Food Analysis Lab · W8

Class proficiency test and data QC

Interlaboratory Proficiency Testing & Statistical Quality Control

Food Analysis Lab · 4-hour session
Pooled blind-sample results (moisture/ash/protein/fat) · Carbohydrate by difference · Grubbs · ANOVA · z-score · Control chart · Measurement uncertainty · Release decision

Pooled blind samplesGrubbs outlier testOne-way ANOVAz-scoreShewhart control chartGuard-band release decision

Before we start, watch a factory video

Butter measured at 80.3% ± 0.4%,
can this batch be released?

Butter factory line (07:04–08:17, note the milk-fat spec ≥ 80%) ↗

Think first: The measured 80.3% is already above the 80% spec, but this number carries an uncertainty of ±0.4% — the true milk-fat content lies between 79.9% and 80.7% with about 95% confidence (U with k=2). Is “the measured number is above spec” the same as “this batch truly conforms”? At the end of class we will answer this with a rule called the guard band.

Today’s task: Pool the shared blind-sample data that the 5 groups produced in W2–W6 and cross-check one another, just like a proficiency test between testing laboratories.
Optional extra practice: Food Analysis Statistics site ↗ — mini-games on control charts, outliers, the uncertainty puzzle, the release officer, and ANOVA; do them and class will make more sense.

Today’s learning goals

After W8 you should be able to…

Pool the 5 groups’ data from W2 (moisture)/W3 (ash)/W5 (protein)/W6 (fat), and calculate carbohydrate by difference
Understand why the uncertainty of the difference method is the root sum of squares of the other four — it inherits everyone’s errors
Use the Grubbs test to check for outliers within groups and at class level, and know how weak this test is at n=3

Use one-way ANOVA to separate repeatability (s_r) from reproducibility (s_R)
Calculate the z-score and classify it into three levels: |z| ≤ 2 / 2 < |z| < 3 / ≥ 3
Read the ±2s / ±3s limits of a Shewhart control chart and spot rule violations
Build a simple uncertainty budget and make a release decision with the guard band rule

Mini-game ①Random error or systematic error?0 / 8

Sort these 8 phenomena from today’s data into “random error” or “systematic error”.

Introduction to measurement uncertainty

Where does the number after ± come from?

Uncertainty: a parameter describing the range in which the true value may lie
Steps: list the sources → convert each to a standard uncertainty u (standard-deviation form) → combine by root sum of squares into u_c → multiply by the coverage factor k to get the expanded uncertainty U
If you only know “within ±a” (tolerance, scale graduation): rectangular distribution, u=a/√3
k=2 corresponds to about a 95% confidence level
Combine by root sum of squares: errors partly cancel, so they cannot simply be added

Statistics Ch06–08 ↗

Carbohydrate: by difference

Why does its uncertainty add up everyone’s errors?

Carbohydrate % = 100 − (moisture % + ash % + protein % + fat %)

u_carb = √( u_moisture² + u_ash² + u_protein² + u_fat² )

The difference method has no experimental step of its own — it just adds and subtracts other people’s results. For that reason, the errors of moisture, ash, protein, and fat are all carried into it, combined by root sum of squares (uncertainties cannot be added directly). So carbohydrate is always the item with the largest uncertainty in proximate analysis (moisture, ash, protein, fat, carbohydrate) — today you will see it calculated.

Errors get passed down too: This class’s fat uses the direct Soxhlet (unhydrolyzed) result from W6. The 5-group mean is only 21.28%, versus the dairy reference method Röse–Gottlieb value of 25.7% — about 4.4 percentage points lower. The difference method adds this underestimate straight onto carbohydrate, inflating it to about 44% (database: about 37–38%). This is a method bias, not random error — however small U_carb is, it cannot cover it.

Class data summary

Blind-sample results of the 5 groups (illustrative data)

Group ⇅	Moisture % ⇅	Ash % ⇅	Protein % ⇅	Fat % ⇅	Carb (by diff.) % ⇅

Click a column header to sort. Moisture/ash/fat are each group’s triplicate mean (fat = direct Soxhlet from W6); protein is a single determination per group (n=1) because the W5 digestion block holds only 8 tubes. The high carbohydrate comes from the underestimated fat (error propagation). Illustrative data — not real measurements.

4-hour lab schedule

Pool → Check → Retest → Statistics → Decide

Time	Contents
0:00–0:30	Explain the pooling rules: long-format input, significant figures, source of the illustrative data (W2/W3/W5/W6)
0:30–1:30	Each group uploads its data; Colab compiles the class blind-sample table and runs the Grubbs outlier check
1:30–3:00	Groups flagged as outliers or with \|z\| > 2 redo that item (moisture or ash); the other groups do the ANOVA and uncertainty calculation
3:00–3:40	Class discussion: which error sources are largest? Cross-check against the W1–W7 calibration and standardization records
3:40–4:00	Release-decision practice: applying the guard band rule

Only moisture or ash is retested: redoing protein (Kjeldahl) and fat (Soxhlet) takes too long to fit in 4 hours —if either of these has a problem, just record it today and leave it for the root-cause discussion; do not redo it on the spot.

