RD statistics

How Cloud QC uses these statistics

“RD to use” on the Precision charts offers ARD%, CV%, HARD%, MPRD% (the default) and RDHYSLOP. In every formula a is Sample 1 (the original) and b is Sample 2 (the duplicate).

MPRD% and RDHYSLOP keep their sign, so the charts can show a bias between original and duplicate; ARD%, CV% and HARD% are absolute.

The interpretation bands below are general guides. Cloud QC’s charts draw control lines at ±2 and ±3 standard deviations of the pairs shown (or their robust MAD equivalents), and the Overview judges each check stage against its own limit on the absolute MPRD% (standard limits ±30% for field, ±20% for coarse and ±10% for pulp duplicates).

Only pairs whose mean is at least the threshold (the “Threshold factor” × the lower detection limit) are used, since at very low concentrations small absolute differences give large relative differences.

The formulas

Statistic	Formula	a = 2.50, b = 2.40
ARD%	|a − b| / a × 100	4.00
CV%	|a − b| / (√2 × (a + b) / 2) × 100	2.89
HARD%	|a − b| / (a + b) × 100	2.04
MPRD%	(a − b) / ((a + b) / 2) × 100	+4.08
RDHYSLOP	(a − b) / (√2 × (a + b) / 2) × 100	+2.89

The different statistics produce different percentage values from the same pair. All of them measure agreement between duplicate results, but each uses a different normalisation and therefore has a slightly different interpretation.

Absolute Relative Difference (ARD%)

ARD% is one of the most common statistics used to evaluate agreement between duplicate measurements. It expresses the absolute difference between two results as a percentage of the original result:

ARD% = |2.50 − 2.40| / 2.50 × 100 = 0.10 / 2.50 × 100 = 4.0

A lower ARD% indicates better agreement and therefore better analytical precision. It is widely applied to laboratory duplicates, field duplicates, check assays, sample preparation duplicates and routine precision monitoring.

Coefficient of Variation (CV%)

CV% is a standardised measure of relative variability. For duplicate measurements it expresses the standard deviation of the pair as a percentage of the mean, so precision can be compared across different concentration ranges:

Mean = (2.50 + 2.40) / 2 = 2.45    CV% = 0.10 / (1.414 × 2.45) × 100 = 2.89

It is used in laboratory precision studies, duplicate sample monitoring, method validation and inter-laboratory comparisons.

Half Absolute Relative Difference (HARD%)

HARD% scales the duplicate difference by the sum of the two results:

HARD% = 0.10 / 4.90 × 100 = 2.04

Because a + b is twice the pair mean, HARD% is exactly half of the absolute MPRD%, so its values are half as large: a HARD% limit of 10% corresponds to an MPRD% limit of ±20%.

Mean Pair Relative Difference (MPRD%)

MPRD% is a symmetric duplicate precision statistic based on the average of both results. Because neither value is treated as the reference measurement, it is often preferred for routine duplicate precision assessment:

MPRD% = (2.50 − 2.40) / 2.45 × 100 = +4.08

Cloud QC keeps the sign: positive when the original (a) is higher, negative when the duplicate (b) is. A run of one sign on RD v Sequence shows a bias between original and duplicate; the Overview uses the absolute value. (MPRD% without its sign is sometimes called bl%; Cloud QC does not offer it separately.)

Hyslop Relative Difference (RDHYSLOP)

From Hyslop, N.P. and White, W.H. (2009), Estimating Precision Using Duplicate Measurements, Journal of the Air & Waste Management Association, 59(9), 1032–1039.

RDHYSLOP is the difference between the pair, divided by √2 and by the pair mean, as a percentage, with its sign kept. Dividing by √2 turns the difference of two results into the standard deviation of one, so for a single pair it is the CV% with a sign:

RDHYSLOP = 0.10 / (1.4142 × 2.45) × 100 = +2.89

Values near zero indicate strong duplicate agreement. The sign shows which result was higher: positive when the original (a) is higher, negative when the duplicate (b) is. Hyslop and White estimate the precision of a whole set of pairs as the root mean square of these values; Cloud QC charts each pair’s value.

General guide

Absolute value	Assessment
< 5%	Excellent agreement
5–10%	Acceptable agreement
> 10%	Investigation may be required

These bands are a general guide only: what is acceptable depends on the check stage, the element and its concentration.