# Quantitative analysis of uncertainty in measurement

### Significant figures

Non-zero digits in data are always considered significant. Leading zeros are never significant whereas following zeros and zeros between non-zero digits are always significant. For example, 075.0210 contains six significant figures with the zero at the beginning not considered significant. A whole number may be a counting number or a measurement and determination of significant figures varies in the literature. For the purpose of the
*VCE Physics Study Design*, whole numbers will have the same significant figures as number of digits, for example 400 has three significant figures while 400.0 has four.

Using a significant figures approach, one can infer the claimed accuracy of a value. For example, 400 is closer to 400 than 399 or 401. Similarly 0.0675 is closer to 0.0675 than 0.0674 or 0.0676.

Columns of data in tables should have the same number of decimal places, for example, measurements of lengths in centimetres or time intervals in seconds may yield the following data: 5.6, 9.2, 11.2 and 14.5. Significant figure rules should then be applied in subsequent data analysis.

Calculations in physics often involve numbers having different numbers of significant figures. In mathematical operations involving:

- addition and subtraction, the student should retain as many digits to the right of the decimal as in the number with the fewest significant digits to the right of the decimal, for example: 386.38 + 793.354 - 0.000397 = 1179.73
- multiplication and division, the student should retain as many significant digits as in the number with the fewest significant digits, for example: 326.95 x 10.2 ÷ 20.322 = 164.

Intermediate results in calculations should retain at least one significant figure more than such analysis suggests until the final result is ascertained.

### Determining uncertainty in measured data

Whenever a measurement is made, there will always be uncertainty in the result obtained. Sources of variation in the data generated include contributions from both systematic and random effects. The uncertainty of a measurement can be expressed as the interval within which the true value can be expected to lie, with a given level of confidence or probability, for example, ‘the temperature is 20 °C ± 2 °C,’. This is an uncertainty of 2 °C.

The uncertainty may sometimes be estimated by understanding the instruments used (for example, the uncertainty in reading a scale may be estimated as ± half the smallest scale division or the instrument specifications may state a nominal figure such as 2%) and considering the effect that any outside disturbances may have (for example, the temperature sensor is exposed to a flow of cool air from a nearby air conditioner).

When determining the average and uncertainty of a set of readings, the average is the simple mean (possibly with outliers ignored) while the uncertainty should be an estimate of the spread of readings.

Where several readings are averaged, the average should have the same number of decimal places as the uncertainty. For example, if the rebound heights of a basketball are measured to the nearest centimetre and yield the set of results: 60 ± 0.5, 62 ± 0.5, 59 ± 0.5, 60 ± 0.5, 61 ± 0.5, then the average rebound height is 60.4 cm with a maximum of 62 and a minimum of 59. The larger difference of these two values from the mean is 62 - 60 = 2 cm, so the reading now becomes 60.4 ± 2. Since the average has more decimal places than the uncertainty, the number recorded should be 60 ± 2 cm.

### Propagation of uncertainty

There are various ways to represent uncertainty. For VCE Physics, students should represent uncertainties as absolute uncertainties, as in the preceding example of rebound height with h = 60 cm and Δh = 2 cm. Proportional uncertainties are sometimes used, expressed as a percentage, for example, Δh/h = 2 cm/ 60 cm = 0.033 or 3% (to 1 significant figure). Tables of results usually include absolute uncertainties.

When adding or subtracting quantities, these absolute uncertainties are added. Hence the difference between 62 ± 2 cm and 52 ± 2 cm is 10 ± 4 cm. When multiplying or dividing quantities, proportional uncertainties are added. This is more advanced and beyond the expectations of VCE Physics.

For any other mathematical treatment of variables, students may generally substitute the lowest and the highest data points to determine the range. An example of the uncertainty in the gradient of a linear trend line could be found by comparing the gradients of the steepest and least steep trend lines that could reasonably be fitted to the relevant data.