature and ten to other change on the original-price basis. The two transactions alone do not establish that split.
The second explanation is an invented additive decomposition, not a valuation method. Its purpose is to expose the extra information needed before calling a repeated sale a pure comparable-quality price change.
Likewise, a house divided into two separately traded units cannot be matched casually with one unit’s later price. The identity and bundle of assets differ. A shared address may be insufficient to establish that the same economic object was sold.
Keep an identity note beside each pair: same unit, same included rights and relevant recorded changes. If the evidence is missing, acknowledge the limitation rather than inventing an adjustment.
The Statistics guide provides the wider route for measurement and inference. A careful matched record improves comparability while preserving the assumptions it cannot independently verify.
CHAPTER 12 OF 20 · Preserve evidence and scope
12. Distinguish a repeated-sale sample from every home
Only homes with suitable repeated observations enter a particular matched sample. Long-held homes, new construction and records lacking reliable identity may be absent. Their absence can affect what the result represents.
Suppose our four matched homes belong to a larger district of twenty. Sixteen have no usable repeated transaction in the chosen interval. The observed ratios describe the four pairs directly. Extending the result to all twenty requires an appropriate representativeness argument or model.
The number sixteen is not evidence that the missing homes had zero change. A missing price ratio and a ratio of one are different records. Filling the gaps with one would add a substantive assumption while making the table appear complete.
Selection can also vary through time. More transactions can add information while changing which kinds of properties are represented. A larger sample is not automatically the same target population measured more precisely.
Report the sample scope alongside the statistic. How many usable pairs, from which property types, over which intervals, under which exclusions? That information helps readers judge the claim’s reach.
The practical lesson is proportionate confidence: matching solves a particular comparison problem, while sampling and coverage remain separate evidence questions. A good explanation names the problem improved and the uncertainty still present.
CHAPTER 13 OF 20 · Audit the interpretation
13. Avoid turning a sample result into a causal story
Even a well-defined matched-home measure describes price movements rather than automatically identifying their causes. Changes can relate to wider demand, financing conditions, local amenities, supply, expectations and many other influences.
Imagine a fictional transport connection opens between the two transaction dates. A matched home’s price rises. That sequence alone does not establish how much of the rise the connection caused, or whether another factor moved prices in the opposite direction.
The Urban Economics root’s worked comparison explains why a credible counterfactual matters. Matching property identity is not a substitute for identifying what would have happened without the intervention.
Keep three sentences separate. The observation states the recorded paired prices. The measurement states the chosen ratio or summary. The causal claim states why a change occurred and requires additional evidence.
This separation makes analysis more useful. A decision-maker can understand a measured pattern without being asked to accept an unsupported explanation for it. It also leaves room to revise the explanation when new evidence arrives.
For learners, ask which extra information would distinguish two proposed causes. Repeating the same matched-price calculation with more decimal places will not answer that causal question. Repair the missing comparison rather than the arithmetic that was already correct.
CHAPTER 14 OF 20 · Audit the interpretation
14. Check prices, areas and the bundle being sold
A whole-property price and a price per square metre answer different questions. Dividing by area can help describe a unit measure, but it does not hold every other quality constant or ensure that different area definitions are compatible.
Suppose a fictional home sells for 200 credits with a stated area of 100 area units. Its price per area unit is two. A second record gives 220 credits and 110 area units, also two per unit. The total price rose while the supplied unit-price measure stayed unchanged.
What happened physically? The records alone do not establish whether the home expanded, the area definition changed or an earlier measurement was corrected. Those possibilities have different meanings for a repeated-property comparison.
Use the same definition before comparing floor area, land area or another area basis. An external plot measure should not silently replace an internal floor measure merely because both use square metres.
Similarly, a sale may include associated parking, storage or other rights under the actual arrangement. A comparison needs to understand the bundle recorded, rather than treating the displayed price as a fully standardised object.
The classroom habit is to name numerator, denominator and identity together. A tidy ratio becomes more useful when another reader can tell exactly which price and which area produced it.
CHAPTER 15 OF 20 · Audit the interpretation
15. Treat revisions as changes in the evidence set
A statistical series can change when additional records, corrections or methods are incorporated. A revised past value need not mean the original publisher fabricated a result. It can reflect information that was unavailable or treated differently earlier.
For a paper illustration, calculate the equal-home mean ratio from A and B only: it is 1.10. Add C and D, whose ratios are 0.95, and the four-home mean becomes 1.025. The earlier two-home calculation remains reproducible for its smaller dataset.
That is not an explanation of any particular official revision. It demonstrates why a summary’s version and underlying sample matter. A comparison between last month’s preliminary number and this month’s revised history can mix information sets.
Keep the release date, reference period, series identifier and revision status when using actual published statistics. If a chart changes, identify whether the source corrected a record, added observations, changed seasonal treatment or revised methodology.
Do not infer the reason from the revised number alone. The publisher’s explanation is the appropriate source for that question.
For teaching, save both fictional versions and write what changed. A reliable record preserves the route from input to result, rather than overwriting an old value and leaving another reader unable to explain why their earlier calculation differs.
