Measurement Uncertainty in Weighing and Dilution

Before you start

You need partial derivatives (or the patience to compute them numerically), the stock arithmetic of the formula as a linear system, and the bench perspective of batch errors. The blog's equipment list covers choosing a balance.

From readability to standard uncertainty

A reading displayed to 0.01 g is known only to within ±0.005 g from rounding alone. With no reason to prefer any value in that interval, the GUM treats it as a rectangular distribution of half-width a, whose standard deviation is u = a/√3 = 0.00289 g. That is a Type B evaluation, from knowledge of the instrument. A Type A evaluation uses repeated measurement: weigh the same check weight ten times and take the standard deviation. Real balances also have repeatability, linearity and calibration terms in their specification; include them when you have them. This lesson uses resolution alone, which gives a lower bound.

A weighing by difference uses two readings (tare and gross), so u_w = √2 · 0.00289 = 0.00408 g.

Propagation through a dilution

A dilution's concentration is c = m₁/(m₁ + m₂), material over total. Its sensitivity coefficients are

∂c/∂m₁ = m₂/(m₁+m₂)² and ∂c/∂m₂ = −m₁/(m₁+m₂)².

For 1.00 g of material and 9.00 g of solvent: c = 0.100, the coefficients are 0.09 and −0.01 per gram, and

u(c) = √[(0.09 × 0.00408)² + (0.01 × 0.00408)²] = 0.00037, a relative uncertainty of 0.37%.

Propagation into a formula line

Now weigh 0.05 g of that 10% dilution into a formula. The weighing's relative uncertainty is 0.00408/0.05 = 8.16%. The pure material delivered is a = m·c, and for a product, relative uncertainties combine in quadrature:

u_rel(a) = √(8.16%² + 0.37%²) = 8.17%

The dilution hardly matters; the small weighing is everything. Weigh 0.5 g of a 1% dilution instead (same pure amount) and the weighing contributes 0.82%. This is the quantitative argument for stock solutions and for the practical minimum weight: not a rule of thumb, but a factor of ten in uncertainty.

Report the result with an expanded uncertainty: 0.050 g ± 0.008 g (k = 2). With k = 2 and an approximately normal distribution, the interval covers about 95% of plausible values; with a rectangular-dominated budget, say so.

Where it stops

• Resolution is not accuracy. A balance that reads to 0.001 g may be repeatable only to 0.002 g; use its specification or your own check-weight data.
• Not every error is random. Evaporation while weighing, static on plastic, draughts and drops left in a pipette are biases, which the budget does not capture unless you measure them.
• Correlation. Lines weighed on one balance with one calibration error share a bias; independent-quadrature combination then understates the uncertainty of their ratio.

In professional practice

Quality laboratories write uncertainty budgets for release tests; compounding operations set weighing tolerances per line. Knowing where uncertainty comes from tells you which lines need a finer balance, which need a premix, and which are safely weighed directly.

Common mistakes

• Adding uncertainties linearly. Independent ones combine in quadrature.
• Using readability as the whole uncertainty. It is the smallest part of a real budget.
• Quoting an expanded uncertainty without k. The interval means nothing without its coverage factor.

Key terms

Standard uncertainty

The uncertainty of a result expressed as a standard deviation. For a reading known only to lie within ±a with equal likelihood, it is a/√3.

Sensitivity coefficient

The partial derivative of a result with respect to one input: how much the result changes per unit change in that input.

Expanded uncertainty

The standard uncertainty multiplied by a coverage factor k, commonly 2, giving an interval expected to contain a large fraction of plausible values, about 95% when the distribution is roughly normal.