Topic 3: Uncertainty and Comparison Thresholds

Learning Objectives

By the end of this knowledge development session, student divers will be able to:

1. Distinguish accuracy from precision, and give an example of an instrument that has one without the other.
2. Explain why the digits shown on a digital display are not a statement of how well a quantity is known.
3. Explain why a set of instruments that are individually inaccurate can still measure a difference precisely, and state the condition under which this fails.
4. Describe the cross-calibration procedure used in this course and state what quantity it produces.
5. Define u(T), the standard uncertainty of a single corrected reading, and explain why it must be measured rather than taken from a specification sheet.
6. Explain why u(T) is not itself the threshold for a difference, and state how a comparison threshold is built from it.
7. State how many figures to report for an absolute value and for a difference, and explain why the two differ.
8. State the reporting rule for a difference smaller than its applicable threshold, and explain why an instrument reading of zero is not the same as an absence of the thing being measured.

Presentation Notes

a. Accuracy and precision are different, and the difference is the whole topic. Accuracy is how close a reading is to the truth. Precision is how close repeated readings are to each other. A watch running exactly four minutes fast is precise and inaccurate: every reading is wrong by the same amount, and it will tell you the length of a dive perfectly. A watch that gains and loses at random is inaccurate and imprecise, and it is useless for both.

b. A display is not a measurement. A digital instrument will show you as many digits as its display format allows. Those digits are produced by arithmetic inside the instrument — an analogue signal converted to a number, then run through a calibration formula — and the arithmetic will happily produce six decimal places from a sensor that cannot justify one. Nothing in that chain knows how good the sensor is. Its uncertainty is a property of the physical device: how it was calibrated, how far it has drifted, how it was handled, how long it has been in the water. No amount of computation adds information the sensor did not supply. A logger displaying 10.540544 °C is reporting the result of a calculation. It is not telling you the temperature to six decimal places, and copying all six into a report is a claim the instrument never made.

c. Two numbers, two different meanings. The loggers used in this course carry a stated accuracy of about ±0.5 °C and a resolution of about 0.04 °C.

Accuracy says a logger may sit half a degree from the truth. If the water is really 10.00 °C, a logger reading 9.54 °C is simply wrong by 0.46, and the last two digits tell you nothing whatever. For stating what the temperature is, resolution buys you nothing.

Resolution says something narrower and more useful: the same logger, in unchanged water, will keep reporting nearly the same number. That is worthless for saying what the temperature is, and essential for detecting what changes.

d. Why differences behave better than absolutes. Think of each logger as carrying a fixed offset — a private amount by which it reads high or low. You do not know its size or its sign, but it stays with the instrument. Now take two readings from the same logger, morning and afternoon, and subtract one from the other. The offset appears in both readings and cancels out of the difference. What survives is only the wobble, which is small.

So a change of 0.2 °C can be resolved by an instrument that does not know the absolute temperature to better than half a degree. This is the four-minutes-fast watch again: it cannot tell you the time, and it can time a dive to the second.

e. And here is where it fails. The offsets only cancel within one instrument. Compare two different loggers and you are left with the difference between their two offsets, which can be a full degree.

Hang three uncalibrated loggers on a line in water that is uniform from top to bottom, and they may report a spread of a whole degree — and you would draw a thermocline that does not exist. Straight from the specification sheet, an uncalibrated array cannot be trusted to measure a gradient at all. This is not a defect of inexpensive instruments; it is what "±0.5 °C accuracy" means, and it applies to any set of sensors bought off a shelf.

f. The fix is to measure the disagreement instead of assuming it away. Before the dive, put every logger — and every dive computer in the team — into one container of well-mixed water. Leave them long enough to settle. Every instrument is now in the same water at the same temperature, so any difference between their readings is the instruments, not the water.

Read the whole group three times, five minutes apart. Not once. The first reading establishes each instrument's offset — its departure from the group mean. The second and third test whether that offset holds.

Why three and not one. Suppose you read the group once and define each instrument's offset as its distance from the average. Now subtract those offsets from those same readings. Every instrument lands exactly on the average, every time, without exception — because that is what you just defined the offset to be. The disagreement has not been measured; it has been defined away. An array assessed that way appears to resolve arbitrarily small differences, and every comparison in the Station Report would pass.

The second and third readings are not rigged in that way. Apply the offsets from reading one to the numbers from readings two and three, and what is left over is real: it is the amount by which each instrument's offset failed to stay the same over ten minutes. That is the quantity that matters, because an instrument whose offset wanders on a deck will wander during a dive, and no calibration can rescue it.

g. Use seawater from the site, and keep it in the shade. Two reasons, both of which will otherwise corrupt the result. A logger calibrated in cool tap water and then dropped into warm sea enters the water several degrees off and spends the first part of the dive chasing — and all the loggers chase together, so any agreement you see during that period means nothing. Filling the container from the site removes almost all of that. Second, a dark instrument sitting in tropical sun reads the sunlight falling on it as much as the water around it. Offsets recorded in a sunlit bucket do not describe how the instruments behave once submerged, which defeats the purpose of recording them. Shade the container, and leave the instruments in it until entry rather than laying them out on a hot deck.

h. Shade is not the same as stable, and the difference matters. A shaded bucket on a warm deck still warms, slowly and steadily. That looks harmless — everything is in the same water, so everything should chase together — but the instruments do not respond at the same speed. Published testing of dive computers found response time constants ranging from about seventeen seconds to over five minutes, depending on where the sensor sits and what the housing is made of. In water whose temperature is climbing, a fast instrument tracks it and a slow one lags behind, and the gap between them looks exactly like an offset.

