A temperature table shows 19°C, then 21°C, then 20°C. The smallest recorded value is 19°C. That does not, by itself, establish that the temperature stayed at or above 19°C between observations.
The short answer is to distinguish the minimum of the recorded samples from the minimum over the complete period. Extra assumptions can sometimes bound a missed dip. They must be stated and justified for the task; a smooth line drawn between observations does not supply them.
This guide uses original fictional temperature records and an explicitly assumed mathematical rate limit. The assumed limits are teaching conditions, not verified limits on how quickly real air temperature can change. The calculations are not forecasts, warning criteria or a basis for deciding that an outdoor activity is safe. Begin with the Weather subject guide for observation, forecasting and the wider evidence-to-action chain.
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ROUTE 3 · CHAPTERS 9–14
Read uncertainty honestly
Examine thresholds, new readings, errors and missing values.
Chapter contents
Name the observation · Chapters 1–4
Derive the bounds · Chapters 5–8
Read uncertainty honestly · Chapters 9–14
- 9. Interpret a threshold without inventing an excursion
- 10. Add an observation and see which uncertainty narrows
- 11. Compare a simple gap allowance with the sharper bound
- 12. Include measurement uncertainty without hiding its role
- 13. Check uncertain timestamps and uncovered edges
- 14. Preserve missing-value and quality information
Practise and compare · Chapters 15–18
Choose the next step · Chapters 19–21
A sampled minimum is the smallest value in the supplied list after the relevant quality and missing-data rules have been applied. A period minimum concerns the temperature throughout a defined time window. The quantities can differ because the list may not observe every moment.
Suppose a fictional sensor reports exactly at midnight, 02:00 and 05:00. Its three values are 19, 21 and 20 degrees Celsius. Taking the smallest of those three gives 19. This arithmetic does not describe what happened at 00:30 or 03:45.
A published daily-minimum product can have its own instrument, aggregation and observation conventions. Do not assume it was created simply by taking the minimum of an hourly table that happens to be available alongside it.
NOAA’s GHCN-Daily documentation, for example, distinguishes minimum temperature from temperature at observation and includes a measurement flag identifying certain values derived from hourly reports. Read the actual product’s metadata before comparing it with another series.
For a learner, write the claim in full: the smallest supplied reading in this five-hour record is 19°C. That statement is supported. Changing it to the temperature never fell below 19°C adds information that the three numbers have not established.
CHAPTER 2 OF 21 · Name the observation
2. Fix the time window before inspecting the values
The main example covers exactly five hours, from time t = 0 to t = 5. It is not a complete calendar day. The first and last readings occur at the window endpoints, which will matter when we later bound the gaps.
Use hours consistently. The first gap is two hours, from t = 0 to t = 2. The second is three hours, from t = 2 to t = 5. A table with three rows does not have three equal observation intervals.
| Time in the fictional window | Supplied temperature | Gap to the next reading |
|---|---|---|
| t = 0 h | 19°C | 2 h |
| t = 2 h | 21°C | 3 h |
| t = 5 h | 20°C | End of window |
A different time zone or observation-day convention can move a reading across a reporting boundary. That changes the set being summarised even if none of the measured values changes. Preserve the timestamps and their interpretation rather than relying only on a date label.
Also separate an observation’s time from when a file was downloaded. A record retrieved this morning can describe an earlier night. Current retrieval does not make the measurement current.
The first practical check is therefore modest: identify the window, its endpoints, the actual observation times and any uncovered edge periods. Only then choose the calculation that answers the stated question.
CHAPTER 3 OF 21 · Name the observation
3. Construct two different histories through the same readings
One compatible fictional history rises steadily from 19 to 21 over the first two hours, then falls steadily to 20 over the next three. Its minimum over the window is 19.
Another history first dips below 19, then rises to reach 21 at t = 2. It can still reach 20 at t = 5. Without a restriction on the intervening change, the table alone does not distinguish the two histories.
This is an information problem rather than a calculator problem. Reporting more decimal places for the same three values does not reveal the unobserved times. Nor does a plotting program know which of many compatible paths the atmosphere followed.
A straight connector is one interpolation choice. It can be useful for visual orientation if its role is clear. Calling its minimum the measured period minimum would turn a drawing convention into an observation.
