Find the question you need
Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Name the shared measurement
Chapters 1–2. Draw the data flow and choose the counting model.
Follow the uncertainty
Chapters 3–5. Calculate covariance, differences and averages.
Practise and continue
Chapters 11–12. Work new count records and choose a reading route.
Full chapter index · Practice and worked checks · Photonics subject root
A laboratory measures several light signals and subtracts the same background estimate from each. The corrected values look like separate results. Their uncertainty, however, now contains a shared ingredient. Treating them as independent can exaggerate uncertainty in a difference and understate it in an average.
This guide follows that shared ingredient through original photon-counting examples. It develops a different question from dividing time between gross and background measurements: what happens after one background measurement is reused? Return to the Photonics root for the wider light-to-information chain.
Chapter index
Name the shared measurement · Chapters 1–2
1. Draw the data flow before combining results
2. State the counting model and units
Follow the uncertainty · Chapters 3–5
3. Calculate the shared covariance
4. Subtract two corrected readings
5. Average readings that share a correction
Change the correction · Chapters 6–8
6. Compare shared and independently measured backgrounds
7. Change the correction coefficients
8. Distinguish shared estimation from physical drift
Check the evidence · Chapters 9–10
9. Report the information needed for reuse
10. Answer the useful questions
Practise and continue · Chapters 11–12
11. Practise with a new acquisition record
12. Choose a connected reading route
CHAPTER 1 OF 12 · Name the shared measurement
1. Draw the data flow before combining results
Let g₁ and g₂ denote measured gross count rates and b the measured background rate. Define corrected rates x₁ = g₁ − b and x₂ = g₂ − b. Each gross observation has its own fluctuations, but both corrected results contain exactly the same measured b.
Imagine that the background estimate happens to be high by one count per second. Both corrected results move down by one. Imagine it is low by one: both move up by one. This common movement creates positive covariance, even if the original gross and background observations were independent.
The dependence comes from data processing; it need not mean photons in the two gross measurements interacted. Keep a record of which raw observation feeds which result. Two copies of the same background number do not become two independent measurements because they occupy different spreadsheet cells.
CHAPTER 2 OF 12 · Name the shared measurement
2. State the counting model and units
Assume separate acquisition intervals give independent Poisson counts G₁, G₂ and B. Their durations t₁, t₂ and t_b are treated as known. Count rates are g₁ = G₁/t₁, g₂ = G₂/t₂ and b = B/t_b. The optical background is assumed stable and applicable to both gross measurements.
For a Poisson count with mean μ, variance is μ. A usual plug-in variance estimate for a measured rate is observed count divided by duration squared: v₁ = G₁/t₁², v₂ = G₂/t₂² and v_b = B/t_b². These have units of rate squared. Standard uncertainties are square roots of variance estimates.
These are counting contributions, not a complete instrument uncertainty. Detector dead time, drift, afterpulsing, sample changes, timing uncertainty and calibration can require other terms or another model. Very small counts also need suitable interval methods; a square-root estimate alone does not guarantee a stated coverage probability.
Because the three raw observations are independent, Var(x₁) = v₁ + v_b and Var(x₂) = v₂ + v_b. Cov(x₁, x₂) = v_b: both contain −b, and multiplying the two negative coefficients gives a positive contribution.
Use invented observations G₁ = 10,000 over 100 s, G₂ = 9,000 over 100 s and B = 500 over 50 s. The measured rates are 100, 90 and 10 counts/s. Thus x₁ = 90 and x₂ = 80 counts/s.
| Quantity | Value | Plug-in variance, (counts/s)² |
|---|---|---|
| Gross rate g₁ | 100 counts/s | 1.00 |
| Gross rate g₂ | 90 counts/s | 0.90 |
| Shared background b | 10 counts/s | 0.20 |
| Corrected x₁ | 90 counts/s | 1.20 |
| Corrected x₂ | 80 counts/s | 1.10 |
The covariance is 0.20 (counts/s)². The corresponding correlation coefficient is 0.20/√(1.20 × 1.10) ≈ 0.174. This moderate correlation is sufficient to change a combined uncertainty calculation; it should not be discarded because the individual uncertainty bars look small.
