Math Bridge: RSSI distance uncertainty

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Math BridgeUX DesignStruggle-friendly runway

How does -75 dBm become 4.37 metres—and a wide band?

Invert the log-distance rule, then carry each uncertain dB into metres.

UX Uma, the guideUX Uma guides
The one targetCalculate RSSI range and its first-order uncertainty.
The chapter case-59 dBm at 1 m; n=2.5; -75 dBm; 4-6 dB spread.
What it buys youA privacy claim that treats inferred location as a band.

A field team faces an unresolved physical question: How does -75 dBm become 4.37 metres—and a wide band? They must answer it before changing measured rssi on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is measured rssi. The middle card applies this page's relationship. The green card is estimated distance. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Measured RSSI changes estimated distance An input card leads through the page relationship to the estimated distance result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Equal dB changes are multiplicative, so a fixed RSSI error becomes a wider spatial band farther from the beacon.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for measured rssi is -75.

  2. 2

    Name the relationship. d=1x10^[(-59-(-75))/(10x2.5)]=10^0.64=4.37 m relative sensitivity=ln(10)/(10x2.5)=9.21% per dB metre sensitivity=4.37x0.0921=0.402 m per dB 4-6 dB spread→1.61-2.41 m first-order uncertainty

  3. 3

    Substitute the chapter fixture. Set measured rssi to -75. The page ledger gives estimated distance as 4.365 m.

  4. 4

    Read the result. Keep m beside the value. Use it only inside the technical boundary on this page.

Predict, then change measured rssi

Try Predict the direction of estimated distance. Move one control, calculate, then check your prediction.

-75
Chapter baseline
Estimated distance

Observe Equal dB changes are multiplicative, so a fixed RSSI error becomes a wider spatial band farther from the beacon. Reset the control to -75 and compare estimated distance.

Explain Only measured rssi moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only measured rssi moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only measured rssi. It replays the chapter fixture; it is not a field prediction.

1. Begin with a calibrated power reading

RSSI is received power on a logarithmic dBm scale, not direct distance. A reference reading at one metre and a path exponent describe how power is expected to fall in one environment.

UX Uma: The system assumes a physical model, then inverts it; uncertainty follows that same inversion.

2. Name every algebra move

1

Find the dB dropΔR=RSSI0−RSSI.

2

Scale by the environmente=ΔR/(10n).

3

Undo the logarithmd=d0×10e.

4

Differentiate for sensitivityΔd/d≈ln(10)ΔRSSI/(10n).

5

Convert fraction to metresΔd=d×relative error.

3. Reproduce the -75 dBm estimate

d=1×10^[(-59−(-75))/(10×2.5)]=10^0.64=4.37 m
relative sensitivity=ln(10)/(10×2.5)=9.21% per dB
metre sensitivity=4.37×0.0921=0.402 m per dB
4-6 dB spread→1.61-2.41 m first-order uncertainty

The output is not a precise circle around a person. It is one model's range estimate before geometry, calibration drift, body shadowing, and inference risks.

4. Try the RSSI reading

TryMove received power while keeping the one-metre calibration and building exponent fixed.

Measured RSSI
Estimated distance
Relative error / dB
Metres / dB
4 dB spread
6 dB spread

ObserveThe percentage sensitivity per dB stays fixed for this n, but the uncertainty in metres grows with estimated distance.

ExplainEqual dB changes are multiplicative, so a fixed RSSI error becomes a wider spatial band farther from the beacon.

Technical boundaries.

This first-order error propagation is local and assumes the calibration model is valid.

Radio
Multipath, bodies, orientation, hardware, and interference change RSSI
Statistics
A 4-6 dB spread is not automatically a confidence interval
Privacy
Repeated weak signals and joined datasets can reveal more than one estimate

Measure distributions and govern collection, retention, access, inference, and consent.

5. Test inference, not just ranging

Sample multiple people, devices, orientations, rooms, motion states, and time periods. Check what an observer can infer from sequences and joined records.

6. Record the privacy state

Store calibration, exponent, RSSI distribution, device, site, timestamp, estimator, uncertainty, purpose, consent, access, retention, deletion, and retest triggers.

7. Check yourself

Why is -75 dBm not a distance by itself?
Answer: Distance appears only after applying a reference calibration and path exponent.
Why does one dB mean about 9.21% here?
Answer: The inverted logarithm has local fractional sensitivity ln(10)/(10n), with n=2.5.
Does a wide uncertainty band make the data private?
Answer: No. Repeated, combined, or contextual readings may still reveal location or behaviour.
Honesty boundary.

The -59 dBm calibration, n=2.5, -75 dBm reading, and 4-6 dB spread reproduce the chapter's teaching case.

4.37 m
Model estimate, not measured ground truth
First order
Small-error approximation around one operating point
Privacy
Uncertainty does not remove identifiability or duty of care

Correct, not complete: RSSI error propagation does not qualify a privacy control.

Return to Location Privacy and Consent with the maths unpacked →