1.2 Uncertainties and errors
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Uncertainties in measurement
No experimental quantity can be absolutely accurate when measured – it is always subject to some degree of uncertainty. We will look at the reasons for this in this section.
There are two types of error that contribute to our uncertainty about a reading – systematic and random.
Systematic errors
These types of errors are due to the system being used to make the measurement. This may be due to faulty apparatus. For example, a scale may be incorrectly calibrated either during manufacture of the equipment, or because it has changed over a period of time. Rulers warp and, as a result, the divisions are no longer symmetrical. A timer can run slowly if its quartz crystal becomes damaged (not
because the battery voltage has fallen – when the timer simply stops).
The zero setting on apparatus can drift, due to usage, so that it no longer reads zero when it should – this is called a zero error.
Often it is not possible to spot a systematic error and experimenters have to accept the reading on their instruments, or else spend significant effort in making sure that they are re-calibrated by checking the scale against a standard scale. Repeating a reading never removes the systematic error. The real problem with systematic errors is that it is only possible to check them by performing the same task with another apparatus. If the two sets give the same results, the likelihood is that they are both performing well; however, if there is disagreement in the results a third set may be needed to resolve any difference.
In general we deal with zero errors as well as we can and then move on with our experimentation. When systematic errors are small, a measurement is said to be accurate.
Uncertainty when using a 300 mm ruler may be quoted to ±0.5 mm or ±1 mm depending on your
view of how precisely you can gauge the reading. To be on the safe side you might wish to use the
larger uncertainty and then you will be sure that the reading lies within your bounds.You should make sure you observe the scale from directly above and at right angles to the plane of the ruler in order to avoid parallax errors.
In each of uncertainty is quoted to the same precision (number of decimal places) as the reading – it is essential to do this as the number of decimal places is always indicative of precision. When we write an energy value as being 8 J we are implying that it is (8 ± 1) J and if we write it as 8.0 J it implies a precision of ±0.1 J.
Random errors
Random errors can occur in any measurement, but crop up most frequently when the experimenter has to estimate the last significant figure when reading a scale. If an instrument is insensitive then it may be difficult to judge whether a reading would have changed in different circumstances. For a single reading the uncertainty could well be better than the smallest scale division available. But, since you are determining the maximum possible range of values, it is a sensible precaution to use this larger precision. Dealing with digital scales is a problem – the likelihood is that you have really no idea how precisely the scales are calibrated. Choosing the least significant digit on the scale may severely underestimate the uncertainty but, unless you know the manufacturer’s data regarding calibration, it is probably the best you can do.
When measuring a time manually it is inappropriate to use the precision of the timer as the uncertainty in a reading, since your reaction time is likely to be far greater than this. For example, if you timed twenty oscillations of a pendulum to take 16.27 s this should be recorded as being (16.3 ± 0.1) s. This is because your reaction time dominates the precision of the timer. If you know that your reaction time is greater than 0.1 s then you should quote that value instead of 0.1 s.
The best way of handling random errors is to take a series of repeat readings and find the average of each set of data. Half the range of the values will give a value that is a good approximation to the statistical value that more advanced error analysis provides. The range is the largest value minus the smallest value.
Readings with small random errors are said to be precise (this does not mean they are accurate, however).
Source: IB DP Physics Course Companion by David Homer and Michael Bowen-Jones.
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