# Limitation of Physical Measurements

## Key Terms

Along with the above, you need to know these definitions and be comfortable with using them in the right circumstance.

**Precision:** The precision of a measurement is the degree of exactness (sometimes the number of significant figures) to which the measurement of a quantity or value can be obtained and reproduced consistently.

• If a reading is constant when repeated, the precision of the measurement will be the precision of the instrument.

• If a reading fluctuates the precision of the measurement (its mean value) is given by half the maximum range of the readings.

**Repeatability:**__ __A measurement is repeatable if the original experimenter repeats the investigation using same method and equipment and obtains the same results.

N.B. “*the same*” results implies identical, but in reality “*the same*” means that random error will still be present in the results.

**Resolution:**__ __The smallest change in the quantity being measured (input) of a measuring instrument that gives a perceptible change in the reading. e.g. a typical mercury thermometer will have a resolution of 1°C, but a typical digital thermometer will have a resolution of 0.1°C.

**Accuracy:** Accuracy is a measure of confidence in an accurate measurement, often expressed as an upper and lower limit of the measurement based on the uncertainty in the measurement (eg g = 9.8 ± 0.3 m s-2)

## Random and Systematic Errors

You are required to know the difference between random and systematic errors.

A random error occurs when there is no pattern or bias. Readings with random errors vary in an unpredictable manner with no discernable pattern or trend. The effect of random variations in measurements of a quantity is reduced by taking more readings and finding a mean value.

A systematic error occurs in measurements where the errors show a pattern or a bias or a trend. Systematic errors can result from an instrument calibration error (eg zero errors), from incorrect use or reading of instruments (eg parallax errors) or be caused by another factor changing the quantity in an unknown or unrecognised manner.

In essence, random errors are random and systematic errors follow a pattern.

## Uncertainties

There are a few different types of uncertainty: absolute, fractional and percentage. Let’s take a look at each one.

Firstly, an uncertainty of a measurement is an expression of the spread of values which are likely to include the accepted value. For these physics specifications, the uncertainty in a measurement is taken as half the range from the lowest to the highest value obtained.

The uncertainty in a measurement is expressed as a ± value attached to the mean value.

For example, if you measure a length of a piece of string. You take two measurements and in the first you get 25cm and the second you get 24cm. To find the uncertainty, using what we said above we do (25+24)/2 = 24.5cm. So our ± value is 25-24.5 and 24-24.5. In both cases we get a result of 0.5cm. Therefore you measurements are 25 cm ± 0.5cm and 24 cm ± 0.5cm.

The absolute uncertainty is the ±0.5cm.

The fractional uncertainty is the uncertainty by the value that you measure. So the fractional uncertainty for our first measurement in the example above is 0.5/25 = 0.02. You would be correct in thinking that the larger the measured value, the smaller the fractional uncertainty. This is true and is one way we decrease the uncertainty in an experiment! We take a reading over a larger measurement.

The percentage uncertainty is the fractional uncertainty x100%. So in the above example it would be 2%.

The percentage uncertainty is useful when we need to combine uncertainties.

If we are adding or subtracting measurements together - we add the absolute uncertainties.

If we are multiplying or dividing measurements together - we add the percentage uncertainties.

If we are raising a measurement to a power - we multiply the percentage uncertainty.

## Uncertainty on a graph

Look at the graph and information below. Place close attention to the fact that there is a minimum line and a maximum line of best fit.

- A car traveled 600. m ± 12 m in 32 ± 3 s. What was the speed of the car? (Include uncertainties)
- Your answer should include: 19 / 2

Explanation:*19m/s ± 2m/s* - Five people measure the mass of an object. The results are 0.56 g, 0.58 g, 0.58 g, 0.55 g, 0.59 g. How would you report the measured value for the object’s mass?
- Your answer should include: 0.57g / 0.02g

Explanation:*0.57g ± 0.02g* - The radius of a circle is measured to be 3.5 cm ± 0.2 cm. What is the area of the circle with its uncertainty?
- Your answer should include: 38cm2 / 4cm2

Explanation:*38 cm2 ± 4 cm2*