CALIBRATION FORMULAS & DEFINITIONS
Offline reference — Uncertainty Calculator 3.3

Symbols: x_i = instrument reading; n = number of repeated readings at one stable reference; r = actual reference; d = digital display increment; k = coverage factor. Use compatible units throughout.

1. MEAN
x̄ = Σx_i / n
Average indication. Example: 25.01, 25.02, 25.03 bar → 25.02 bar.

2. INDICATION ERROR / DEVIATION
e = x̄ − r
Signed difference from reference. At r = 25 bar and mean 25.02 bar, error = +0.02 bar. Deviation here means indication error, not standard deviation.

3. CORRECTION
C = −e = r − x̄
Example: correction = −0.02 bar. Applying a correction does not eliminate measurement uncertainty.

4. RELATIVE DEVIATION
100 × e / |r| percent
Example: 0.02 / 25 × 100 = 0.08%. Undefined at r = 0. This is not percent of full scale.

5. SAMPLE STANDARD DEVIATION
s = √[Σ(x_i − x̄)² / (n − 1)], n ≥ 2
Describes spread of repeated observations, not their signed error. For the readings above, s = 0.01 bar.

6. REPEATABILITY STANDARD UNCERTAINTY
Mean result: u_repeat = s / √n.
Individual indication: u_repeat = s.
Established-repeatability option: use a validated standard uncertainty from a prior study directly. One current reading cannot estimate sample standard deviation.

7. REFERENCE CERTIFICATE
u_ref = U_certificate / k_certificate
Copy expanded U and coverage factor k from the reference certificate. Example: U = 0.04 bar and k = 2 → u_ref = 0.02 bar. Point overrides replace top defaults only for that point. Do not invent certificate values.

8. DIGITAL RESOLUTION
u_resolution = d / √12
For rounding within ±d/2 with a rectangular distribution. Example: increment 0.01 bar → u_resolution ≈ 0.00288675 bar. Choose another model if this assumption does not fit the instrument.

9. MANUAL DISTRIBUTIONS
Standard u: entered value directly.
Rectangular bounds ±a: u = a / √3.
Triangular bounds ±a: u = a / √6.
Digital increment d: u = d / √12.
Sensitivity coefficient c_i: converts each input's uncertainty to its effect in the output quantity; contribution is c_i × u_i.

10. COMBINED STANDARD UNCERTAINTY
Calibration mode: u_c = √(u_ref² + u_resolution² + u_repeat² + u_other²).
Manual budget: u_c = √(u_ref² + Σ(c_i × u_i)²).
These app models assume independent contributions. Exclude effects already counted when entering other u. Correlated contributions require covariance terms and a separately justified model; the app does not calculate them.

11. EXPANDED UNCERTAINTY
U = k_final × u_c.
Example: u_ref = 0.02, d = 0.01, s = 0.01, n = 3, other u = 0, k_final = 2 gives U ≈ 0.0420317 bar.
Uncertainty describes dispersion associated with a result; it is not its indication error. k = 2 does not guarantee a particular coverage probability for every budget. The app does not determine effective degrees of freedom or select k for you.

12. CALIBRATION POINTS
point_i = minimum + i × (maximum − minimum)/(N − 1), i = 0 to N−1.
Percentage position = 100 × i/(N−1).
N = 5: 0%, 25%, 50%, 75%, 100%.
N = 9: 0%, 12.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%, 100%.
Use the actual reference value at each planned point.

13. TOLERANCE, ACCURACY AND TRACEABILITY
Tolerance: permitted limits specified for the instrument or task. It is distinct from uncertainty. This app does not automatically issue pass/fail decisions or apply guard bands.
Accuracy: qualitative closeness to the true quantity value; avoid treating it as a substitute for error or uncertainty.
Traceability: connection of a measurement result to a reference through a documented calibration chain, each contributing uncertainty. Software alone cannot establish this.

SCOPE
This reference covers the formulas implemented by this app. Specialized temperature, electrical, pressure, mass, torque, dimensional and chemical methods may need further corrections and uncertainty terms. Unit selection labels values; it does not convert them. Validate each method before reporting results.

14. WORKED PROCEDURE EXAMPLES
The procedure PDFs include hypothetical datasets and calculated budgets, not job readings. For a pressure example at reference 50 psi, readings 50.02, 50.03, 50.03, 50.03, 50.04 psi give mean 50.03, error +0.03, correction -0.03, sample s=0.00707107 and uA=0.00316228 psi. With certificate U=0.02 psi at k=2, resolution 0.01 psi, independent rectangular drift +/-0.01 psi and environment +/-0.005 psi, uc=0.01264911 psi and U=0.02529822 psi at k=2. These assumptions are not defaults for real work.

15. LINEAR TRANSFER EXAMPLES
For configured input limits L,H and output limits A,B: expected output = A + (input-L)*(B-A)/(H-L). Output error = measured output - expected output. Input-reference sensitivity magnitude = abs((B-A)/(H-L)); convert input uncertainty to output units before combining it. Include the output measuring reference separately. Calibration mode assumes direct same-unit comparison; it does not automatically perform this transfer. Use a correctly constructed manual budget for the uncertainty calculation.

16. COVERAGE AND MODEL LIMITS
Example PDF effective degrees of freedom use the Welch-Satterthwaite approximation: nu_eff = uc^4 / sum[(ci*ui)^4/nu_i]. The printed examples assign nu=4 to repeatability and infinite degrees of freedom to the other reliably known contributions. The app does not automatically select k. Procedure coverage calculations are teaching examples, not proof of a specified coverage probability.

Prepared by Sahnwaj
Built by Sahnawaj Alam