For a classroom exercise, ask two learners to draw different continuous curves through the same three points. One should have a dip between readings; the other should not. Then ask which extra assumption would rule out one drawing.
The answer should name the assumption, such as a specified bound on the rate of change. Saying the first drawing looks smoother is not a measurement and does not establish a physical limit.
For the main mathematical exercise, assume that the fictional temperature T cannot change by more than 2°C per hour over any part of the window. In symbols, the absolute temperature difference between any two times is at most 2 times their separation in hours.
Call the bound L = 2°C per hour. It limits both cooling and warming. It is stronger than saying that the average change between two recorded readings was no more than two degrees per hour.
For example, the observed change from 19 to 21 over two hours averages 1°C per hour. A path could still cool and warm much faster between those endpoints while producing that same average. The endpoint average does not prove the assumed bound.
In mathematics, this type of absolute-change condition is called a Lipschitz bound. You do not need the term to use the relationship, but you do need its meaning: every relevant subinterval must satisfy the limit, not only the gaps whose endpoints were sampled.
The numerical value here is invented for teaching. Before applying a bound to a real system, the relevant evidence would have to support it for the variable, instrument, exposure and conditions involved. The three readings alone cannot validate it.
Keep this sentence beside the calculation: the resulting range is conditional on the supplied rate bound. That prevents a correct model result from being mistaken for an unconditional weather observation.
CHAPTER 5 OF 21 · Derive the bounds
5. Check that the readings can satisfy the model
Before bounding an unseen minimum, check whether the observations are compatible with the assumed rate limit. Across a gap of h hours, endpoint temperatures a and b must satisfy |b − a| ≤ Lh.
For the first gap, the temperature difference is 2°C and the allowed total change between endpoints is 2 × 2 = 4°C. The endpoints are compatible. For the second, the difference is 1°C and the allowed amount is 2 × 3 = 6°C. Those endpoints are compatible too.
This check is necessary but does not prove that the real intervening path respected the bound. It only establishes that at least one path could connect the supplied endpoints without exceeding it.
Now change a fictional first gap to a rise from 19 to 24 in two hours with the same L. The difference is 5, greater than 4. No path satisfying that bound can connect those exact readings over the supplied interval.
Do not repair the contradiction by quietly applying the formula anyway. Investigate the assumptions and records: the rate limit, values, times, units or measurement uncertainty may need attention. The model should reveal incompatible evidence rather than conceal it.
For a learner, state the compatibility inequality in words before substituting. That small step explains why the check belongs before a more elaborate calculation of what might have happened between the readings.
In the first two-hour gap, the starting temperature is 19 and the ending temperature is 21. At time t hours after the start, the rate assumption prevents temperature from being below 19 − 2t.
The ending reading gives another restriction. To reach 21 by t = 2 without warming faster than the limit, the temperature at t cannot be below 21 − 2(2 − t). Simplifying gives 17 + 2t.
Both restrictions must hold at once. The lower permitted temperature at each time is therefore the larger of 19 − 2t and 17 + 2t. Using only the first endpoint would miss the restriction imposed by having to reach the second.
The two lines meet when 19 − 2t = 17 + 2t. This gives t = 0.5 hours and temperature 18°C. The lowest compatible point in the gap is therefore 18 under the assumed model.
One compatible path falls from 19 to 18 in half an hour, then rises to 21 over the remaining one and a half hours. Both slopes have magnitude 2°C per hour. This construction shows that the bound can actually be reached in the ideal exercise.
The straight rising path has minimum 19. The same exact endpoints and rate bound therefore permit a first-gap minimum anywhere within the supported range from 18 to 19, rather than identifying one unique hidden temperature.
For an interval of length h with exact endpoint temperatures a and b, and a positive rate bound L, the two lower lines are a − Lt and b − L(h − t).
Their intersection occurs at t = (a − b + Lh)/(2L). If the endpoints are compatible, this time lies within the interval. Substituting it into either line gives the lowest compatible interval minimum:
Lower bound on the interval minimum = (a + b − Lh)/2.
The interval minimum cannot exceed the smaller endpoint, because the endpoints are included in the interval. A straight path between compatible endpoints attains that upper value. The supported range is therefore from (a + b − Lh)/2 to min(a,b).
