Statistics for analysts

Why it matters

Every analytical result is a number with uncertainty. When a lab replaces an old method with a faster one, changes an instrument, or compares its results with another lab, someone must answer: do the two methods give the same results, or is there a real difference? Answering that with an eyeball is how labs drift. Answering it with the right statistical test, and the right sample design, is a core skill for analytical chemists. This lesson builds on statistics-fundamentals and lab-data-quality.

Describing replicates

For n replicate results x1 ... xn:

• Mean (x-bar): the best estimate of the value.
• Standard deviation (s): the spread, with n - 1 in the denominator.
• Relative standard deviation (RSD, %): 100 x s / mean. Useful for comparing precision across concentrations.
• Confidence interval of the mean: mean +/- t x s / sqrt(n), where t comes from the t table for n - 1 degrees of freedom at the chosen confidence (95 % is typical).

Accuracy vs precision

• Precision is how close replicates are to each other (random error).
• Accuracy is how close the result is to the true value. Bias (systematic error) is the difference between the mean and a reference value, often measured with a certified reference material (CRM).

A method can be precise and biased at the same time. Statistical tests help you separate a real bias from random scatter.

Choosing the test

The paired design is the workhorse of method comparison. If you measure six different real samples with both methods, the samples have very different concentrations, so the between-sample spread would swamp any method difference in a two-sample test. Pairing removes that: you analyze the differences for each sample.

Worked example: a paired comparison

Six fictional samples (mg/L) by the reference method A and a new method B:

• Mean difference d-bar = 0.283 mg/L; s_d = 0.147 mg/L; n = 6.
• t = d-bar / (s_d / sqrt(n)) = 0.283 / 0.0601 = 4.71
• Critical t (two-tailed, 95 %, 5 degrees of freedom) = 2.571.

Because |t| = 4.71 > 2.571, the difference is statistically significant: method B reads about 0.28 mg/L high. The next question is practical: is 0.28 mg/L acceptable for the method's purpose? Statistics tells you a difference is real; the acceptance criteria (set before the study) tell you whether it matters.

Beyond the t-test

• When comparing methods across a wide concentration range, labs also use regression of B against A (slope near 1 and intercept near 0 means agreement) and Bland-Altman plots of difference vs mean. Ordinary least squares assumes A has no error; Deming or Passing-Bablok regression handle error in both methods.
• Outlier tests (Grubbs, Dixon's Q) exist, but an outlier is only rejected with a documented reason, never just because it is inconvenient (see lab-data-quality).
• In Python, scipy.stats.ttest_rel performs a paired t-test. The activity uses only the standard library so you see the formula.

Common mistakes

• Using a two-sample test on paired data, which hides a real bias under sample-to-sample variation.
• Confusing statistical significance with practical importance.
• Running many tests and reporting only the one with p < 0.05.
• Comparing methods on too few samples, or only at one concentration.

Quick recap

• Mean, s, RSD and confidence intervals describe replicates.
• Bias is systematic; precision is random scatter.
• Same samples measured by two methods: use a paired t-test on the differences.
• Decide acceptance criteria before the study.

Try it

Open the Do tab. A Python method-comparison script is written except for the paired t statistic.