Type A Statistics and Repeatability

Measurement uncertainty evaluation in modern calibration is governed internationally by the Guide to the Expression of Uncertainty in Measurement (GUM), published as JCGM 100:2008 (formerly ISO/IEC Guide 98-3). The GUM establishes a fundamental distinction between two methods of evaluating uncertainty components:

1. Type A Evaluation: The method of evaluation of uncertainty by the statistical analysis of a series of observations.
2. Type B Evaluation: The method of evaluation of uncertainty by means other than the statistical analysis of series of observations.

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The Mathematical Basis of Type A Evaluation

Repeated observations under the same stabilized setup can support a repeatability study, but those conditions alone do not establish independence. Check autocorrelation, drift, timing, and resetting or reloading effects. The following formulas assume independent observations representing the intended measurand; correlated samples require an appropriate analysis.

The Sample Mean (Arithmetic Average)

The best estimate of the true value of the measurand $\mu$ based on $n$ independent observations is the arithmetic mean $\bar{x}$:

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i = \frac{x_1 + x_2 + \dots + x_n}{n}$$

Statistically, $\bar{x}$ is an unbiased, minimum-variance linear estimator under equal-variance, independent observations of the population mean $\mu$.

Sample Variance and Bessel's Correction

The dispersion of the individual observations about their sample mean is quantified by the sample variance $s^2$:

$$s^2 = \frac{1}{n - 1} \sum_{i=1}^n (x_i - \bar{x})^2$$

Why divide by $n - 1$ instead of $n$? If deviations were calculated from the true population mean $\mu$ (which is unknowable in practical calibration), dividing by $n$ would yield an unbiased estimate: $\sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2$. However, because the sample mean $\bar{x}$ is calculated directly from the sample data, the sum of deviations $\sum_{i=1}^n (x_i - \bar{x})$ is algebraically forced to equal zero. This mathematical constraint removes one degree of freedom from the system.

Calculating deviations from $\bar{x}$ systematically underestimates the true dispersion because the data points cluster more tightly around their own sample average than around the true population mean $\mu$. Introducing Bessel's correction (dividing by $n - 1$ rather than $n$) mathematically corrects for this downward bias, producing an unbiased estimator of population variance.

Sample Standard Deviation

The sample standard deviation $s$ is the positive square root of the sample variance:

$$s = \sqrt{s^2} = \sqrt{\frac{1}{n - 1} \sum_{i=1}^n (x_i - \bar{x})^2}$$

Alternative computational formula (ideal for hand calculators):

$$s = \sqrt{\frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n - 1}}$$

Degrees of Freedom ($\nu$)

The number of degrees of freedom represents the number of independent pieces of information available to estimate the variance:

$$\nu = n - 1$$

Degrees of freedom reflect the reliability of the standard deviation estimate. When $n$ is small (e.g., $n = 3, \nu = 2$), the estimate of $s$ is statistically volatile. When $n \ge 30$ ($\nu \ge 29$), the sample standard deviation $s$ generally becomes more stable as sample size increases, without a universal cutoff that proves convergence toward the true population standard deviation $\sigma$.

Standard Uncertainty of the Mean (Standard Error)

When reporting calibration results, the value assigned to the measurand is typically the calculated mean $\bar{x}$. For independent, identically distributed observations with finite variance, the mean has a standard deviation that is smaller than the standard deviation of the individual readings by a factor of $1/\sqrt{n}$.

The standard uncertainty of the mean $u(\bar{x})$, often designated $s(\bar{x})$, is calculated as:

$$u(\bar{x}) = \frac{s}{\sqrt{n}} = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n(n - 1)}}$$

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The Critical Calibration Distinction: $s$ versus $s/\sqrt{n}$

One of the most frequent errors made by calibration technicians and tested on the ASQ CCT exam is confusing the sample standard deviation ($s$) with the standard uncertainty of the mean ($s/\sqrt{n}$) in an uncertainty budget.

Pooled Standard Deviation ($s_p$)

A small repeatability sample provides limited degrees of freedom. When an independent normally distributed Type A contribution dominates, a Student t coverage factor can provide the required coverage; for ν=3, the two-sided 95% factor is approximately 3.18. The larger factor reflects limited information and is not needless inflation. Evaluate the combined model and effective degrees of freedom rather than applying a blanket rule to every budget.

To overcome this limitation without taking dozens of readings during routine calibrations, metrologists use a pooled standard deviation ($s_p$) derived from a historical measurement assurance program (MAP) or control chart on a check standard.

Mathematical Formulation

When $m$ independent series of repeated measurements are performed on stable check standards under identical repeatability conditions, each run $j$ having $n_j$ observations and sample variance $s_j^2$, the pooled sample variance is the degrees-of-freedom weighted average:

$$s_p^2 = \frac{\sum_{j=1}^m (n_j - 1) s_j^2}{\sum_{j=1}^m (n_j - 1)} = \frac{\nu_1 s_1^2 + \nu_2 s_2^2 + \dots + \nu_m s_m^2}{\nu_1 + \nu_2 + \dots + \nu_m}$$

The pooled standard deviation is the square root:

$$s_p = \sqrt{\frac{\sum_{j=1}^m (n_j - 1) s_j^2}{\sum_{j=1}^m (n_j - 1)}}$$

If all $m$ series have the exact same sample size $n$ (so $n_j = n$ and $\nu_j = n - 1$), the formula simplifies to the root-mean-square of the sample variances:

$$s_p = \sqrt{\frac{1}{m} \sum_{j=1}^m s_j^2}$$

Effective Degrees of Freedom for Pooled Data

The pooled degrees of freedom equals the total number of observations minus the number of series:

$$\nu_{\text{pooled}} = \sum_{j=1}^m (n_j - 1) = \sum_{j=1}^m n_j - m = N - m$$

Operational Example of Pooling

Suppose a laboratory records $m = 10$ historical calibration runs over 6 months using $n = 4$ observations per run on a reference standard.

• Total observations: $N = 40$
• Degrees of freedom: $\nu_{\text{pooled}} = 10 \times (4 - 1) = 30$

Because $\nu_{\text{pooled}} = 30$, the laboratory can claim $\nu = 30$ for its Type A repeatability component during a routine 4-reading calibration, improving the reliability of the repeatability estimate. For an isolated normally distributed component with 30 degrees of freedom, a 95% Student t factor is about 2.042; the complete budget determines effective degrees of freedom and coverage.