Errors and uncertainty

Error and Uncertainty

All readings, data, results or other numerical quantities taken from the real world by direct measurement or otherwise are subject to uncertainty. This is a consequence of not being able to measure anything exactly. Uncertainty cannot be avoided but it can be reduced by using 'better' apparatus. The uncertainty on a measurement has to do with the precision or resolution of the measuring instrument. When results are analyzed it is important to consider the affects of uncertainty in subsequent calculations involving the measured quantities.

If you are unlucky (or careless) then your results will also be subject to errors. Errors are mistakes in the readings that, had the experiment been done differently, been avoided. It is perfectly possible to take a measurement accurately and erroneously! Unfortunately it is not always possible to know when you are making an error (otherwise you wouldn't make it!) and so good experimental technique has to able to guard against the affect of errors

Types of Error:

• Human Error: Errors introduced by basic incompetence, mistakes in using the apparatus etc. Reduced by repeating the experiment several times and comparing results to those of other similar experiments, by ensuring results seem reasonable
• Systematic Error: Error introduced by poor calibration or zero point setting of instruments such as meters - this may cause instrumentation to always 'under read' or 'over read' a value by a fixed amount. Reduced by plotting graphs, the relationships between two quantities often depends on the way in which they change rather than their absolute values. A systematic error would manifest itself as an intercept on the y-axis other than that expected. In the A Level course this is most commonly experienced with micrometers (that don't read zero when nothing is between the jaws) and electrical meters that may not rest at zero
• Equipment Error: Error introduced by the mis-functioning of equipment. The only real check is to see if the results seem reasonable and 'make sense' ... take time to stop and think about what the instruments are telling you ... does it seem okay?
• Parallax Error: Error introduced by reading scales from the wrong angle i.e. any angle other than at right angles! Some meters have mirrors to help avoid parallax error but the only real way to avoid parallax error is to be aware of it

Estimating uncertainty

Estimating the uncertainty on a reading is an art that develops with experience. There are two rules of thumb:

Firstly, take repeat readings. If there is a spread of readings then the uncertainty can be derived from the size of the spread of values. What you are doing in effect is seeing how repeatable the results are and this will give an order of magnitude idea of the uncertainty likely on any given reading. (See the section on dealing with averages below). For example, if three readings of time are 42s, 47s and 38s then the average is just over 42s with the other two readings being about 4s away from the average ... so use 42s ± 4s. The uncertainty is taken as 4s

Secondly, if the results are repeatable to the precision of the measuring apparatus then the uncertainty is taken as half of the smallest reading possible. For example, when measuring something with a ruler marked off in mm, the uncertainty is ± 0.5mm. When using a normal protractor the uncertainty on the angle is ± 0.5 degrees etc

Average values

If the experiment generates many repeat readings (as any really good experiment should) then there is a way to analyze the results and obtain a good value for the associated uncertainty:

1. Take an average of the results
2. Work out the deviation of each result from the average
3. Average the deviations (ignore any minus signs) - this is the uncertainty

For example:

Voltage (V)	Deviation from average
2.0	0.1
2.2	0.1
1.8	0.3
1.9	0.2
2.6	0.5
2.3	0.2
1.7	0.4
2.4	0.3
2.2	0.1
1.9	0.2
Av = 2.1	Av = 0.24

Thus we use V = 2.1 ± 0.2 Volts

Combining uncertainty

In many equations two or more values are combined mathematically so it is important to know what happens to the uncertainties. The uncertainty on a value can be expressed in two ways, either as an 'absolute' uncertainty or as a 'percentage' uncertainty. The absolute uncertainty is the actual numerical uncertainty, the percentage uncertainty is the absolute uncertainty as a fraction of the value itself. Consider our previous example:

Voltage = 2.1 ± 0.2
The quantity = 2.1 V
Absolute uncertainty = 0.2 V (it has units)
Percentage uncertainty = 0.2 / 2.1 = 0.095 = 9.5% (no units as its a ratio)