These are logical bounds under the model, not a confidence interval. They do not assign probabilities to possible paths, and they do not say that the midpoint is the best estimate.
For the first gap, (19 + 21 − 2 × 2)/2 = 18, matching the direct construction. An independent derivation and substitution agree.
If L = 0, compatible exact readings must be equal and the model requires constant temperature. Handle that case directly instead of dividing by zero in the intersection-time formula.
The formula is useful because every symbol has a clear role. Keep the temperature unit, time unit, interval length and rate assumption visible whenever it is transferred to another example.
Apply the same method to the second gap, from 21 at t = 2 to 20 at t = 5. Its length is three hours. The lower compatible minimum is (21 + 20 − 2 × 3)/2 = 17.5°C.
Within that gap, the two lower lines meet 1.75 hours after t = 2, at t = 3.75. A V-shaped path reaching 17.5 there can return to the supplied endpoint without exceeding the model rate.
| Interval | Lower compatible minimum | Upper compatible minimum |
|---|---|---|
| t = 0 to 2 h | 18°C | 19°C |
| t = 2 to 5 h | 17.5°C | 20°C |
| Complete five-hour window | 17.5°C | 19°C |
The window’s actual minimum is the smaller of its interval minima. Its lowest compatible value is 17.5, from the second gap. Its largest compatible value is the smallest observed temperature, 19, achieved by joining the observations with straight segments.
It would be wrong to average 18 and 17.5 and call that the window minimum. These values are lower bounds for different intervals, not repeated estimates of one common measurement.
Check the full construction. Use the lower V in the second interval and a compatible straight line in the first. This produces a complete path through all readings whose minimum is 17.5. The shared observation at t = 2 remains consistent in both pieces.
CHAPTER 9 OF 21 · Read uncertainty honestly
9. Interpret a threshold without inventing an excursion
Suppose a fictional learning question asks whether temperature fell below 18.5°C during the five-hour window. The sampled minimum of 19 does not prove that it stayed above that threshold.
Under the assumed rate model, the supported period-minimum range is 17.5 to 19. Some compatible paths cross below 18.5, while the straight-segment path does not. The threshold event is possible but not established by these data and assumptions.
Now ask whether temperature fell below 17°C. Under the exact same rate-bound model, the lower bound 17.5 rules that out. The conclusion remains conditional: it holds only if the observations and rate condition are valid for the entire window.
Ask instead whether it reached 19°C or below. The exact reading of 19 at the first endpoint establishes that inclusive statement in the model. Replacing at or below with strictly below changes the question.
These distinctions matter in reports. Possible, demonstrated and excluded under an assumption are different claims. A precise calculation can still support an incorrect decision if its wording changes the threshold or drops the condition.
The numerical thresholds here are arbitrary classroom questions. They are not weather warnings, ecological limits or product-storage criteria. For a consequential real threshold, the governing measurement method and responsible professional process determine what evidence is sufficient.
CHAPTER 10 OF 21 · Read uncertainty honestly
10. Add an observation and see which uncertainty narrows
Add a fourth exact fictional reading: 20°C at t = 3.5 hours. The earlier first gap remains unchanged. The old three-hour gap is now split into two gaps of one and a half hours.
From t = 2 to 3.5, the lower bound is (21 + 20 − 2 × 1.5)/2 = 19°C. From t = 3.5 to 5, it is (20 + 20 − 2 × 1.5)/2 = 18.5°C.
The lowest compatible minimum over the full window is now the smallest of 18, 19 and 18.5, which is 18. The upper remains 19. The added reading narrows the supported range from 17.5–19 to 18–19.
It does not identify the exact minimum. The first interval still permits a dip to 18 between its endpoints. More information in one part of the record does not remove every unresolved part elsewhere.
Also notice that a new reading’s value matters, not only its existence. A different admissible value at t = 3.5 could impose a different restriction. Sampling design and the observed data work together.
For a learner, ask which previous extreme path has been excluded. The path dipping to 17.5 at t = 3.75 cannot also pass through the new 20°C reading at t = 3.5 while respecting the same rate bound. That explanation connects the improved range with the information that produced it.
CHAPTER 11 OF 21 · Read uncertainty honestly
11. Compare a simple gap allowance with the sharper bound
For exact readings at both endpoints of every gap, a simpler observation is available. Any time inside a gap is at most half that gap’s duration from one endpoint. If the largest gap is H and the smallest reading is m, the model gives the conservative lower bound m − LH/2.
