Chemical measurements are inherently subject to variability, and statistics provides the language and tools to describe, interpret, and draw conclusions from that variability. Descriptive statistics summarize data sets using measures of central tendency — the mean (average) and median (middle value) — and measures of dispersion such as the standard deviation (s) and variance (s²). The normal (Gaussian) distribution describes many natural sources of random error; approximately 68% of measurements fall within ±1s of the mean, 95% within ±2s, and 99.7% within ±3s.
Confidence intervals express the range within which the true population mean is expected to lie at a given probability level (typically 95%). The interval is calculated as x̄ ± t · s / √n, where t is the Student’s t-value for the desired confidence and degrees of freedom. Hypothesis testing uses the t-test to compare a sample mean with a reference value (one-sample t-test) or to compare two sample means (two-sample and paired t-tests). The F-test compares two variances to determine whether their difference is statistically significant.
Analysis of variance (ANOVA) extends the t-test to compare three or more group means simultaneously. One-way ANOVA partitions the total variance into between-group and within-group components. The F-ratio (between-group variance divided by within-group variance) tests the null hypothesis that all group means are equal. Post-hoc tests such as Tukey’s HSD identify which specific pairs differ significantly.
Outlier detection is critical because a single aberrant value can distort statistical conclusions. Grubbs’ test identifies one outlier at a time by comparing the maximum deviation from the mean against a critical Z-value. Dixon’s Q-test evaluates whether the smallest or largest value in a small data set (n ≤ 30) is discordant. Suspected outliers should never be discarded arbitrarily — they require documented justification and should only be removed if a physical or procedural cause is confirmed.
Calibration curves relate instrument response y to analyte concentration x through linear regression based on the least-squares criterion: minimizing Σ(yᵢ − ŷᵢ)². The regression yields slope m, intercept b, and the correlation coefficient r². Unknown concentrations are predicted by interpolating their response on the regression line. The limit of detection (LOD) is the smallest concentration distinguishable from the blank, typically calculated as 3.3 · σ/S, where σ is the standard deviation of the blank and S is the slope. The limit of quantification (LOQ) is set at 10 · σ/S, representing the lowest reliable quantitative measurement.