In all of the following examples we consider combing 2 values:

and

Addition: The uncertainty on the sum of the two values is the sum of the absolute uncertainties

Subtraction: The uncertainty on the difference of the two values is the sum of the absolute uncertainties

Multiplication: The uncertainty on the product of the two values is the sum of the percentage uncertainties

Division: The uncertainty on the division of the two values is the sum of the percentage uncertainties

There are two important points to note here:

1. If two values that are very similar are subtracted then the uncertainty becomes very large ... this can render the results of an experiment meaningless. For example, consider 4.0 ± 0.1 - 3.5 ± 0.1 = 0.5 ± 0.2. The percentage uncertainty on the individual values is about ± 2.5% whereas the percentage uncertainty on the result is ± 40%

2. Be very careful to convert percentage uncertainties back to absolute values after combining the various values. Only the absolute uncertainty has any real physical meaning

Error bars on graphs

Having taken measurements and calculated the associated uncertainties, it is often necessary to plot these values graphically. Uncertainties are represented as 'error bars' on graphs - although this is a misleading title, it would be better to call them 'uncertainty bars'. Error bars are simply a line used to represent the possible range of values, the line or curve drawn through the points can pass through any part of the error bar. The graph below shows how the error bars are drawn. The values on the x-axis are shown with a constant absolute uncertainty, the values on the y-axis are shown with a percentage uncertainty (and so the error bars gets bigger)

What to plot?

The art of analysing experimental data is knowing what to plot, in most experiments it is not enough to simply plot the recorded values directly, instead some more appropriate graph is needed.

It is always the case that a linear graph gives the most useful analysis and so the data is manipulated to give the required linear relationship

The mathematical relationship for a linear relationship is y = mx + c

In a Physical situation each of these quantities has physical meaning and appropriate units - this includes the gradient and the y-intercept. Don't forget to include units when calculating values from a 'Physics' graph!

Formula	plot y-axis	plot x-axis	Notes
y = mx + c	y	x	Gradient = m, y-intercept = c
y = kx2	y	x2	Gradient = k
y = k / x	y	1 / x	Gradient = k
y = k / x2	y	1 / x2	Gradient = k
y = ekx	ln(y)	x	Gradient = k
y = k sqrt(x)	y2	x	Gradient = k2

Examples

For the dynamics equation s= ½at² (u=0) used to determine the value of g by free fall
plot s (y-axis) vs t² (x-axis)
which will be a linear graph with a gradient of ½a

For the nuclear physics equation for gamma ray intensity R = k / (x + x₀)² where R = rate, x = distance, k & x₀ are constants
plot x (y-axis) vs 1 / sqrt(R) (x-axis)
which gives a linear relationship with Gradient = sqrt(k) and y-intercept = -x₀. To see why, re-arrange the equation to make x the subject (i.e. x = ....)

For the dynamics equation v² = u² + 2as
plot v² (y-axis) vs s (x-axis)
which gives a linear relationship with gradient = 2a and y-intercept = u²

Measurement and uncertainties

 State the fundamental units in the SI system.

Many different types of measurements are made in physics. In order to provide a clear and concise set of data, a specific system of units is used across all sciences. This system is called the International System of Units (SI from the French "Système International d'unités").

The SI system is composed of seven fundamental units:

Figure 1.2.1 - The fundamental SI units
Quantity	Unit name	Unit symbol
mass	kilogram	kg
time	second	s
length	meter	m
temperature	kelvin	K
Electric current	ampere	A
Amount of substance	mole	mol
Luminous intensity	candela	cd

Note that the last unit, candela, is not used in the IB diploma program.

1.2.2 Distinguish between fundamental and derived units and give examples of derived units.

In order to express certain quantities we combine the SI base units to form new ones. For example, if we wanted to express a quantity of speed which is distance/time we write m/s (or, more correctly m s-1). For some quantities, we combine the same unit twice or more, for example, to measure area which is length x width we write m2.

