Keep trustworthy lab records, query LIMS data, run QC with Westgard rules, build calibration curves, estimate uncertainty, and work within SOPs, CAPA and accreditation.

Lab Data & Quality

## Why it matters

How do you know today's results are right? Every analytical run in a clinical, pharmaceutical or environmental lab includes **quality control (QC) samples**: materials with a known, stable expected value that are tested exactly like real samples. If the QC results behave, the run is accepted. If they do not, the patient or batch results from that run are held. Medical laboratory technologists make this call many times a shift, and getting it wrong in either direction is costly: releasing bad results harms patients, and rejecting good runs wastes time and samples.

## The Levey-Jennings chart

A **Levey-Jennings chart** plots each QC result over time against lines for the expected mean and for ±1, ±2 and ±3 standard deviations (SD). The mean and SD come from the lab's own repeated measurements of that QC lot (often at least 20 results over several days), not just the manufacturer's insert.

To compare results on the same scale, convert each one to a **z-score**:

```text
z = (result - mean) / SD
```

**Worked example.** A glucose control has mean 100 mg/dL and SD 2.5 mg/dL. A result of 106 mg/dL has z = (106 - 100) / 2.5 = 2.4: between 2 and 3 SD above the mean.

If results follow a normal distribution, about 95 percent fall within ±2 SD and about 99.7 percent within ±3 SD. So one result beyond 2 SD happens by chance about 1 time in 20, which is why a single 2 SD result is treated only as a warning.

## Westgard rules

James Westgard's **multirule QC** combines several rules, each sensitive to a different kind of error. The names read as "number of results, and limit": 2_2s means two results beyond 2 SD.

| Rule | Violated when | Usually points to |
|---|---|---|
| **1_2s** (warning) | One result beyond ±2 SD | Triggers inspection with the other rules |
| **1_3s** (reject) | One result beyond ±3 SD | Random error or a large systematic error |
| **2_2s** (reject) | Two consecutive results beyond 2 SD **on the same side** | Systematic error (shift) |
| **R_4s** (reject) | One result above +2 SD and the other below -2 SD (range over 4 SD) | Random error |
| **4_1s** (reject) | Four consecutive results beyond 1 SD on the same side | Systematic error (gradual shift) |
| **10_x** (reject) | Ten consecutive results on the same side of the mean | Systematic error (bias, drift) |

Strictly, R_4s is applied to two control levels within the same run, while 2_2s, 4_1s and 10_x can be applied within a run and across consecutive runs. In this lesson's activity we apply every rule to a single control's sequence of results, to keep it simple.

Rules for your lab are chosen when the method is validated, based on how good the method is compared with its allowed error. Some labs use fewer rules for very precise methods. Westgard's site explains the full approach: https://www.westgard.com/mltirule.htm (Checked on 2026-10-23.)

## Worked example: judging a run

The last five results (as z-scores, oldest first) for a control are:

```text
0.4, 1.1, -0.3, 2.3, 2.6
```

- The last result (2.6) is beyond 2 SD: **1_2s warning**, so check the other rules.
- 1_3s? No, 2.6 is below 3.
- 2_2s? The last two results, 2.3 and 2.6, are both above +2 SD. **Yes: reject.**
- This pattern suggests a systematic shift: perhaps a new reagent lot, a calibration problem or a deteriorating control.

## What to do when a rule fails

1. **Do not report** patient or batch results from the run.
2. **Do not simply repeat the control** until it passes. That hides real problems.
3. Investigate: reagents and lots, calibration, instrument maintenance, the control material itself (stored correctly, in date), recent changes.
4. Fix the cause, recalibrate if needed, rerun controls, then rerun the affected samples.
5. **Document** the failure, the investigation and the corrective action.

In clinical labs, QC rules and records are required under CLIA (42 CFR Part 493: https://www.ecfr.gov/current/title-42/chapter-IV/subchapter-G/part-493, checked on 2026-10-23) and checked by accrediting bodies such as CAP.

## Common mistakes

- Treating a 1_2s warning alone as a rejection, or ignoring it entirely.
- Rerunning a failed control until it passes.
- Using the manufacturer's range instead of the lab's own mean and SD.
- Ignoring slow trends because no single result looks bad (the 4_1s and 10_x rules catch these).

## Quick recap

- z = (result - mean) / SD puts every QC result on the same scale.
- 1_2s warns; 1_3s, 2_2s, R_4s, 4_1s and 10_x reject.
- Random-error rules: 1_3s, R_4s. Systematic-error rules: 2_2s, 4_1s, 10_x.
- On rejection: hold results, investigate, fix, rerun, document.

## Try it

Open the **Do** tab. A Python Westgard checker already handles most rules; you write the 2_2s rule.


Plot QC on a Levey-Jennings chart and flag runs that break Westgard rules.