It is not one. It is an artefact of the warming, and it will be a different size tomorrow in different weather — which makes it worse than useless, because it is stable enough within one session to look real.

The test takes five minutes. Read the whole group, wait five minutes, read it again. If the group mean has moved by more than the spread you are trying to measure, the bath is still warming and the offsets are not yet trustworthy. Wait, or make the bath more stable.

An insulated cooler is the practical answer and costs very little. A large volume of water in an insulated box holds its temperature long enough for every instrument to reach it, whatever the deck is doing. A bare bucket in shade is often adequate on a cool morning and often not on a hot afternoon — and the five-minute test tells you which day you are having.

i. Subtract the offsets and the accuracy problem largely disappears. What remains is not limited by how close each logger sits to the truth, but by how consistently it reads — which is the far better number. The array still does not know the absolute temperature of the sea to better than half a degree. It does not need to. Every question in this course is about a difference: between two depths, between two positions, between morning and afternoon.

j. Repeat the calibration after the dive. Instruments drift, and a logger that took a knock may have moved. The difference between the before and after offsets tells you whether anything changed during the dive, and gives you a second estimate of the same quantity.

k. The spread that remains is u(T), and it is not yet a threshold. After removing the offsets, the instruments still will not agree perfectly at the second and third readings. That residual disagreement — measured, on your equipment, on that day — is the standard uncertainty of a single corrected reading. It is the single most important number your team produces, and the single easiest to misuse.

Every comparison contains two uncertain values. A difference between two readings is therefore more uncertain than either reading, and the two uncertainties combine in quadrature. Multiply by the course's coverage factor of 2 and you have the threshold that a difference must actually clear:

C = 2 × √(u₁² + u₂²)

Where both readings carry the same u(T), that is 2√2 × u(T), or about 2.8 times the bath result. Comparing a difference against u(T) itself would understate the bar by a factor of nearly three, and it is the most likely mistake a team will make.

Four quantities, four jobs, and the course keeps them apart deliberately:

Supplement A.8 gives the arithmetic, and shows explicitly why the first bath reading cannot be used to compute u(T).

l. It must be measured, not looked up. A specification sheet describes a population of instruments under laboratory conditions. It does not describe your loggers, in a bucket, on a boat, at your site, today, with the surge you had and the handling they got. A measured u(T) accounts for all of that. A quoted one accounts for none of it, and is usually optimistic.

m. Every measurement in this course has an uncertainty, not just temperature. Two divers independently judging the distance at which a target disappears will not agree; the difference between them is the between-observer spread for that observation, and it is a property of human eyes rather than of electronics. Two readings of the same light channel, two estimates of surge period, two depth records of the same station — all of them wobble, and all of them can be characterized the same way: repeat the observation when nothing has changed, and see how much the answer moves.

n. How many figures to write down. The rule is that precision follows the quantity, not the instrument. The same logger justifies different precision in different fields of the same report:

The last row is worth dwelling on. An uncalibrated array cannot report a gradient to any number of figures. Rounding does not rescue it.

o. Capture everything; round only when you report. Write down what the instrument shows, at full displayed precision, in the raw record. Rounding at the moment of transcription is irreversible, and it destroys the small differences the whole course exists to detect. Round when you report, to the uncertainty of the quantity you are reporting. Full capture, honest reporting.

p. The reporting rule follows directly, and it is not optional. A difference larger than its applicable threshold may be reported as a difference. A difference smaller than that threshold is reported as unresolved — not as a small change, not as a trend, not as "slightly warmer." The Station Report carries the appropriate uncertainty for every reported value and the applicable threshold for every comparison, and a value without one is incomplete.

q. This is harder than it sounds, because the data will tempt you. Your instruments will hand you numbers with several decimal places. It is natural to read meaning into the last one. The discipline of this course is to look at the noise floor first and the difference second, and to say "unresolved" out loud when that is the honest answer. A student who reports an unresolved result correctly has done better work than one who reports a change that is not there.

r. Zero is a reading, not an absence. Every instrument has a concentration or a magnitude below which it simply cannot see, and it reports that as zero. A nitrate test kit designed for aquarium water, dipped into clear water over a healthy reef, will read zero every time — not because there is no nitrate, but because there is roughly a hundred times less than the kit can detect. The correct statement is never "there is none." It is "below the detection limit of this method," and the detection limit is quoted alongside it. The same applies to a visibility measurement where the target never disappears: the answer is "greater than the distance we reached," which is a real result and not a failure.

s. Why this is the centre of the course. Everything that follows — the temperature profile, the light profile, the clarity measurement, the re-occupation and the comparison — is a claim that something is different from something else. Without a measured noise floor, none of those claims can be evaluated, by you or by anyone reading your report. With one, even a null result is informative. This is the difference between collecting numbers and doing science, and it costs twenty minutes with a bucket.