Measurements and Uncertainties

Uncertainties and ErrorsResolution This is the smallest change in a quantity that can be measured by an instrument.

E.g. What is the resolution ofa. A 30cm rulerb. An analogue wristwatchc. A thermometer

We prefer high resolution instruments:- A micrometer screw gauge measures to a

resolution of 1/100 mm.- A vernier calliper measures to a resolution of

1/10mm.

Error and uncertaintyAn error that is repeated in all readings taken is called a systematic error. Poor instrument calibration leading to a ‘zero error’ is an example of a systematic error.

Random errors occur often during repeated experiments, causing the reading to vary above and below the true value. If values have to be estimated, there is often a random error.

Q. Which of these two types of errors can be reduced by taking an average of repeated readings?

Precision and AccuracyIf repeated measurements always give very similar results, they are precise – they have small random error and low uncertainty. However, this does not mean they are accurate.

If repeated measurements are fairly near to the true value, they are accurate – the systematic error is small. However, they may not be precise.

Accurate but imprecise

Precise but inaccurate

Ruler scale

True value

To reduce error:- Always repeat readings if possible.- Take readings at eye level- Use take at least 6 sets of readings - Check for a zero error before starting

Determining Errors and UncertaintiesOn a single reading, the (+ or -) uncertainty could be considered equal to either the instrument resolution or the human error, depending which is largest.

Q. Measure the length of your pen using a ruler. What is a) the uncertainty b) the % uncertainty

If an average reading is calculated from a few readings, the uncertainty equals ½ x the range:

E.g. In an experiment a toy car was rolled repeatedly down a ramp in these times…

8.12s, 8.16s, 8.08s, 8.14s, 8.10s

Average time = 8.12s, however we are not certain that this is the true value.

Range = 8.16 – 8.08 = 0.08 so uncertainty = +/-0.04

Time = 8.12 +/- 0.04 s

(Note: With multiple oscillations, when dividing to find T you also divide the absolute uncertainty)

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