In the original record, m = 19, L = 2 and H = 3. The simple lower bound is 19 − 2 × 3/2 = 16°C. It is valid under these assumptions, but weaker than the interval-specific lower bound of 17.5.
Why is it weaker? It combines the smallest reading anywhere with the largest gap anywhere, even though the long gap has warmer endpoints. The sharper calculation preserves the actual endpoint pair for each interval.
The simple expression can still help explain sampling. For equally spaced endpoint-covered observations h hours apart, the maximum model allowance below the sampled minimum is at most Lh/2. With L = 2 and h = 0.5 hours, that allowance is 0.5°C.
This is not a promise about a real sensor or atmosphere. It shows how an assumed bound and observation spacing interact in the classroom model.
Ask learners to compare valid with sharp. A bound can be correct yet leave avoidable uncertainty because it discards useful relationships. Choosing a tighter supported bound is different from merely printing more decimal places.
CHAPTER 12 OF 21 · Read uncertainty honestly
12. Include measurement uncertainty without hiding its role
The earlier readings were exact mathematical inputs. Real measurements have uncertainty, and a display rounded to a tenth of a degree does not automatically establish the underlying value exactly.
For an isolated two-hour example, suppose the readings 19 and 21 each represent supplied admissible ranges of plus or minus 0.1°C. Retain the assumed L = 2. Treat all endpoint combinations in those ranges as allowed, subject to compatibility.
The lowest compatible interval minimum uses endpoints 18.9 and 20.9: (18.9 + 20.9 − 4)/2 = 17.9°C. The greatest compatible interval minimum uses the smaller of the largest endpoint values, which is 19.1°C. A compatible straight path with those upper endpoints attains that value.
The supported range becomes 17.9–19.1 for this isolated exercise. Do not report it as though it were the exact-reading range with decorative error bars. The endpoint information itself has changed.
The supplied ±0.1 is an admissible bound here, not a standard deviation or a confidence level. Those other interpretations need different reasoning. An instrument’s displayed resolution alone should not be silently promoted into a complete accuracy specification.
For a practical record, state what uncertainty information was supplied, where it came from and which model conditions remain separate. This keeps measurement uncertainty distinct from the missing-time ambiguity that persists even with exact readings.
CHAPTER 13 OF 21 · Read uncertainty honestly
13. Check uncertain timestamps and uncovered edges
Interval length also affects the bound. If exact endpoint temperatures 19 and 21 are separated by an admissible time gap from 1.9 to 2.1 hours, the lowest compatible minimum under L = 2 uses the largest gap.
That lower result is (19 + 21 − 2 × 2.1)/2 = 17.9°C. The upper remains the smaller endpoint, 19. The wider permitted time gap leaves more opportunity for an unseen dip under the same rate condition.
Keep dependencies in view. If two timestamps share the same clock offset, their difference may be unchanged. Treating their absolute uncertainties as two unrelated errors could exaggerate the uncertainty in the interval. Use the actual timing model rather than adding every stated number automatically.
Uncovered window edges are another issue. If the first reading is at 01:00 but the stated window begins at midnight, an endpoint-covered formula does not automatically describe that first hour. It needs its own evidence and a clearly applicable model.
Similarly, a final reading before the window ends leaves an unobserved tail. A daily-looking date label does not make that tail disappear. Identify which portions of the intended period are actually constrained.
The check is simple: draw the full requested window, place the observations on it, and mark every interval including the edges. A formula applied to the wrong window can be numerically perfect and still answer the wrong question.
CHAPTER 14 OF 21 · Read uncertainty honestly
14. Preserve missing-value and quality information
A missing value is not a low temperature. A sentinel code is not an extreme weather event. Before calculating a sampled minimum, interpret the source’s missing-data and quality conventions.
GHCN-Daily’s documentation identifies −9999 as a missing-value code and specifies TMIN in tenths of degrees Celsius. Those conventions belong to that format; another source can use different units and markers.
In a fictional parsing exercise using those stated conventions, values 190, 210 and −9999 do not mean three ordinary temperatures whose minimum is −9999°C. The valid supplied temperatures are 19 and 21°C, with one missing entry. Preserve the missing time in the observation record.
