How to calculate repeatability and intermediate precision

Repeatability uses an analysts × replicates design and intermediate precision a days × replicates design; a well-planned days × replicates study gives both at once.

• Repeatability: each analyst measures the same sample n times under repeatability conditions. The standard deviations of the groups are pooled: s_r = √(Σ(n_i − 1)s_i² / Σ(n_i − 1)). Eurachem recommends 6–15 replicates per material.
• Intermediate precision: on each day, two or three measurements are made under repeatability conditions, and the analyst, instrument and calibration are deliberately varied between days. Eurachem recommends 6–15 groups for this design.
• Sample: stable and homogeneous, with a matrix and concentration close to routine samples. Use separate levels for the low and high ends of the working range.

For a balanced design with p days and n replicates per day, the calculation is a one-way analysis of variance (ANOVA):

• Within-day mean square: MS_within = ΣΣ(x_ij − x̄_i)² / (p(n − 1))
• Between-day mean square: MS_between = n · Σ(x̄_i − x̄)² / (p − 1)
• Repeatability: s_r = √MS_within
• Between-day component: s_between = √((MS_between − MS_within) / n), set to zero if the difference is negative
• Intermediate precision: s_R(w) = √(s_r² + s_between²)
• Relative standard deviation RSD = s / x̄ × 100; limits r = 2.8 · s_r and R = 2.8 · s_R(w)
• F = MS_between / MS_within; if F exceeds F_crit, the days differ significantly

r is the limit that the difference between two results obtained under the same conditions is expected not to exceed with 95% probability (√2 × 1.96 ≈ 2.8). If the number of replicates differs between days, use n_0 = (N − Σn_i² / N) / (p − 1) instead of n, where N is the total number of values.

In the example, a nitrate control sample at about 25 mg/L is measured three times on each of three days, with a different analyst and a fresh calibration each day.

The grand mean is x̄ = 25.150 mg/L. The ANOVA table:

The results follow in five steps:

• F = 0.2001 / 0.0220 = 9.10; F_crit(0.95; 2; 6) = 5.14. The difference between days is too large to be explained by repeatability.
• s_r = √0.0220 = 0.1483 mg/L; RSD_r = 0.1483 / 25.150 × 100 = 0.59%
• s_between = √((0.2001 − 0.0220) / 3) = 0.2437 mg/L
• s_R(w) = √(0.1483² + 0.2437²) = 0.2853 mg/L; RSD_R(w) = 1.13%
• r = 2.8 × 0.1483 = 0.42 mg/L; R = 2.8 × 0.2853 = 0.80 mg/L

A single standard deviation of all nine values, which is the square root of the total row (√0.0665 = 0.258 mg/L), understates the day effect; ANOVA separates the two sources. The example is kept small to show the arithmetic: the degrees of freedom of s_between are the number of days minus one, here 2. In a real study, the 6–15 groups that Eurachem recommends give a more reliable estimate.

Outlier tests flag a value; removing it from the calculation needs a technical reason as well. In ISO 5725-2, a value above the 5% critical value is a straggler and a value above the 1% critical value is an outlier.

• Cochran's test looks at the share of the largest group variance in the total: C = s²_max / Σs_i². In the example, C = 0.163² / (0.140² + 0.163² + 0.141²) = 0.40. For p = 3 and n = 3 the critical values are 0.871 at 5% and 0.942 at 1%, so nothing is flagged.
• Grubbs' test looks at how many standard deviations the most extreme value lies from the mean: G = |x − x̄| / s. For the day means G = 1.14, and the critical value at 5% is 1.15. With three values G can be at most 1.155, so Grubbs' test rarely flags anything in this design; a meaningful test needs more groups.

The cause of a flagged value is investigated: dilution, calibration, instrument log, sample preparation. If a technical cause is found, the value is set aside with the reason recorded and the original entry is kept; if not, the value stays in the calculation.

When verifying a standard method, you compare your s_r with the value the method gives; in validation, you compare it with the acceptance criterion written beforehand.

• Verification (ISO/IEC 17025 §7.2.1.5): before using a standard method, the laboratory shows that it can achieve the required performance. If the standard gives a repeatability limit r, convert it to a standard deviation with s_r = r / 2.8 and compare it with your own s_r.
• Validation (§7.2.2.1): for non-standard methods, laboratory-developed methods and standard methods used outside their intended scope or otherwise modified, precision is established together with its acceptance criterion.
• The Horwitz equation predicts the between-laboratory RSD expected at a given concentration: PRSD_R (%) = 2 · C^(−0.1505), where C is the mass fraction and 1 mg/kg = 10⁻⁶. The ratio of the RSD you found to this value is the HorRat, a quick check of whether the precision is reasonable for the concentration.

The relative standard deviation of intermediate precision can be used directly as u(Rw) in a Nordtest calculation; in this example u(Rw) = 1.13%. In the measurement uncertainty guide, 14 months of data from the same control sample give 1.85%.

The difference is expected: a three-day study does not yet cover new stock solutions, reagent lots or recalibration of the instrument. That is why Nordtest recommends u(Rw) data covering at least a year, ideally more than 60 results. A short study is a starting value until control data build up. You can try both calculations on your own data with the measurement uncertainty Excel template.

PureField LIMS is laboratory software that works in line with ISO/IEC 17025 and ISO 15189. In its measurement uncertainty module, repeatability and intermediate precision are separate study types.

• Repeatability: k operators × n replicates (at least 2). For each group it calculates n, mean, s and RSD, then the pooled s_r, RSD_r and r = 2.8 · s_r.
• Intermediate precision: at least 2 groups (days, operators or instruments) × replicates, and groups may have different numbers of replicates. The screen shows the ANOVA table with F and F_crit, s_r, the between-group s, s_R(w), RSD_R(w) and R = 2.8 · s_R(w).
• Cochran and Grubbs results are flagged as a straggler (5%) or an outlier (1%). Every value stays in the calculation, and flags are reviewed in the evaluation.
• Under each level there is a chart of values by group: points for the values, a thick line for the group mean and a dashed line for the grand mean.
• For mass fraction units (mg/kg, µg/L, %), the Horwitz PRSD_R and HorRat are shown for information.
• The study owner invites the analysts; each analyst enters values in their own column, and the statistics are recalculated on the server after every save. A saved value is corrected by the person who entered it, with a reason, and the previous value stays in the record history.
• An approved intermediate precision study can be selected as the u(Rw) source of a Nordtest study; its RSD_R(w) is copied into the Nordtest study.