## Experimental Error and Uncertainty

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Study Text: "Essential Mathematics and Statistics for Science", 2nd ed, G Currell and A A Dowman (Wiley-Blackwell)

### QVA (questions and video answers) Tutorials

Experimental errors and uncertainties Study Text: Sections 1.2 (p2), 8.2 (p217), 8.3 (p224)
Experimental uncertainty and the normal distribution Study Text: Section 8.1 (p212)
Uncertainty and confidence interval of replicate measurements Study Text: Section 8.2 (p217)

### Introduction

An experimental result is a ‘best estimate’ of the true value being measured, and
• Error = True value Experimental result
The true value, and hence the actual error, is never usually known.

The uncertainty in an experimental result (often just called the experimental error) is an estimate of the unknown possible error.

In any experimental measurement there are two main types of possible errors:

Random errors – errors that change randomly if the measurement is repeated under the same conditions. The magnitude of this possible error is described by the precision of the measurement.

Systematic errors – errors that remain constant if the measurement were to be repeated under the same conditions. The magnitude of this possible error is described by the bias in the measurement. Many chemists use the term accuracy to describe bias separately from precision.

Good experimental design is used to reduce the bias in the experiment, possibly by using standards for comparison and by making measurements under different conditions to convert systematic errors into random errors.

Replicate measurements (repeated measurements under the same conditions) and their analysis by statistical methods are used to quantify and counteract the effect of random errors.

See Study Text: Section 1.2

### Describing uncertainties/errors in a data value

Standard deviation uncertainty is used to describe the uncertainty in repeated measurements giving an estimate of the variation in the individual measurments. The variations in many experimental measurements follow a normal distribution of values. It is expected that 68% of such replicated measurements would fall within the range of plus or minus one standard deviation,
e.g. 3.72 ±0.06(sd) implies that repeated measurements would follow a normal distribution with 68% of values falling between 3.66 and 3.78.

Standard uncertainty, or standard error of the mean, is the standard deviation uncertainty of the calculated average value of a set of replicate measurements. It allows for the fact that repeating a measurement n times will reduce the random uncertainty in the average of those values by a factor of √n.
e.g. the standard uncertainty in the average of 5 repeated measurements, each with a standard deviation uncertainty of 0.06 will be 0.06/√5 ≈ 0.027.

Limiting uncertainty (sometimes called tolerance) gives the maximum range within which the true value is expected to lie. Limiting values often include possible systematic errors.
e.g. A 10 cm3 class A graduated pipette has a volume of 10.00 ±0.03 cm3 which implies that the true value will lie somewhere between 9.97 cm3 and 10.03 cm3.

Non-symmetrical uncertainty is described by giving both the upper and lower possible errors,
e.g. 0.36 (+0.5,-0.2)

Error bars are used on a graph to represent the above uncertainties in data points. It is essential that the type of representation is clearly shown on the graph,
e.g. graphs sometimes have error bars to represent the uncertainty using two standard deviations.

Confidence interval is the range of values within which it is possible to state, with a specific confidence (typically 95%), that the true value will lie. Note that confidence interval is not used to describe the possible variation of the experimental measurements (use standard deviation for this), but is used with the final result to express a confidence range for the true value being measured.

See Study Text: Sections 8.2 and 8.3