Certain combinations or SI units can be rather long and hard to read, for this reason, some of these combinations have been given a new unit and symbol in order to simplify the reading of data.
For example: power, which is the rate of using energy, is written as kg m2 s-3. This combination is used so often that a new unit has been derived from it called the watt (symbol: W).

Below is a table containing some of the SI derived units you will often encounter?

Table 1.2.2 - SI derived units
SI derived unit	Symbol	SI base unit	Alternative unit
newton	N	kg m s-2	-
joule	J	kg m2 s-2	N m
hertz	Hz	s-1	-
watt	W	kg m2 s-3	J s-1
volt	V	kg m2 s-3 A-1	W A-1
ohm	Ω	kg m2 s-3 A-2	V A-1
pascal	Pa	kg m-1 s-2	N m-2

1.2.3 Convert between different units of quantities.

Often, we need to convert between different units. For example, if we were trying to calculate the cost of heating a litre of water we would need to convert between joules (J) and kilowatt hours (kW h), as the energy required to heat water is given in joules and the cost of the electricity used to heat the water is a certain price per kW h.

If we look at table 1.2.2, we can see that one watt is equal to a joule per second. This makes it easy to convert from joules to watt hours: there are 60 second in a minutes and 60 minutes in an hour, therefor, 1 W h = 60 x 60 J, and one kW h = 1 W h / 1000 (the k in kW h being a prefix standing for kilo which is 1000). 

1.2.4 State units in the accepted SI format.

There are several ways to write most derived units. For example: meters per second can be written as m/s or m s-1. It is important to note that only the latter, m s-1, is accepted as a valid format. Therefor, you should always write meters per second (speed) as m s-1 and meters per second per second (acceleration) as m s-2. Note that this applies to all units, not just the two stated above.

1.2.5 State values in scientific notation and in multiples of units with appropriate prefixes.

When expressing large or small quantities we often use prefixes in front of the unit. For example, instead of writing 10000 V we write 10 kV, where k stands for kilo, which is 1000. We do the same for small quantities such as 1 mV which is equal to 0,001 V, m standing for milli meaning one thousandth (1/1000).

When expressing the units in words rather than symbols we say 10 kilowatts and 1 milliwatt.

A table of prefixes is given on page 2 of the physics data booklet.

1.2.6 Describe and give examples of random and systematic errors.

Random errors
A random error, is an error which affects a reading at random.
Sources of random errors include:

·         The observer being less than perfect

·         The readability of the equipment

·         External effects on the observed item

Systematic errors

A systematic error, is an error which occurs at each reading.
Sources of systematic errors include:

·         The observer being less than perfect in the same way every time

·         An instrument with a zero offset error

·         An instrument that is improperly calibrated

1.2.7 Distinguish between precision and accuracy.

Precision
A measurement is said to be accurate if it has little systematic errors.
 

Accuracy
A measurement is said to be precise if it has little random errors.

A measurement can be of great precision but be inaccurate (for example, if the instrument used had a zero offset error). 

1.2.8 Explain how the effects of random errors may be reduced.

The effect of random errors on a set of data can be reduced by repeating readings. On the other hand, because systematic errors occur at each reading, repeating readings does not reduce their affect on the data.

1.2.9 Calculate quantities and results of calculations to the appropriate number of significant figures.

The number of significant figures in a result should mirror the precision of the input data. That is to say, when dividing and multiplying, the number of significant figures must not exceed that of the least precise value.

Example:
Find the speed of a car that travels 11.21 meters in 1.23 seconds.

11.21 x 1.13 = 13.7883

The answer contains 6 significant figures. However, since the value for time (1.23 s) is only 3 s.f. we write the answer as 13.7 m s-1.

The number of significant figures in any answer should reflect the number of significant figures in the given data.

1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.