QC charts and Westgard rules

Chapter 3 Levey-Jennings Charts & Westgard Rules

Bio-Rad Blackboard - Levey-Jennings Charts

## Why it matters

Most quantitative lab methods do not measure concentration directly. An HPLC measures peak area, a spectrophotometer measures absorbance, an ICP measures emission intensity. A **calibration curve** turns that signal into a concentration. How well you build it decides how accurate your results are, and its statistics tell you the smallest amounts you can reliably detect and report. Method validation guidelines (such as ICH Q2 for pharmaceuticals) require labs to show both.

## Building a calibration curve

1. Prepare a **blank** and at least five standards that span the expected range of samples.
2. Measure each standard (often in replicate).
3. Fit a straight line by **least-squares linear regression**: signal = slope x concentration + intercept.
4. Check the fit: look at the **residuals** (measured signal minus the line's predicted signal). They should scatter randomly around zero. A curved pattern means the response is not linear over that range.
5. Use the line to calculate sample concentrations: c = (signal - intercept) / slope.

**Correlation coefficient r (or r squared)** is often reported, and many methods require, for example, r squared of at least 0.99. But a high r squared does not prove linearity: a slightly curved response can still give r squared of 0.995. Always look at the residuals.

## Limit of detection and limit of quantitation

- **Limit of detection (LOD):** the lowest concentration that can be reliably distinguished from zero. Below it, you report "not detected".
- **Limit of quantitation (LOQ):** the lowest concentration that can be measured with acceptable precision and accuracy. Between LOD and LOQ, you can say the analyte is present but cannot give a reliable number.

One approach in ICH Q2 uses the standard deviation of the response (sigma) and the slope (S) of the calibration curve:

```text
LOD = 3.3 x sigma / S
LOQ = 10 x sigma / S
```

Sigma can be the **residual standard deviation** of the regression line, the standard deviation of the y-intercept, or the standard deviation of blank measurements. The residual standard deviation for n points is:

```text
s_res = sqrt( sum of squared residuals / (n - 2) )
```

The n - 2 is because the line used up two parameters (slope and intercept).

### Worked example

Lead standards (µg/L) and instrument signals:

| Concentration | 0 | 1 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|---|
| Signal | 0.010 | 0.122 | 0.228 | 0.452 | 0.668 | 0.889 |

A least-squares fit gives a slope of about 0.1098 per µg/L and an intercept of about 0.0106. The residuals are small, and the residual standard deviation is about 0.0019. So:

```text
LOD = 3.3 x 0.0019 / 0.1098 = about 0.057 µg/L
LOQ = 10  x 0.0019 / 0.1098 = about 0.17 µg/L
```

A sample with a calculated concentration of 0.12 µg/L would be reported as detected but below the LOQ; one at 3.2 µg/L is reported as a number. (Your activity will compute these values exactly.)

Calculated LOD and LOQ are estimates. Validation guidelines expect them to be **confirmed** by analysing samples near those levels.

## Good practice

- **Bracket your samples.** Do not report results above the highest standard; dilute and rerun.
- **Do not force the line through zero** unless the method was validated that way.
- **Run check standards** during the run (for example every 10 samples) to show the calibration still holds.
- **Use independent standards** for QC: prepared from a different stock or lot than the calibration standards, so a stock error does not hide itself.
- **Record everything:** standard preparation, lot numbers, the fitted equation, r squared and residuals.

## Common mistakes

- Trusting r squared without looking at residuals.
- Extrapolating below the lowest or above the highest standard.
- Using the same stock solution for calibration standards and QC.
- Reporting numeric results below the LOQ as if they were reliable.
- Dropping an inconvenient standard from the curve without a documented reason.

## Quick recap

- Fit signal against concentration with least squares; check residuals, not just r squared.
- c = (signal - intercept) / slope, within the calibrated range only.
- LOD = 3.3 x sigma / S and LOQ = 10 x sigma / S (ICH Q2 approach), with sigma often the residual SD.
- Confirm LOD and LOQ experimentally, and use independent QC standards.

ICH quality guidelines (Q2 covers analytical validation): https://www.ich.org/page/quality-guidelines (Checked on 2026-10-23.)

## Try it

Open the **Do** tab. The Python code fits the curve with `statistics.linear_regression` and computes the residual SD; you add the LOD and LOQ formulas.


Fit a calibration curve, check its residuals, and compute the limit of detection and limit of quantitation.

Calibration curves

Spectrophotometers, calibration curves and Beer's Law

Chapter 5: Limit of Detection and Limit of Quantitation | CHM 214 | 040

## Why it matters

No measurement is exact. A result of "0.995 mg/mL" really means "somewhere close to 0.995, within some range". **Measurement uncertainty** quantifies that range. It tells a client whether a result is clearly inside a specification, or close enough to the limit that the difference may not be real. Accredited testing and calibration labs (ISO/IEC 17025) must estimate it for their methods, and medical labs accredited to ISO 15189 must too.

