Lesson 1 · 25 min

Errors, the mean and the Gaussian distribution

Two titrations differ by 0.08 mL — which is right? Random error spreads results into a bell curve, and averaging n of them tightens the mean by 1/√n — but does nothing for systematic error.

In this lesson

Distinguish random from systematic error and say which one repeated measurement can reduce
Compute the mean, standard deviation, relative standard deviation and standard error of the mean
Use the Gaussian distribution to estimate the fraction of results within μ ± σ and μ ± 2σ
Explain why the scatter of the mean shrinks as 1/√n and plan the number of replicates accordingly

You titrate the same hydrochloric acid twice with the same burette and read end points of 24.53 mL and 24.61 mL, 0.08 mL apart. Which one is right? Both, and neither: every measurement carries random error. The lesson on measurement separated accuracy from precision and taught you to write significant figures; this unit goes further — using statistics to say exactly how far a result can be trusted.

Predict first

Each titration has random error σ. If you average 4 titrations instead, by what factor does the random scatter of the mean shrink?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

To 1/4 To 1/2 It does not shrink To 1/16

Two kinds of error

Random error is sometimes positive, sometimes negative, and unpredictable in size: judging the last estimated digit, small temperature fluctuations, electronic noise in an instrument. It scatters replicate results and sets the precision; averaging many measurements reduces it.
Systematic error pushes every result the same way: a mis-calibrated burette, an indicator that changes colour away from the equivalence point, a standard solution whose concentration was written down wrongly. It shifts all results together and sets the accuracy; no number of replicates reveals it — only comparison with a reference material or an independent method does.

There is also a third kind, gross error (a misread scale, a spill, an arithmetic slip). It is not a matter for statistics: find the cause in your lab notes and repeat the measurement.

Mean, standard deviation and the Gaussian

If there is only random error, and it comes from the sum of many small independent causes, a large number of replicates falls on the bell-shaped Gaussian (normal) distribution: its mean μ is the centre and its standard deviation σ the width. About 68.3 % of results fall within μ ± σ, 95.4 % within μ ± 2σ and 99.7 % within μ ± 3σ. In the lab we only have a few results, so we estimate μ by the mean $\bar{x}$ and σ by the sample standard deviation $s$ .

y = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^2/2\sigma^2},\qquad s_{\bar{x}} = \frac{s}{\sqrt{n}}

Left: the Gaussian probability density. Right: the standard error of the mean — how much the mean itself scatters.

$\mu,\ \sigma$: mean and standard deviation of the population (true but unknown)
$\bar{x},\ s$: mean and sample standard deviation of n results (divisor n − 1)
$s_{\bar{x}}$: standard error of the mean
$s/\bar{x}$: relative standard deviation (RSD), often in %

Random error, systematic error and the standard error of the mean σ/√n

Replicate simulator: why measure more than once

Random error σ / mL · 0.06 Systematic error (bias) / mL · +0.10 Replicates per set n · 4

This set of 4 titration end points

Mean x̄: 25.138 mL
Standard deviation s: 0.053 mL
Standard error s/√n: 0.027 mL
Error of the mean x̄ − true: +0.138 mL

Single results (2000)
Means of n (1000 sets)

Of 2000 single results, 67.4 % fall within μ ± σ and 95.3 % within μ ± 2σ

Simulated (400 sets per n)
σ/√n

Spread of the mean against n (log scales)

No amount of averaging removes a systematic error: the mean converges on 25.00 + bias, not on the true value.

These titrations have a systematic error of +0.10 mL. Raise n to 25: the mean becomes ever steadier — at the wrong value. Then set the bias back to 0 and compare how the two kinds of error affect this one set and the long-run distributions.

Worked example

Statistics of five titrations

Five replicate titrations give end points of 24.53, 24.61, 24.48, 24.57 and 24.55 mL. Find the mean, standard deviation, relative standard deviation and standard error of the mean.

Concept check

Which of these finds and reduces systematic error?

Increasing the replicates from 3 to 10 Analysing a certified reference material and comparing Recording one more digit Dropping the result furthest from the mean

Try the calculation

Each titration has a standard deviation of about 0.05 mL. How many replicate titrations do you need, at least, for the standard error of the mean to be 0.02 mL or less?

Key takeaways

Random error sets precision and can be reduced by repetition; systematic error sets accuracy and can only be found with reference materials or independent methods.
In a Gaussian distribution about 68 % of results lie within μ ± σ and 95 % within μ ± 2σ.
The standard error of the mean is s/√n: doubling the precision takes four times as many measurements.
A small standard deviation does not make a result accurate.

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Data and references

D. C. Harris, Quantitative Chemical Analysis, 9th ed., W. H. Freeman (2016), chapters 3 and 4.
JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), BIPM.
J. N. Miller, J. C. Miller, Statistics and Chemometrics for Analytical Chemistry, 6th ed., Pearson (2010), chapters 1–2.