Chemistry Labs

Analytical chemistry

Errors and statistical treatment of data

Separate random dispersion from systematic bias and report a defensible uncertainty with every result.

1 Lower secondary2 Upper secondary3 Undergraduate4 Advanced5 Research

IntuitionIntuition: the measurement idea

A number alone is not a measurement result. Replicates show random variation, while a stable offset shows bias; statistics turns both into a defensible uncertainty.

The normal distribution of repeated results: move the mean off the true value to bias the method, widen the spread to lose precision. The dashed curve is the mean of 9 measurements — the scatter narrows by a factor of 3.

SchoolSchool level: signal and result

Statistics describes measurement variability through the mean, the sample standard deviation, confidence intervals and tests for systematic difference.

Definition:

Separate random dispersion from systematic bias and report a defensible uncertainty with every result.

First identify the measurand, choose a signal that responds to it, and compare the sample with a calibrated standard or a validated model.

xˉ=∑ixin,s=∑i(xi−xˉ)2n−1\bar{x}=\dfrac{\sum_i x_i}{n},\qquad s=\sqrt{\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}}

Example: Worked analytical example

Calculate the analyte result from the stated measurement and method relation.

Solution

Replicates 10.02, 9.98, 10.01, 10.04 and 9.95 mg/L give mean 10.00 mg/L and sample s ≈ 0.034 mg/L.

Method checkpoints
StagePurpose
PrepareControl matrix and contamination
MeasureAcquire a calibrated response
ValidateCheck recovery and uncertainty

UndergraduateUniversity: quantitative method

For a single quantity, propagate uncertainty; for many observations, model random error separately from systematic effects. Report a confidence interval or standard uncertainty with the result.

Calibration, selectivity, sample preparation and uncertainty belong to the method itself, not to afterthoughts. Report units, conditions and the calibration range.

xˉ=∑ixin,s=∑i(xi−xˉ)2n−1\bar{x}=\dfrac{\sum_i x_i}{n},\qquad s=\sqrt{\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}}

Which statement correctly distinguishes random error from systematic error?

What does the sample standard deviation ss quantify?

References