Dropping the missing row without retaining its timestamp can conceal a longer gap. The minimum of the available readings may remain easy to calculate while the evidence about the complete period becomes weaker.
Quality flags also deserve attention. A value that fails a relevant check should not be treated as interchangeable with a verified reading merely because it occupies the same numeric column. Follow the dataset’s actual guidance and describe the filtering choice.
This is where programming and scientific interpretation meet. A minimum function can execute correctly on the wrong inputs. The useful result requires a valid variable, unit, time basis and treatment of missing or questionable observations before the mathematical comparison begins.
Work through a changed fictional record before reading the next chapter. The supplied exact temperatures are 14°C at t = 0, 16°C at t = 1 and 15°C at t = 3 hours. Assume a bound L = 3°C per hour throughout this three-hour window.
Find the sampled minimum. Check compatibility over both gaps. Calculate the lowest compatible interval minimum for each gap, then the supported minimum range over the full window.
Next add an exact reading of 15.5°C at t = 2. Recalculate the affected intervals. Explain whether the added observation narrows the whole-window range and which remaining interval now controls the lower bound.
Then compare the original record with the loose global allowance using its largest gap. Do not assume that a simple formula and the interval-specific formula must give identical lower bounds.
Finally, consider a fictional statement that the temperature definitely stayed above 13°C. Is that established under the original three-reading model? Does the answer change after the additional reading? Be careful with definitely and with the phrase under the assumed model.
Write a forward check for any claimed extreme path. It should pass through the supplied values and respect the stated slope limit. A lower number produced by arithmetic is not a valid bound construction if the proposed path cannot reach the next observation in time.
CHAPTER 16 OF 21 · Practise and compare
16. Check the changed record and its threshold claim
The sampled minimum is 14°C. The first endpoint difference is 2 across one hour, within the allowance of 3. The second difference is 1 across two hours, within the allowance of 6. Both gaps are compatible with the assumed limit.
The first lower interval minimum is (14 + 16 − 3 × 1)/2 = 13.5°C. The second is (16 + 15 − 3 × 2)/2 = 12.5°C. The original whole-window range is therefore 12.5–14°C.
With the added 15.5°C reading at t = 2, the split intervals have lower bounds (16 + 15.5 − 3)/2 = 14.25 and (15.5 + 15 − 3)/2 = 13.75. The first interval’s 13.5 is now the lowest, so the improved whole-window range is 13.5–14.
| Independent record | Lower compatible minimum | Upper compatible minimum |
|---|---|---|
| Original three readings | 12.5°C | 14°C |
| Added t = 2 reading | 13.5°C | 14°C |
| Loose original allowance, H = 2 | 11°C | Not a replacement for the interval calculation |
The loose allowance is 14 − 3 × 2/2 = 11. It remains a valid but less informative lower bound.
Originally, staying above 13 was not established: a compatible dip to 12.5 was possible. With the additional reading and the same valid model assumptions, the lower bound 13.5 excludes a dip to 13 or below within the defined window. Neither result is an unconditional conclusion about an actual night.
If a learner equates the sampled minimum with the period minimum, return to two different paths through the same points. Ask what the observations share and what they leave open. More arithmetic will not repair the missing distinction.
If the learner calculates the interval formula but ignores an impossible endpoint change, revisit compatibility. The method applies to a class of admissible paths. An empty class is a contradiction to investigate, not a source of a meaningful range.
If the learner uses the average observed slope as a guaranteed rate limit, construct a rapid dip and recovery between unchanged endpoints. The same average can conceal very different short-term behaviour.
If they average interval lower bounds, ask where the minimum over a union of time intervals occurs. It is the smallest relevant value, not an average of the interval summaries.
If the maths is correct but the conclusion becomes an actual weather warning or safety decision, return to the model boundary. The exercise supplied an invented rate bound and fictional observations. It did not validate the observing system or provide the governing real-world decision criteria.
Finish the repair with a new record and a sentence identifying what additional evidence would matter most. A learner who chooses the right next question has transferred the principle beyond one memorised formula.
CHAPTER 18 OF 21 · Practise and compare
18. Compare observation spacing under a stated mathematical aim
Suppose a paper exercise permits four exact observations across a five-hour window, including both endpoints. Before their values are known, the aim is to minimise the largest gap between readings under the same assumed rate bound.
