The importance of error analysis The aim of

The importance of error analysis • The aim of error analysis is to quantify and record the errors associated with the inevitable spread in a set of measurements, and to identify how we may improve the experiment. • In the physical sciences experiments are often performed in order to determine the value of a quantity. However, there will always be an error associated with that value due to experimental uncertainties. • We can never be certain what the exact value is, but the errors give us a characteristic range in which we believe the correct value lies with a specified likelihood.

What is a measurement? • A measurement tells us about a property of something. It might tell us how heavy an object is, or how hot, or how long it is. • A measurement gives a number to that property. • Measurements are always made using an instrument of some kind. Rulers, stopwatches, weighing scales, and thermometers are all measuring instruments. • The result of a measurement is normally in two parts: a number and a unit of measurement, e. g. • ‘How long is it? . . . 2 metres. ’

What is not a measurement? There are some processes that might seem to be measurements, but are not. • comparing two pieces of string to see which is longer is not really a measurement. • Counting is not normally viewed as a measurement. • a test is not a measurement: tests normally lead to a ‘yes/no’ answer or a ‘pass/fail’ result.

An accurate vs. a precise measurement • A precise measurement is one where the spread of results is ‘small’, either relative to the average result or in absolute magnitude • An accurate measurement is one in which the results of the experiment are in agreement with the ‘accepted’ value

An accurate vs. a precise measurement Both are Precise measurements (relatively narrow histogram) Fig. 1. 2(a) the center of the histogram is close to the dashed line, hence we call this an accurate measurement.

An accurate vs. a precise measurement imprecise and accurate measurement imprecise and inaccurate measurement

Based on the discussion of precision and accuracy, we can produce the following taxonomy of errors, each of which is discussed in detail below: • random errors—these influence precision; • systematic errors—these influence the accuracy of a result; • mistakes—bad data points.

Random errors • Much of experimental physics is concerned with reducing random errors. • The signature of random errors in an experiment is that repeated measurements are scattered over a range • The smaller the random uncertainty, the smaller the scattered range of the data, and hence the more precise the measurement becomes. • The best estimate of the measured quantity is the mean of the distributed data, and as we have indicated, the error is associated with the distribution of values around this mean • The distribution that describes the spread of the data is defined by a statistical term known as the standard deviation

Systematic errors • Systematic errors cause the measured quantity to be shifted away from the accepted, or predicted, value. • Measurements in which this shift is small (relative to the error) are described as accurate. • Systematic errors are reduced by estimating their possible size by considering the apparatus being used and observational procedures

Mistakes • These are similar in nature to systematic errors, and can be difficult to detect. • A 'mistake' is usually accidental, you know it is wrong. • An 'error' is usually made due to the lack of knowledge and is more formal than 'mistake'. • Machines never make mistakes, but rather they make errors.

Mistakes (Examples) • Writing 2. 34 instead of 2. 43 in a lab book is a mistake and if not immediately corrected is very difficult to compensate for later. • In 1999 the failure of the NASA Mars Climate Orbiter was attributed to confusion about the value of forces: some computer codes used SI units, whereas others used imperial. • A Boeing 767 aircraft ran out of fuel mid-flight in 1983; a subsequent investigation indicated a misunderstanding between metric and imperial units of volume.

Uncertainty of measurement • Uncertainty of measurement is the doubt that exists about the result of any measurement. Y • you might think that well-made rulers, clocks and thermometers should be trustworthy, and give the right answers. But for every measurement - even the most careful - there is always a margin of doubt. • In everyday speech, this might be expressed as ‘give or take’. . . e. g. a stick might be two metres long ‘give or take a centimetre’.

Uncertainty of measurement How should we interpret an uncertainty in a measurement? • Reporting a quantity as the best estimate ± the error should be regarded as a statement of probability. • The scientists who performed the measurements and analysis are confident that the Avogadro constant is within the range 6. 022 141 49 × 1023 mol− 1 ≤ NA ≤ 6. 022 142 09 × 1023 mol− 1. • They cannot be certain that the Avogadro constant is within the limits quoted, but the measurements lead them to believe that there is a certain probability of its being so.

Uncertainty of measurement two numbers are really needed in order to quantify an uncertainty. Ø One is the width of the margin, or interval. Ø The other is a confidence level, and states how sure we are that the ‘true value’ is within that margin. Ex) We might say that the length of a certain stick measures 20 centimetres plus or minus 1 centimetre, at the 95 percent confidence level. This result could be written: 20 cm ± 1 cm, at a level of confidence of 95% The statement says that we are 95 percent sure that the stick is between 19 centimetres and 21 centimetres long.

Error versus uncertainty • Error is the difference between the measured value and the ‘true value’ of the thing being measured. • Uncertainty is a quantification of the doubt about the measurement result

The importance of uncertainty of measurement • to make good quality measurements and to understand the results. • There are other more particular reasons for thinking about measurement uncertainty: • calibration - where the uncertainty of measurement must be reported on the certificate • test - where the uncertainty of measurement is needed to determine a pass or fail • tolerance - where you need to know the uncertainty before you can decide whether the tolerance is met