Example 3 — Load cell, hysteresis + repeatability

Exercises: the up/down sweep hysteresis detector, the triplicate-point repeatability detector, and another simple-acceptance-vs-shared-risk divergence.

3.0 Setup

3.1 Calibration wizard settings

Type external, Regression degree: 1 (explicit — this example is about hysteresis/repeatability, not degree selection), distribution Normal, confidence 95%, decision rule Shared risk (tolerance + U).

Data points — an ascending sweep, a descending sweep back down (for hysteresis), and one extra reading at 0 N (three total readings at 0 N, for repeatability):

The math

Fit (degree 1, all 12 points): $a = 0.99833$, $b = -0.76668$.

Max error = 0.863215 N, at point 11 (reference 100 N, measured 101.8 N) → %FS error = $0.863215/500 \times 100\% = \mathbf{0.1726\%}$.

Hysteresis — Open Gauge groups points by reference value and takes the largest spread of measured values within any group where the reference value repeats with both an ascending and descending segment present:

Largest span = 1.2 N (at 100 N) → hysteresis = 1.2.

Repeatability — only 0 N has 3+ readings (0.2, 0.3, 1.0):

$$\bar{x} = \frac{0.2 + 0.3 + 1.0}{3} = 0.5, \qquad s^2 = \frac{(0.2-0.5)^2 + (0.3-0.5)^2 + (1.0-0.5)^2}{3-1} = \frac{0.09+0.04+0.25}{2} = 0.19$$ $$\text{repeatability} = \sqrt{0.19} = 0.435890\ \text{N}$$

Uncertainty budget:

• Combined: $u_c = \sqrt{0.487526^2 + 0.057735^2} = \mathbf{0.490933\ N}$.
• $\nu_{\text{eff}} \approx \mathbf{10.3}$.
• Coverage factor $k = \Phi^{-1}(0.975) \approx 1.96$ (from Confidence 95%, not entered).
• Expanded: $U = 0.490933 \times 1.96 = 0.962211 \approx \mathbf{0.96\ N}$ (2 sig figs).

Decision rule — try both and compare. The accuracy spec is ±0.15% FS, so the tolerance is flat across the range: $\text{tolerance} = 0.0015 \times 500\ \text{N} = 0.75\ \text{N}$. The sweep's largest error (0.863215 N, at reference 100 N) exceeds that:

This is a genuinely realistic case: a load cell whose raw error exceeds its tight ±0.15% FS spec, but whose overall measurement uncertainty is large enough that, under a shared-risk rule, the lab and customer agree it's still acceptable. A good test of both "real fails happen" and "the rule you pick changes the outcome."

Expected results