The difference is d = x₁ − x₂ = (g₁ − b) − (g₂ − b) = g₁ − g₂. Under the identical additive correction, the shared estimate cancels exactly from the difference.
Using covariance gives the same answer: Var(d) = Var(x₁) + Var(x₂) − 2Cov(x₁, x₂) = v₁ + v₂. For the invented data, d = 10 counts/s and estimated variance is 1.90 (counts/s)², giving standard uncertainty about 1.378 counts/s.
If the corrected values were incorrectly treated as independent, the variance would be 1.20 + 1.10 = 2.30, giving about 1.517 counts/s. That calculation counts the shared background fluctuation twice even though it cancels algebraically.
Cancellation of the numerical correction does not prove that the true physical backgrounds were equal during both measurements. If they changed, g₁ − g₂ can contain a background difference. The algebra establishes what cancels in the model, not that the model fits the experiment.
CHAPTER 5 OF 12 · Follow the uncertainty
5. Average readings that share a correction
For the average a = (x₁ + x₂)/2, the background does not cancel: a = (g₁ + g₂)/2 − b. Its variance is (v₁ + v₂)/4 + v_b.
Our example gives a = 85 counts/s, variance 1.90/4 + 0.20 = 0.675 (counts/s)² and standard uncertainty about 0.822 counts/s. Incorrectly treating x₁ and x₂ as independent gives (1.20 + 1.10)/4 = 0.575, or about 0.758 counts/s. This time the error understates uncertainty.
For n independent gross measurements with equal variance v, sharing one background estimate with variance v_b, the arithmetic mean has variance v/n + v_b. Repeating gross measurements reduces the first term. It does not repeat the background observation or divide its uncertainty by n.
The limiting standard uncertainty √v_b is a floor only within this design and fixed-background model. It is not an unavoidable physical limit: obtaining additional suitable background evidence can change it.
CHAPTER 6 OF 12 · Change the correction
6. Compare shared and independently measured backgrounds
Suppose four gross observations each have variance 1, and the shared background has variance 0.2. Their mean variance is 1/4 + 0.2 = 0.45. If each gross observation instead has its own independent background estimate with variance 0.2, the mean variance is (1 + 0.2)/4 = 0.30.
| Design or calculation | Mean variance, (counts/s)² | Standard uncertainty, counts/s |
|---|---|---|
| Four gross observations, one shared background | 0.45 | About 0.671 |
| Four gross observations, four independent backgrounds of variance 0.2 each | 0.30 | About 0.548 |
| Four gross observations, background known without uncertainty | 0.25 | 0.500 |
This comparison gives the independent-background design four background acquisitions at the stated individual precision. It does not hold total counting time fixed. If a fixed total background time is divided among four independent Poisson acquisitions under identical rates and no overhead, each is less precise; the apparent advantage must be recalculated.
The measurement-time guide is the next route when total experimental time, rather than the uncertainty of an already defined set of acquisitions, is the constraint.
Sometimes background is scaled differently for two results. Write x₁ = g₁ − a₁b and x₂ = g₂ − a₂b, with a₁ and a₂ treated as known coefficients. Their shared covariance is a₁a₂v_b. The difference contains −(a₁ − a₂)b, so its variance is v₁ + v₂ + (a₁ − a₂)²v_b.
Use the same invented raw observations but a₁ = 1 and a₂ = 2. Corrected rates are now 90 and 70 counts/s, and their difference is 20 counts/s. The variance of that difference is 1 + 0.9 + 0.2 = 2.1, giving standard uncertainty about 1.449 counts/s.
The background no longer cancels because the coefficients differ. This could represent a justified scaling in a defined measurement model; it is not permission to choose coefficients to improve an answer. If coefficients have uncertainty or share calibration inputs, include their contributions and correlations too.