Grubbs outlier test

Within group (n=3) too weak vs class level (n=5) can detect it

Within group: G3 moisture triplicate

Values: 2.70, 2.74, 3.30% (★ = suspected outlier)

G = |x−x̄| / s = 1.1526

G_crit(n=3, α=0.05) = 1.1543 (computed from the formula, not looked up in a table)

At n=3, Grubbs almost cannot detect anything: the theoretical upper bound of G is (n−1)/√n = 1.1547 — no matter how extreme the outlier, G can never exceed this number. And G_crit (1.1543) is only 0.0004 below the bound, which means with only 3 replicates, nothing can be judged significant unless the other two values are nearly identical — within a group you must rely on checking the records.

Class level: single protein results of the 5 groups

Group	Result %	G value
G1	25.930	0.2982
G2	26.170	0.6390
G3	26.030	0.4402
G4	24.470	1.7751
G5	26.000	0.3976

G_crit(n=5, α=0.05) = 1.715; G4’s G=1.7751 exceeds the critical value, so it is judged an outlier.

G > G_crit → outlier; α=0.05 = upper limit of 5% probability of wrongly flagging a normal value. Statistics Ch05 ↗

What ANOVA asks

Is the between-group difference real, or just shaky hands?

🔁

Repeatability s_r

Spread of repeats by the same person, same instrument, in a short time (within group)

🏫

Reproducibility s_R

Spread that also includes differences between groups/laboratories; s_R≥s_r

➗

F = between-group ÷ within-group

MSB = variance between group means, MSW = variance among replicates within groups; large F → the between-group difference is too large to be explained by repeatability error alone

🎯

p and F_crit

F>F_crit (p<0.05) → significant difference between groups

Statistics Ch16 ↗ Ch17 ↗

One-way ANOVA

Repeatability (s_r) vs reproducibility (s_R): using ash as an example

s_R = √(s_r² + s_L²)

Within-group variance (F-test denominator, MSW)	s_r = 0.0259% (repeatability)
Between-group variance (related to the F-test numerator, MSB)	s_L = 0.0545% (between groups/laboratories)
Combined	s_R = 0.0604% (reproducibility)
F = MSB / MSW	F = 14.24 (F_crit(0.05,4,10)=3.478, p=0.0004)

F is far above the critical value, p<0.05: the differences among the 5 groups’ ash means cannot be explained by within-group repeatability (s_r) alone — there is a real systematic gap between groups (different weighing habits, judgment of crucible constant weight…). This is exactly why reproducibility s_R must include the between-group variance, rather than looking only at your own group’s replicates.

z-score

Score each group’s protein result with the z-score

z = (x_group − x_assigned) / σ_pt

x_assigned = median (robust assigned value) of the 5 groups’ single results = 26.000%
σ_pt = predefined target standard deviation (fitness-for-purpose: the tolerable spread set in advance for the intended use) = 0.40% (not the contaminated class SD)
|z| ≤ 2: satisfactory 2 < |z| < 3: questionable |z| ≥ 3: unsatisfactory
G4 has z = -3.83, in the unsatisfactory zone — the same group that the class-level Grubbs test flagged on the previous slide, so the two corroborate each other

Illustrative data. If σ_pt is too large, truly problematic groups get “hidden”; if too small, there are too many false flags — set it with the method’s practical precision in mind.

Mini-game ②Order the steps of evaluating a proficiency test— / 6

Use ▲▼ to arrange the 6 steps of “evaluating a proficiency test (PT)” into the correct order.

How to read a control chart

Measure the same QC sample once in every run

QC check sample: a stable sample with a known value, measured together in every analytical run
Center line = historical mean; ±2s warning lines (about 95% of points inside); ±3s action lines (about 99.7%)
Westgard 1_3s: one point beyond ±3s → stop reporting, investigate
2_2s: two consecutive points beyond 2s on the same side → systematic shift
Use the historical baseline for the limits, not the current batch’s own data

Statistics Ch03 ↗

Shewhart control chart

QC check sample: 12 analytical runs, spot the violations

Control limits use the established baseline (mean=26.0%, s=0.30%, not calculated from these 12 points); run 8 exceeds +3s (1_3s violation (Westgard rule): a single point beyond the action limit); runs 10 and 11 are two consecutive points beyond −2s on the same side (2_2s violation) — illustrative data.

Measurement uncertainty budget

Example: protein (Kjeldahl)

Uncertainty source	Relative standard uncertainty
Weighing	0.0115%
Titrant conc.	0.3000%
Titration volume	0.1155%
Conversion factor 6.38	0.2715%
Repeatability	0.5875%
Combined u_c (RSS)	0.7227%

u_c = √(Σ components²) U = k·u_c (k=2)

Combined standard uncertainty u_c = 0.7227% (relative)
Corresponding absolute value u_c(abs) ≈ 0.1879% (at the assigned value 26.000%)
Expanded uncertainty U(k=2) ≈ 0.3758% — this is where the number after ± comes from in a report like 26.0% ± 0.4%.