Four observations create three gaps whose lengths sum to five hours. At least one gap must therefore be at least 5/3 hours. Equal spacing achieves that largest-gap value, with observations at 0, 5/3, 10/3 and 5 hours.
With the fictional L = 2, the simple maximum allowance below that schedule’s own sampled minimum is L times the largest gap divided by two, or 5/3°C. The original three-reading schedule had a largest gap of three hours and a corresponding allowance of 3°C. Adding a reading at 3.5 reduced the largest gap to two hours and that allowance to 2°C.
These comparisons concern the stated worst-gap criterion, not an actual observing-system recommendation. The sampled values can differ between schedules, and the sharper interval-specific bound also uses those values. Device response, operating constraints and the physical validity of the assumed rate limit remain outside this paper optimisation.
The useful transfer is to name the aim before calling one schedule better. Minimising the largest unobserved gap is a precise mathematical objective; it does not automatically optimise every scientific or practical purpose.
CHAPTER 19 OF 21 · Choose the next step
19. Choose repair, stabilisation or extension practice
The repair route begins with timestamps, units and the distinction between a reading and a continuous period. Use a short table and two drawn paths. The learner should be able to explain the missing-time issue before learning any envelope calculation.
The stabilisation route adds an explicitly supplied rate bound, checks endpoint compatibility and works each unequal interval separately. Ask the learner to construct a path attaining a bound and a different path producing a higher minimum.
The extension route introduces measurement ranges, uncertain intervals, edge coverage and dependencies. Keep these models separate enough that the learner can say which assumption changed the result. More complicated notation should clarify the evidence rather than conceal an unsupported physical premise.
For classroom work, use fictional data or suitable published records within their documented scope. No unsupervised weather exposure or new sensor installation is required. A useful project can begin entirely with the source’s metadata and a small table.
For transfer, ask whether the same reasoning could apply to a different continuously changing quantity with a justified change bound. Then ask which parts would have to be re-established. The algebra may travel; the numerical rate limit does not travel automatically.
The standard is an explanation with its conditions still attached. A correct bound should make the uncertainty easier to see, not give a sparse record a false appearance of continuous observation.
Is the smallest recorded temperature wrong?
It can be perfectly correct as a sampled minimum. The error is describing it as the complete period minimum without the required observing or modelling evidence.
Does a straight line between readings prove what happened?
No. It is an interpolation choice unless the task supplies evidence supporting that path. Other compatible histories may contain lower values.
Can the observed average slope validate the rate bound?
No. A modest endpoint average can conceal faster cooling and warming between observations. The assumed limit concerns every relevant subinterval.
Are these ranges confidence intervals?
No. They are finite admissible bounds under stated assumptions. They do not attach probabilities to unseen paths or include every source of real measurement uncertainty.
Does a new observation always identify the exact minimum?
No. It can remove some possibilities while leaving others. Check which interval still controls the remaining uncertainty.
Can the classroom rate limit be used for local weather decisions?
No. Its value is invented to teach reasoning. Actual observations, forecasts, warnings and activity procedures need their appropriate authoritative sources and responsibilities.
CHAPTER 21 OF 21 · Choose the next step
21. Continue through the relevant weather and sampling guides
Understand the observation and the decision
The Weather root connects measurements with forecasts and action. Forecast Reliability, Sharpness and Decision Thresholds examines repeated forecast probabilities, a different question from bounding an unseen value between observations.
Compare related sampling problems
Kinematics: Sampled Position Data, Hidden Reversals and Measurement Error owns the motion and distance problem. Time–Temperature Integrity owns product-condition history and its professional disposition boundary. The present lesson derives a conditional weather-record minimum envelope; it does not replace either specialist guide.
Check the source and transfer the method
NOAA NCEI’s GHCN-Daily readme was checked on 22 October 2026 for variable definitions, units, missing-value conventions and measurement flags. The temperatures, rate bounds, envelopes and exercises in this article are original mathematical illustrations. NOAA does not supply or endorse the fictional rate limits used here.
Continue through Education: learning that transfers into life and How X Works. The next useful habit is to ask which part of a minimum claim was observed, which part was modelled, and which times remain unresolved.