CHAPTER 16 OF 20 · Audit the interpretation
16. Build a reproducible comparison ledger
A useful ledger has one row per clearly defined transaction or matched pair. Preserve the property identifier, dates, price definition, inclusion decision and reason for any adjustment. Distinguish supplied facts from assumptions.
For each summary, state its target and rule. Transaction mean, fixed-category mix, equal-home mean ratio and basket-total ratio should have separate labels. A column headed average alone leaves too much for the reader to guess.
Calculate an individual pair by hand before aggregating. Check A’s ratio of 110/100 = 1.10, then verify the four-home sums of 720 and 717. These simple checks can reveal an inverted ratio, duplicated record or omitted home.
Retain enough information to reproduce the result after sorting the table. A correct fixed-record aggregate should not change because rows were reordered. If a spreadsheet result changes, inspect references or filters.
Finish with a scope statement. These pairs describe these fictional homes during their stated intervals; they do not establish every home’s change, a causal explanation or a forecast.
For a real review, use suitable authorised data and the publisher’s method. The paper ledger teaches clarity without requiring anyone to disclose personal transaction records in a public classroom or general-purpose tool.
CHAPTER 17 OF 20 · Practise and continue
17. Diagnose the first mistaken comparison
A learner calculates an 85.71% rise from 140 to 260 and says every home appreciated by that amount. The arithmetic is correct; the scope is wrong. Return to the unchanged type prices and changed transaction counts.
Another learner replaces the mean with the median and declares the composition problem solved. Use the 100-to-300 median counterexample. The correction concerns comparability, rather than the formula for a median.
A third learner obtains 2.5% and −0.4167% from the four matched homes and assumes one must be a mistake. Ask for the weights. The first result gives homes equal influence in percentage terms; the second gives earlier-price value its role in a fixed basket.
A fourth learner calls the geometric mean an FHFA index. The repair is methodological scope. Using a similar broad idea does not reproduce an agency’s estimator, selection rules and timing treatment.
Give a new small dataset targeting the first unstable relationship. Change the mix while holding prices fixed, or hold identities fixed while changing the averaging rule. Ask for an explanation before another calculation.
Independent understanding appears when the learner can identify what stayed the same, what changed and why the statistic responded. A longer worksheet of unlabelled averages can conceal the same mistake.
CHAPTER 18 OF 20 · Practise and continue
18. Try a changed-context price comparison
Use two fictional property types, Small at 80 credits and Family at 200. In period A, six Small and four Family homes sell. In period B, four Small and six Family homes sell at unchanged type prices.
The first mean is (6 × 80 + 4 × 200)/10 = 128 credits. The second is (4 × 80 + 6 × 200)/10 = 152. The increase is 24/128 = 18.75%, entirely due to the stipulated sales mix.
Now keep a reference mix of three-quarters Small and one-quarter Family. Its standardised mean is 0.75 × 80 + 0.25 × 200 = 110 credits in both periods. Explain why this does not contradict the actual transaction means.
For a separate matched-home check, take two homes priced 100 and 300 initially, then 110 and 285. The equal-home mean change is 2.5%, while the basket total changes from 400 to 395, a decrease of 1.25%.
Finally remove the assurance that the same property units were matched. Can the pairwise interpretation still be asserted? No. The arithmetic can still be performed, but its intended meaning now lacks the identity evidence.
These changes test the transferable method: define the set, preserve the comparison object and state the weighting rule before interpreting the number.
Is a sale-price mean misleading by definition?
No. It describes the included transactions under its stated construction. Misinterpretation occurs when it is treated as a comparable-home appreciation measure without addressing changing composition.
Does matching the same homes eliminate every quality issue?
No. A property can change between transactions, and the matched sample can differ from the wider stock. Identity, condition, timing and coverage still matter.
Which of the three matched summaries is correct?
Each can be correct for its stated question. Equal-home arithmetic change, geometric change and basket-value change use different rules. Name the intended quantity before choosing one.
Can an index tell me exactly what my own home is worth?
A broad index does not observe every feature or circumstance of a particular transaction. An individual valuation and a population summary are different tasks. This lesson supplies no actual valuation.
Does a revised index prove the old figure was dishonest?
No. Revisions can reflect new or corrected evidence and methodological treatment. Consult the responsible publisher’s account rather than infer a cause from the change alone.
CHAPTER 20 OF 20 · Practise and continue
20. Choose the next urban and evidence route
Understand the city question
Return to Urban Economics for location, accessibility, housing supply, appraisal and causal comparisons. Use the linked Housing and Urban Planning guides when the question concerns living conditions or the organisation of land uses rather than a price-summary method.
Strengthen the measurement
Statistics develops populations, summaries and inference. The distinction between changing composition and changing outcomes also connects with Universities: When Cohort Mix Reverses an Outcome Comparison. The settings differ; both require preserving the group and weighting behind a comparison.
Read the primary methodological context
FHFA's House Price Index FAQ, updated 7 October 2026, explains its weighted repeat-sales approach and distinguishes it from transaction means and medians. FHFA's technical description of its house-price indexes provides historical technical background, published in 1996 and linked from the current FAQ. Neither is implemented by our small invented arithmetic examples.
Source routes checked on 22 October 2026. All datasets and calculations in this lesson are original teaching constructions. Continue through Learning That Transfers into Life and How X Works by asking which items, weights and intervals a headline has actually compared.