Absolute uncertainties
When marking the absolute uncertainty in a piece of data, we simply add ± 1 of the smallest significant figure.

Example:

13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1

Fractional uncertainties
To calculate the fractional uncertainty of a piece of data we simply divide the uncertainty by the value of the data.

Example:

1.2 s ± 0.1

Fractional uncertainty:

0.1 / 1.2 = 0.0625

Percentage uncertainties
To calculate the percentage uncertainty of a piece of data we simply multiply the fractional uncertainty by 100.

Example:

1.2 s ± 0.1

Percentage uncertainty:

0.1 / 1.2 x 100 = 6.25 %

1.2.11 Determine the uncertainties in results.

Simply displaying the uncertainty in data is not enough, we need to include it in any calculations we do with the data.

Addition and subtraction
When performing additions and subtractions we simply need to add together the absolute uncertainties.

Example:

Add the values 1.2 ± 0.1, 12.01 ± 0.01, 7.21 ± 0.01

1.2 + 12.01 + 7.21 = 20.42
0.1 + 0.01 + 0.01 = 0.12
20.42 ± 0.12

Multiplication, division and powers
When performing multiplications and divisions, or, dealing with powers, we simply add together the percentage uncertainties.

Example:

Multiply the values 1.2 ± 0.1, 12.01 ± 0.01

1.2 x 12.01 = 14
0.1 / 1.2 x 100 = 8.33 %
0.01 / 12.01 X 100 = 0.083%
8.33 + 0.083 = 8.413 %

14 ± 8.413 %

Other functions
For other functions, such as trigonometric ones, we calculate the mean, highest and lowest value to determine the uncertainty range. To do this, we calculate a result using the given values as normal, with added error margin and subtracted error margin. We then check the difference between the best value and the ones with added and subtracted error margin and use the largest difference as the error margin in the result.

Example:

Calculate the area of a field if its length is 12 ± 1 m and width is 7 ± 0.2 m.

Best value for area:
12 x 7 = 84 m2

Highest value for area:
13 x 7.2 = 93.6 m2

Lowest value for area:
11 x 6.8 = 74.8 m2

If we round the values we get an area of:
84 ± 10 m2

 Identify uncertainties as error bars in graphs.

When representing data as a graph, we represent uncertainty in the data points by adding error bars. We can see the uncertainty range by checking the length of the error bars in each direction. Error bars can be seen in figure 1.2.1 below:

 State random uncertainty as an uncertainty range (±) and represent it graphically as an "error bar".

In  physics, error bars only need to be used when the uncertainty in one or both of the plotted quantities are significant. Error bars are not required for trigonometric and logarithmic functions.

To add error bars to a point on a graph, we simply take the uncertainty range (expressed as "± value" in the data) and draw lines of a corresponding size above and below or on each side of the point depending on the axis the value corresponds to.

Example:

Plot the following data onto a graph taking into account the uncertainty.

Time ± 0.2 s	Distance ± 2 m
3.4	13
5.1	36
7	64

Table 1.2.1 - Distance vs Time data

Figure 1.2.2 - Distance vs. time graph with error bars 

In practice, plotting each point with its specific error bars can be time consuming as we would need to calculate the uncertainty range for each point. Therefor, we often skip certain points and only add error bars to specific ones. We can use the list of rules below to save time:

·         Add error bars only to the first and last points

·         Only add error bars to the point with the worst uncertainty

·         Add error bars to all points but use the uncertainty of the worst point

·         Only add error bars to the axis with the worst uncertainty

 Determine the uncertainties in the gradient and intercepts of a straight- line graph.

