VKC Notes · Analytical chemistry · Geo-Scientist Paper III

17.5 Errors in Chemical Analysis: Accuracy, Precision, Standard Deviation, Confidence Limits, Q-test, t-test and F-test

★★★ 2021–2025: 11 Mains parts, 75 marks, all in Geo-Scientist Paper III. Every year Q1(a) is a five-mark statistics numerical, and a 10–15 mark t- or F-test comes in Q5. The arithmetic is simple and the marks are certain.

Syllabus line. Geo-Scientist Paper III, A1 Errors in quantitative analysis: “Accuracy and precision, sensitivity, specific standard deviation in analysis, classification of errors and their minimization, significant figures, criteria for rejection of data, Q-test, t-test and F-test, control chart, sampling methods, sampling errors, standard reference materials, statistical data treatment.”

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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes

17.5.1 Accuracy, precision and types of error ★★★

Accuracy: closeness of a result (or the mean) to the true or accepted value; expressed as error. Precision: closeness of replicate results to one another; expressed as s, RSD, variance, range or confidence interval. Good precision does not guarantee accuracy Geo 2022 Q1(a) 5m.SN

(17.5.1)

A balance shows 9.5 g for a 10.0 g mass: E = −0.5 g, E_r = −5.0 %; keep the sign (the reading is low) Geo 2024 Q1(a) 5m.SN
Determinate (systematic) errors have a cause, one direction and can be corrected: instrumental (uncalibrated balance, burette), method (co-precipitation, indicator error, a side reaction), personal (colour judgement, parallax). Constant errors are independent of sample size; proportional ones grow with it, so varying the sample mass tells them apart. Indeterminate (random) errors have no single cause, both signs, follow the normal curve, and are reduced only by replication.SN
Minimising determinate errors: calibrate apparatus, run a blank, analyse a standard reference material (certified composition), use an independent method, vary the sample size, use standard addition.SN

Fig. 17.5.1 Accuracy and precision as shots at a target.

Draw it in 60 s: four bull’s-eyes: tight cluster at centre; tight cluster off-centre; loose cluster round centre; loose off-centre; label each.

Solved PYQ Geo 2021 Q5(a) 10m

A substance has 49.10 ± 0.02 % of A. Analyst 1: 49.01, 49.25, 49.08, 49.14; analyst 2: 49.40, 49.44, 49.42, 49.42. Compare accuracy and precision; name the errors.

Means 49.12 and 49.42 %: errors +0.02 and +0.32 %. s = 0.102 % (RSD 0.21 %) and 0.016 % (RSD 0.033 %).
Analyst 1: accurate, less precise (random errors, both signs, cancel in the mean). Analyst 2: precise, not accurate (a constant one-way determinate error; find it with a blank or standard).SN
Check with Eq. 17.5.6: t = 0.02 × √4/0.102 = 0.39 < 3.18 (ν = 3, 95 %) — no bias; analyst 2: t = 0.32 × 2/0.0163 = 39 ≫ 3.18 — real bias.

Fig. 17.5.2 The two analysts of Geo 2021 Q5(a) on one scale; shaded band = true value ± 0.02.

Draw it in 60 s: a horizontal % scale; a dashed line at 49.10; four scattered dots round it (analyst 1); four tight dots at 49.42 (analyst 2).

17.5.2 Significant figures and propagation of error ★★

Significant figures = all certain digits + the first uncertain one. Leading zeros never count (0.0042: two); trailing zeros after a decimal point count (2.040: four); in 1200 they are ambiguous, so write 1.20 × 10³. Exact numbers (counted, defined) have unlimited figures.SN
Addition/subtraction: keep the fewest decimal places; multiplication/division: the fewest significant figures (better: the result’s relative uncertainty ≈ the largest relative uncertainty); log x: as many decimal places in the mantissa as figures in x (pH 4.35 → [H⁺] = 4.5 × 10⁻⁵). Round only at the end; a final 5 rounds to the even digit.SN

(17.5.2)

17.5.3 Mean, standard deviation and the normal curve ★★★

(17.5.3)

N − 1 = degrees of freedom ν: one is used up in calculating x̄. Variance = s². RSD in per cent = coefficient of variation; in ppt multiply by 1000. Median = middle value of the ordered set (mean of the two middle ones for even N); it is not pulled by an outlier. Range w = highest − lowest Geo 2023 Q1(a) 5m.SN
Fe in ore, 10.5, 10.6, 10.3, 10.6, 10.2, 10.4 mg: x̄ = 10.433 mg, Σ(Δx)² = 0.1333, s = √(0.1333/5) = 0.163 mg, RSD = 1.57 % (15.7 ppt). Dividing by N instead of N − 1 gives 1.43 % (wrong) Geo 2023 Q1(a) 5m.SN
Random errors follow the normal (Gaussian) curve; for a large set (σ known): 68.3 % within ±1σ, 95.4 % within ±2σ, 99.7 % within ±3σ. The mean of N results scatters less, by √N.SN

Fig. 17.5.3 Normal error curve (computed) with the ±σ, ±2σ, ±3σ areas.

Draw it in 60 s: a symmetric bell; mark μ, ±σ, ±2σ, ±3σ below; write 68, 95, 99.7 % on three nested double arrows.

17.5.4 Confidence limits ★★★

(17.5.4)

The confidence interval is the range round the mean within which the true value lies with a stated probability (absent determinate error). Higher confidence → larger t → wider interval; more results → smaller t and smaller √N term → narrower interval.SN

Solved PYQ Geo 2024 Q5(a) 5m

Seven results give x̄ = 14.74 (± 0.38). Confidence interval at 99.0 %? [t₉₉: 4.604 (4), 4.032 (5), 3.707 (6), 3.499 (7)]

Read ± 0.38 as s; ν = N − 1 = 6, so t = 3.707 (the 3.499 offered is for ν = 7: the trap).SN
s/√N = 0.38/2.646 = 0.1436; ts/√N = 0.53; μ = 14.74 ± 0.53, i.e. 14.21 to 15.27 at 99 %.

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