CHAPTER 8 OF 12 · Change the correction
8. Distinguish shared estimation from physical drift
One reused blank is a shared statistical input. A drifting optical background is a changing physical condition. They can occur together, but they require different evidence. A longer blank acquisition may reduce Poisson uncertainty while averaging over a change that makes it less representative of a later sample measurement.
Record acquisition order, durations, detector settings, sample state and environmental changes. Suitable interleaved blanks or checks can investigate stability. Their interpretation still needs a model: interpolated background estimates can share inputs and create a different covariance pattern across corrected results.
Do not replace one unsupported independence assumption with an unsupported assertion of perfect correlation. Identify the actual shared observations and additional physical variations. A covariance model should describe how uncertainty enters, rather than simply supply a convenient coefficient.
A report listing corrected values and individual uncertainties can be insufficient for someone computing an average or difference. Include the raw counts and durations where appropriate, the correction equation, the identity of the reused background and the covariance information or data needed to reconstruct it.
For our two-result example, a compact covariance matrix has diagonal entries 1.20 and 1.10 and both off-diagonal entries 0.20, in (counts/s)². That representation makes both the individual variances and the shared component available.
The general rule for a linear combination y = Σwᵢxᵢ is Var(y) = Σwᵢ²Var(xᵢ) + 2Σᵢ<ⱼwᵢwⱼCov(xᵢ, xⱼ). In this linear model the variance identity is exact when the stated variances and covariances apply. Plug-in estimates remain estimates; nonlinear outputs such as ratios need their own treatment.
Does background subtraction remove background noise?
It corrects an estimated mean contribution. It does not erase fluctuations in the gross count or uncertainty in the correction. A shared correction can cancel from a particular difference without disappearing from every reported quantity.
Should uncertainty always get smaller when correlation is included?
No. The signs and weights matter. Positive covariance reduces the variance of a difference but increases that of a sum or average relative to the same individual variances treated as independent.
Can I use the scatter of corrected replicates to estimate everything?
If every replicate shares one fixed measured correction, their scatter does not reveal the uncertainty in that common correction. Keep the uncertainty budget as well as the replicate scatter.
Does a background-free difference prove a signal changed?
Only within an adequate measurement model. Changes in physical background, gain, alignment or sample conditions can still affect the interpretation. State what was controlled and checked.
CHAPTER 11 OF 12 · Practise and continue
11. Practise with a new acquisition record
Take invented independent counts G₁ = 625 in 25 s, G₂ = 500 in 25 s and B = 100 in 20 s. One shared background estimate is used without scaling. Find the corrected rates, their covariance, the difference and the mean.
Gross rates are 25 and 20 counts/s; background is 5 counts/s. Corrected rates are 20 and 15 counts/s. Gross variance estimates are 625/625 = 1 and 500/625 = 0.8; background variance is 100/400 = 0.25. Corrected variances are 1.25 and 1.05, with covariance 0.25.
The difference is 5 counts/s with variance 1.8, so standard uncertainty is about 1.342 counts/s. The mean is 17.5 counts/s with variance 1.8/4 + 0.25 = 0.70, so standard uncertainty is about 0.837 counts/s.
Now double only the number of comparable gross acquisitions while reusing the original blank. Explain why the common background term is unchanged. Then change the design to newly measured independent blanks and identify which extra acquisition time and stability assumptions your comparison needs.
Counting and optical evidence
Return to Photonics and gross/background time allocation. Use particle counters and counting uncertainty for the underlying rate and subtraction calculations. Follow X-ray photoelectron spectroscopy for a different measurement in which background modelling affects interpretation; its fitted spectral background is not automatically this independent Poisson blank model.
Primary measurement guidance
NIST Technical Note 1297, Appendix A gives uncertainty propagation including covariance. NIST’s study of background subtraction in XPS examines correlations in a specialist setting. These sources support the measurement principles; all numerical count records here are original teaching cases.
Learning transfer
Connect to The Gold Standard of Education and How X Works. The portable skill is tracing shared inputs before combining results, whether the final task is comparing two signals or reporting their average.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.