Reference ID: MET-5BE8 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Theoretical random variance quantifies the intrinsic scatter expected when a finite number of identical particles are sampled from a well-mixed lot. In process engineering it underpins:
Design of sampling protocols for powders, granules, and slurries.
Assessment of whether observed variability is within the natural statistical limit or indicates segregation, agglomeration, or process drift.
Specification of minimum sample masses to achieve a target precision for assays, moisture, or potency.
Methodology & Formulas
Convert mean diameter to centimetres
\( d = D_{\mu m} \times 10^{-4} \)
Number of particles in the sample
\( n = \frac{M_s}{m_p} \)
Theoretical random variance of the mass fraction
\[ \sigma_r^{2} = \frac{P(1-P)}{n} \]
Standard deviation of the mass fraction
\[ \sigma_r = \sqrt{\sigma_r^{2}} \]
Applicability limits for the binomial approximation
Parameter
Lower limit
Upper limit
Remarks
Mass fraction P
0.05
0.95
Binomial variance formula becomes inaccurate outside this range
Number of particles n
10,000
—
Finite-population correction may be significant below this value
Theoretical random variance is the irreducible scatter expected from a perfectly stable process when only common-cause variation is present. It is the baseline noise level predicted by the underlying statistical distribution (e.g., Poisson for counts, chi-square for variances). Knowing this value lets you:
Quantify how much of your observed variation is truly unavoidable
Set realistic control limits without over-reacting to random fluctuations
Judge whether an apparent drift is a special cause or just noise
For a normal model the theoretical variance σ² is estimated from the within-subgroup dispersion. Typical steps:
Collect k subgroups of size n (n = 4–6 is common)
Compute the average range \(\overline{R}\) or the pooled standard deviation \(s_p\)
σ² = (\(\overline{R}\) / d₂)² where d₂ is a control-chart constant for the chosen n, or σ² = \(s_p^2\)
Verify stability with an s-chart; if stable, this σ² is your theoretical random variance
A variance inflation above the theoretical value signals special-cause variation. Investigate systematically:
Check measurement system repeatability & reproducibility first
Look for shifts in raw-material lots, operators, or environmental conditions
Run a components-of-variance study to isolate batch-to-batch vs within-batch sources
Apply a variance test (Levene or Bartlett) across suspected factors to confirm findings
The theoretical value itself is fixed for a given process configuration, but your estimate can shift if:
Process improvements genuinely reduce common-cause variation (e.g., tighter temperature control)
Raw-material specifications tighten, altering the underlying distribution
After any such change, recollect data and recompute σ² to update control limits
Worked Example – Estimating Random Variance in a Micro-Bead Classification Line
A pharmaceutical plant screens 200 µm polymer beads to ensure each dose contains a tightly-controlled number of particles. Because the feed is a dilute suspension, the exact count in every 20 mL aliquot varies. Quality engineers need the theoretical standard deviation of the number of beads, \( \sigma \), so they can set inspection limits.
Knowns
Mean number of beads per aliquot, \( n \) = 994,718.285 (–)
Volume of one bead, \( V_p \) = 4.189 × 10⁻⁹ L
Mass of one bead, \( m_p \) = 5.027 × 10⁻⁶ g
Probability that any bead is counted, \( P \) = 0.250 (–)
Variance factor for random sampling, \( \sigma^2 \) = 1.885 × 10⁻⁷ (–)
Step-by-Step Calculation
Model the counting process as a binomial distribution with parameters \( n \) trials and success probability \( P \).
Theoretical variance of the count is
\[
\sigma^2 = n\,P\,(1-P)
\]
