Mean, Standard Deviation & Variance — Maths for Analytical Chemistry

Summarising a set of repeats

Mean for the centre, standard deviation for the spread

Repeat a measurement and you inevitably get scatter. Two numbers summarise it compactly: the mean (the best central estimate of the true value) and the standard deviation (the typical spread of individual readings around that centre). Together they let you judge both where your results sit and how repeatable they are.

\overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} x _{i}}{n}, s = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}{n - 1}

The denominator $n - 1$ (Bessel’s correction) rather than $n$ reflects that we have estimated the mean from the same data; it gives an unbiased estimate of the population standard deviation. Squaring $s$ gives the variance $s^{2}$ , which is additive for independent measurements.

The relative standard deviation (RSD, also called the coefficient of variation, %CV) expresses the spread as a percentage of the mean: $RSD = (s / \overset{x}{ˉ}) \times 100%$ . An analytical method with RSD below 1 % is considered highly precise; above 5 % raises questions in most contexts.

Key statistics at a glance

Mean (average)

\overset{x}{ˉ} = \frac{\sum x _{i}}{n}

— the centre of gravity of the data.

Median

Middle value when sorted. Robust to outliers; useful when one rogue reading may inflate the mean.

Sample variance

s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

— squared units; additive.

Standard deviation

s = s^{2}

— same units as

x

; the most common measure of precision.

%RSD / %CV

RSD = \frac{s}{x ˉ} \times 100%

— dimensionless; compare across methods.

Standard error of mean

s_{\overset{x}{ˉ}} = s / n

— uncertainty in

\overset{x}{ˉ}

itself; shrinks as

n

grows.

The standard error of the mean (SEM) answers a different question from the standard deviation: while $s$ tells you how much individual readings scatter, the SEM tells you how precisely you know the mean itself:

SEM = s_{\overset{x}{ˉ}} = \frac{s}{n}

Doubling the number of replicates halves the SEM (improves it by $2$ ), not doubles it. When two independent sets of measurements are pooled — for example to compare two labs — the pooled standard deviation weights each by its degrees of freedom:

s_{p} = \frac{( n _{1} - 1 ) s _{1}^{2} + ( n _{2} - 1 ) s _{2}^{2}}{n _{1} + n _{2} - 2}

This is the correct precision estimate when the F-test confirms that both datasets share the same population variance. The pooled s then enters the two-sample t-test.

Common pitfalls

Using n instead of n−1 for a sample. Dividing by n underestimates the true population spread. Always use $n - 1$ (Bessel's correction) unless you have the entire population.
Confusing s with the SEM. Report $s$ when describing how much individual measurements scatter; report $s / n$ when quoting the uncertainty in the mean. Swapping them makes a result appear 4× better (for n = 16) than it is.
%RSD is dimensionless, not a quantity with units. Writing "RSD = 1.2 ppm" conflates relative and absolute uncertainty. RSD is a pure percentage.
Comparing means without accounting for precision. Two means that differ by more than their combined uncertainties may still overlap when you use the correct t-test — always check before claiming a difference is significant.

Quality control

Every analytical result is quoted as a mean of replicates with its standard deviation. These feed confidence intervals, t-tests and F-tests that decide whether two methods agree. A small

s

signals high precision; closeness of

\overset{x}{ˉ}

to the certified value signals high accuracy — and the two are independent. Open the dataset lab to see both in action.

Dataset lab

Drag the six replicate titration volumes — the mean, standard deviation and %RSD update live. Watch the ±1σ band widen and narrow.

Drag the six replicate readings and watch the mean and spread respond

x110.02

x29.96

x310.08

x49.99

x510.05

x69.94

mean x̄10.007

standard deviation s0.054

relative s (%RSD)0.54%

\overset{x}{ˉ} = \frac{1}{n} \sum x_{i}, s = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

The shaded band is ±1 standard deviation around the mean. Spread the points out and s grows; cluster them and s shrinks — a direct picture of precision.

Worked example 1Mean and sample standard deviation of replicate titrations

Five titre volumes (in mL) from a back-titration are: 24.10, 24.15, 24.08, 24.13, 24.09. Calculate the mean and the sample standard deviation.

Worked example 2Population vs sample standard deviation — when does it matter?

The same five titration volumes are analysed by two students. Alice uses $n = 5$ in the denominator; Bob uses $n - 1 = 4$ . Whose result is correct for estimating the true spread, and by how much do they differ?

Worked examples

Precision, accuracy, RSD and the standard error — try each before revealing the solution.

Worked example 3Calculating %RSD for a colorimetric assay

A colorimetric glucose assay on the same sample gives four absorbance readings: 0.412, 0.408, 0.415, 0.411. Calculate the mean, standard deviation and %RSD.

Worked example 4Precision vs accuracy in a Cu²⁺ determination

Three analysts each make four ICP-OES measurements of a 10.00 mg L⁻¹ Cu²⁺ standard. Their means and standard deviations are:

Analyst A: $9.98 \pm 0.03 mg L^{- 1}$
Analyst B: $10.35 \pm 0.04 mg L^{- 1}$
Analyst C: $9.50 \pm 0.20 mg L^{- 1}$

Classify each result as (a) accurate and precise, (b) precise but inaccurate, (c) imprecise.

Worked example 5Standard error and reporting the final result

A student measures the melting point of naphthalene six times: 80.1, 80.3, 79.9, 80.2, 80.0, 80.1 °C. Report the result as mean ± standard error (95 % confidence, $t_{0.025, 5} = 2.571$ ).

Worked example 6Standard deviation vs standard error — the distinction matters

A flame-AAS method for zinc gives eight replicates (mg L⁻¹): 5.12, 5.09, 5.15, 5.11, 5.08, 5.14, 5.10, 5.13. Calculate (a) the sample standard deviation and (b) the standard error of the mean, and explain which to report with the result.

Worked example 7Pooled standard deviation from two analytical runs

Two analysts measure the same lead standard by ICP-OES:

Lab 1: $n_{1} = 5, \overset{x}{ˉ}_{1} = 24.8 μ g L^{- 1}, s_{1} = 0.45 μ g L^{- 1}$
Lab 2: $n_{2} = 6, \overset{x}{ˉ}_{2} = 25.1 μ g L^{- 1}, s_{2} = 0.38 μ g L^{- 1}$

Assuming equal population variances (F-test passed), calculate the pooled standard deviation $s_{p}$ and the standard error of each mean.

ChallengeChallenge — pooled precision + SEM for a method comparison

Two HPLC methods are validated for caffeine in coffee:

Method A (n = 4): 52.1, 51.8, 52.4, 51.9 mg per 100 mL
Method B (n = 5): 51.5, 52.0, 51.7, 51.9, 52.3 mg per 100 mL

(i) Calculate $\overset{x}{ˉ}$ , $s$ , and %RSD for each method. (ii) Calculate the pooled s and the SEM for both means. (iii) Determine which method is more precise. (iv) What is the combined 95 % CI for Method A? ( $t_{0.025, 3} = 3.182$ .)

Check yourself

Four quick questions on mean, standard deviation, degrees of freedom and precision vs accuracy.

Question 1 of 4 · Score 0

The mean of a set of measurements is:

Choose an answer.