2.2.1 Measurements and uncertainties

Okay so this is not exactly content in itself but is sort of like good information to know.

A true value is the value that would be obtained in an ideal measurement. A measurement error is the difference between a measured value and the true value for the quantity being measured. Mistakes are not counted as errors. If you make a mistake you can repeat an experiment without the mistake. However, an experiment will still contain errors no matter if you repeat it. In science, an error is the difference between the result you get and the correct result. They are usually caused by measuring devices even if they are used correctly. They can also be caused by the design of the experiment itself.

Random errors can happen when any measurement is being made. They are measurement errors in which measurements vary unpredictable. They cannot be corrected but we can reduce their effects by making more measurements and finding the mean value. Reasons for this include:

• factors that are not controlled in the experiment
• difficulty in deciding on the reading given by a measuring device.

Systematic errors are measurement errors in which the measurements differ from the true values by a consistent amount each time a measurement is made. Unlike random errors it is possible to correct systematic errors e.g by changing equipment. Reasons for this include:

• the way in which measurements are taken
• faulty measuring devices

Examples include poor contact between a thermometer and the object whose temperature is being measured, a faulty measuring device (e.g calibrated incorrectly/zero errors).

When we talk about obtaining measurements it is vital that we know the difference between precision and accuracy:

• precision regards how close repeated measurements are to each other (the closer together the more precise)
• accuracy regards how close a measurement result is to the true value (the closer the more accurate)

Uncertainties

Okay so because of random and systematic errors basically it is very hard to obtain the same value for a particular measurement. A mean value can be calculated by adding all the values and dividing it by the number of values. The range is the difference between the smallest and largest readings. The uncertainty in the measurement is an interval within which the true value can be expected to lie.

The absolute uncertainty in the mean value of a measurement can be approximated as half the range. It is expressed as a ± value. When you have a single measurement/repeat measurements are equal you approximate the absolute uncertainty to be equal to the resolution of the measuring instrument.

The percentage uncertainty can be calculated from its absolute uncertainty and mean values:

% uncertainty = 100 x absolute uncertainty/mean value

The final uncertainty depends on how quantities are combined:

• adding or subtracting quantities: add the absolute uncertainties for each value
• multiplying or dividing quantities: add the percentage uncertainties for each value
• raising to a power: increase the percentage uncertainty by the magnitude of the power

I'm sure you will have completed this in PAGs, but we can also use error bars/lines of best and worst fit. The absolute uncertainty in the gradient is the positive difference between the gradient of the line of best fit and the gradient of the line of worst fit. The percentage uncertainty can then be calculated using the following equation:

% uncertainty = 100 x absolute uncertainty/gradient of best fit line