MEASUREMENT UNCERTAINTY · BUDGET

Uncertainty Budget Calculator

Combine the four uncertainty components using Root Sum of Squares to get the combined standard uncertainty u_c and expanded uncertainty U — matching the Uncertainty via Control Charts note.

u_c = √( RSD²/n_s + RBE² + RB² + u(ref)² )

RSD — Precision

= (SD / x̄) × 100

SD of measurements Mean (x̄)

Validation study size (n_v) — sets RBE = RSD/√n_v

RSD = 1.46%

RBE = 1.46/√15 = 0.378%

RB — Relative Bias

= ((x̄ − Target) / Target) × 100

Measured mean (x̄) Certified target

RB = +2.50%

u(ref) — Reference Uncertainty

= (U_CoA / k) / Target × 100

U_CoA (±) k (CoA) Certified value

u(ref) = 0.250%

Measurement Settings

Sample replicates (n_s) Replicates per sample (routine) Coverage factor (k) k=2 → ~95% confidence

Result value Unit

RSD/√n_s = 1.46/√1 = 1.463%

Combined Uncertainty — Full Calculation

Formula Trace

u_c = √( 1.463² + 0.378² + 2.500² + 0.250² )

= √( 2.1416 + 0.1428 + 6.2500 + 0.0625 ) = √8.5969

u_c (relative)

2.93%

U = k × u_c (k = 2)

5.86%

Result ± U

102.5 ± 6.011 ppb

(3.01 ppb at k=1)

Variance Budget — % Contribution to u_c²

24.91% RSD/√n_s (Precision)

1.66% RBE (Statistical)

72.70% RB (Bias)

0.73% u(ref) (Reference)

Effect of Sample Replicates (n_s) on U

n_s	RSD/√n_s	u_c (%)	U = 2 × u_c	Result ± U (ppb)	Selected
n = 1	1.463%	2.93%	5.86%	102.5 ± 6.011	← current
n = 2	1.035%	2.74%	5.49%	102.5 ± 5.624
n = 3	0.845%	2.68%	5.36%	102.5 ± 5.489
n = 5	0.654%	2.62%	5.25%	102.5 ± 5.379
n = 10	0.463%	2.58%	5.17%	102.5 ± 5.294

Note: Systematic bias (RB) accounts for 72.70% of the variance budget. Adding more replicates has diminishing returns — the dominant improvement path is to reduce method bias through instrument re-calibration or matrix-matched standards.

Back to Uncertainty via Control Charts