## 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 cm^{3} 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 cm^{3} and 10.03 cm^{3}.

**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