## Key terms

- **Standard uncertainty (u):** the uncertainty of one input expressed as a standard deviation.
- **Relative standard uncertainty:** u divided by the value, for example u(m) / m.
- **Combined standard uncertainty (u_c):** the uncertainty of the final result, combining all inputs.
- **Expanded uncertainty (U):** u_c multiplied by a **coverage factor k**, usually k = 2, which gives a range expected to contain the true value with about 95 percent confidence.

Results are reported as value ± U, with k stated: "0.9950 ± 0.0024 mg/mL (k = 2)".

## Where the input uncertainties come from

**Type A** evaluations use statistics on repeated measurements: the standard deviation of replicates (or of the mean, s / sqrt(n)).

**Type B** evaluations use other information: certificates, manufacturer specifications, tolerances. A tolerance written as ±a is converted to a standard uncertainty by assuming a distribution:

| Information | Assumed distribution | Standard uncertainty |
|---|---|---|
| Certificate gives U with k = 2 | Normal | U / 2 |
| Tolerance ±a, any value equally likely | Rectangular | a / sqrt(3) |
| Tolerance ±a, values near the centre more likely (often glassware) | Triangular | a / sqrt(6) |

## Combining uncertainties

For a result calculated by **adding or subtracting** inputs, combine the **standard uncertainties** in quadrature (square, add, square root):

```text
y = a + b - c      u(y) = sqrt( u(a)^2 + u(b)^2 + u(c)^2 )
```

For a result calculated by **multiplying or dividing**, combine the **relative** uncertainties in quadrature:

```text
y = a x b / c      u(y) / y = sqrt( (u(a)/a)^2 + (u(b)/b)^2 + (u(c)/c)^2 )
```

Quadrature works because independent errors partly cancel: it is unlikely that every input is at its worst in the same direction at once. This assumes the inputs are independent.

## Worked example: a standard solution

You dissolve 100.0 mg of a reference material with purity 0.995 and make it up to 100.00 mL in a Class A volumetric flask.

```text
c = m x P / V = 100.0 x 0.995 / 100.00 = 0.9950 mg/mL
```

Input uncertainties:

| Input | Value | Standard uncertainty | Relative |
|---|---|---|---|
| Mass m | 100.0 mg | 0.05 mg (from the balance calibration) | 0.00050 |
| Purity P | 0.995 | 0.001 (certificate) | 0.00101 |
| Volume V | 100.00 mL | 0.08 / sqrt(3) = 0.046 mL (rectangular) | 0.00046 |

```text
relative u_c = sqrt(0.00050^2 + 0.00101^2 + 0.00046^2) = sqrt(1.47e-6) = 0.00121
u_c = 0.00121 x 0.9950 = 0.00121 mg/mL
U = 2 x 0.00121 = 0.0024 mg/mL (k = 2)
```

Result: **0.9950 ± 0.0024 mg/mL (k = 2)**. The purity contributes most, so a higher-grade reference material would help more than a better balance. An **uncertainty budget** table like the one above shows exactly where to improve.

## Using uncertainty in decisions

A specification is 0.990 to 1.010 mg/mL. A result of 0.9950 ± 0.0024 is clearly inside. A result of 0.9915 ± 0.0024 straddles the lower limit: the true value might be below it. Labs agree a **decision rule** in advance (often with the client) for these cases, as ISO/IEC 17025 requires.

## Common mistakes

- Adding uncertainties directly instead of in quadrature (overestimates).
- Mixing absolute and relative uncertainties in one formula.
- Using a tolerance as if it were a standard uncertainty (forgetting / sqrt(3) or / sqrt(6)).
- Reporting U without saying k.
- Quoting far more digits than the uncertainty justifies. Round U to one or two significant figures and the result to match.

## Quick recap

- Standard uncertainty is a standard deviation; expanded U = k x u_c, usually k = 2.
- Type A from repeats; Type B from certificates and tolerances (a / sqrt(3) rectangular).
- Sums: combine absolute uncertainties in quadrature. Products and quotients: combine relative uncertainties in quadrature.
- An uncertainty budget shows the biggest contributor.

NIST's guide to expressing uncertainty: https://www.nist.gov/pml/nist-technical-note-1297 (Checked on 2026-10-23.)

## Try it

Open the **Do** tab and finish a Python uncertainty calculator: you write the quadrature sum of relative uncertainties.


Estimate standard uncertainties and propagate them through a calculation to an expanded uncertainty.

Measurement uncertainty

Measurement Uncertainty | Metrology Matters

Error propagation formulas + error propagation examples with addition, multiplication, exponents.

Flag runs with Westgard rules, fit calibration curves with LOD and LOQ, and propagate measurement uncertainty.