The standard uncertainty of a rectangular distribution (instrument scale/tolerance) = half-width ÷ √3; components already in standard-deviation form (such as titrant standardization) are used directly.
The repeatability term: each group digested only 1 tube of protein this time (single determination, n=1; see the 8-tube limit in W5), so s_r cannot be estimated from within-group replicates; instead we use 3 replicate determinations (25.95, 26.15, 25.85; illustrative data) that the TA ran on the same milk powder, giving s_r=0.1528%. Because each group reports exactly this one single value, s_r is used directly as the repeatability contribution, with no division by √n (division is only needed when averaging n results; it has nothing to do with the √3 of the rectangular distribution).

Release decision: result ± U vs specification

Back to the opening question: butter 80.3% ± 0.4%, release it?

Guard band rule (used in this course): for an item that must be “≥ spec”, release only when the worst case (result − U) ≥ spec; when the best case (result + U) < spec, do not release; in all other cases (the spec falls inside the result ± U interval) the verdict is cannot decide — retest or negotiate. In this example the worst case 79.90% < spec 80.0%, so even though the point estimate 80.3% is already > spec, the verdict must still be cannot decide — this is exactly why measurement uncertainty matters: it means “conforming” no longer rests on a single number.

How to judge an upper spec limit (≤)

The direction flips

Spec type	Release	Do not release	Cannot decide
Lower limit ≥L	Result−U ≥ L	Result+U < L	Otherwise
Upper limit ≤H	Result+U ≤ H	Result−U > H	Otherwise

Example: potato chips sodium ≤700 mg/100g, measured 690±20 → worst case 710 > 700, best case 670 ≤ 700 → cannot decide

Statistics Ch08 conformity assessment ↗

Mini-game ③This batch: release / do not release / cannot decide?0 / 6

Using the “spec vs result ± U (guard band)” rule, sort the 6 cases into the correct verdict.

Mini-game ④Statistics quick quiz0 / 5

Mini-game ⑤Decision challenge: reading QC results0 / 5

Mini-game ⑥Calculation challenge: if fat uses the reference method, what does carbohydrate become?Unanswered

Group G1: moisture 2.78%, ash 5.98%, protein 25.93%. For fat, do not use direct Soxhlet; use the dairy reference method Röse–Gottlieb value of 25.7%. Calculate carbohydrate % (by difference) and compare it with G1’s carbohydrate in the summary table.

Error traceback map

Which week does today’s anomaly trace back to?

Anomaly	Go back and check
G3 moisture outlier	W2 constant weight / desiccator cooling, W1 balance
G4 protein too low	W4/W5 HCl standardization (the label concentration was used)
Class-wide fat too low	W6 no hydrolysis (method bias)
Between-group ash differences	W3 crucible constant weight, W1 balance
Carbohydrate too high	Fat underestimate propagated through the difference method

Homework

Colab NB08

📊

Complete proximate report table

Compile your group’s moisture/ash/protein/fat/carbohydrate, each with its standard uncertainty and expanded uncertainty U

🎯

Your group’s z-score

Calculate your group’s z-score for protein (or an assigned item) and decide which level it falls in

🔍

Root-cause analysis

If your group is flagged by Grubbs or |z|>2, cross-check the W1–W7 calibration/standardization records to find the most likely error source

🚦

Release-decision practice

Pick any one result, apply the guard band rule to decide release / do not release / cannot decide, and explain your reasoning

Colab NB08: Open NB08 in Colab ↗ · Beginner (no coding): Excel practice workbook EX08 ⬇, Orange workflow OR08 ↗. This report is also a warm-up for the W9 midterm practical — the midterm will directly test your ability to compile data, catch outliers, and calculate uncertainty.

Self-check

By the end of today you should be able to…

✓Calculate carbohydrate by difference and explain why its uncertainty is the root sum of squares of everyone’s
✓Run both the within-group (n=3) and class-level (n=5) Grubbs outlier tests, and explain why n=3 almost cannot detect outliers
✓Distinguish repeatability (s_r) from reproducibility (s_R), and say what the F test in one-way ANOVA compares
✓Calculate the z-score and explain what the three levels |z| ≤ 2 / 2~3 / ≥ 3 each mean
✓Read the ±2s / ±3s limits of a Shewhart control chart and point out the 1_3s and 2_2s violations
✓Build a simple uncertainty budget and calculate the expanded uncertainty U=k·u_c
✓Use the guard band rule (result ± U vs spec) to reach a release / do not release / cannot decide verdict
✓Say which parts of your group’s data today trace back to the calibration and standardization from W1–W7

Tap to tick — make sure you really can do it.