Gradient
To calculate the uncertainty in the gradient, we simply add error bars to the first and last point, and then draw a straight line passing through the lowest error bar of the one points and the highest in the other and vice versa. This gives two lines, one with the steepest possible gradient and one with the shallowest, we then calculate the gradient of each line and compare it to the best value. This is demonstrated in figure 1.2.3 below:

Gradient uncertainty in a graph 

Intercept
To calculate the uncertainty in the intercept, we do the same thing as when calculating the uncertainty in gradient. This time however, we check the lowest, highest and best value for the intercept. This is demonstrated in figure 1.2.4 below:

Figure : Intercept uncertainty in a graph

Measurements 1.1 Uncertainty in measurements

In an ideal world, measurements are always perfect: there, wooden boards can be cut to exactly two meters in length and a block of steel can have a mass of exactly three kilograms. However, we live in the real world, and here measurements are never perfect. In our world, measuring devices have limitations. The imperfection inherent in all measurements is called an uncertainty. In the Physics 152 laboratory, we will write an uncertainty almost every time we make a measurement. Our notation for measurements and their uncertainties takes the following form: (measured value ± uncertainty) proper units where the ± is read ‘plus or minus.’ 9.794 9.796 9.798 9.800 9.802 9.804 9.806 9.801 m/s2 m/s2 26 Purdue University Physics 152L Measurement Analysis 1 Figure 1: Measurement and uncertainty: (9.801 ± 0.003) m/s2 Consider the measurement g = (9.801 ± 0.003) m/s2. We interpret this measurement as meaning that the experimentally determined value of g can lie anywhere between the values 9.801 + 0.003 m/s2 and 9.801 − 0.003 m/s2, or 9.798 m/s2 ≤ g ≤ 9.804 m/s2. As you can see, a real world measurement is not one simple measured value, but is actually a range of possible values (see Figure 1). This range is determined by the uncertainty in the measurement. As uncertainty is reduced, this range is narrowed. Here are two examples of measurements: v = (4.000 ± 0.002) m/s G = (6.67 ± 0.01) × 10−11 N·m2/kg2 Look over the measurements given above, paying close attention to the number of decimal places in the measured values and the uncertainties (when the measurement is good to the thousandths place, so is the uncertainty; when the measurement is good to the hundredths place, so is the uncertainty). You should notice that they always agree, and this is most important: — In a measurement, the measured value and its uncertainty must always have the same number of digits after the decimal place. Examples of nonsensical measurements are (9.8 ± 0.0001) m/s2 and (9.801 ± 0.1) m/s2; writing such nonsensical measurements will cause readers to judge you as either incompetent or sloppy. Avoid writing improper measurements by always making sure the decimal places agree. Sometimes we want to talk about measurements more generally, and so we write them without actual numbers. In these cases, we use the lowercase Greek letter delta, or δ to represent the uncertainty in the measurement. Examples include: (X ± δX) (Y ± δY ) Although units are not explicitly written next to these measurements, they are implied. We will use these general expressions for measurements when we discuss the propagation of uncertainties in Section 4. 1.2 Uncertainties in measurements in lab In the laboratory you will be taking real world measurements, and for some measurements you will record both measured values and uncertainties. Getting values from measuring 0 2 4 6 8 10 12 cm (L ± δL) = (6 ± 1) cm Purdue University Physics 152L Measurement Analysis 1 27 equipment is usually as simple as reading a scale or a digital readout. Determining uncertainties is a bit more challenging since you—not the measuring device— must determine them. When determining an uncertainty from a measuring device, you need to first determine the smallest quantity that can be resolved on the device. Then, for your work in PHYS 152L, the uncertainty in the measurement is taken to be this value. For example, if a digital readout displays 1.35 g, then you should write that measurement as (1.35 ± 0.01) g. The smallest division you can clearly read is your uncertainty. On the other hand, reading a scale is somewhat subjective. Suppose you use a meter stick that is divided into centimeters to determine the length (L ± δL) of a rod, as illustrated in Figure 2. First, you read your measured value from this scale and find that the rod is 6 cm. Depending on the sharpness of your vision, the clarity of the scale, and the boundaries of the measured object, you might read the uncertainty as ± 1 cm, ± 0.5 cm, or ± 0.2 cm. An uncertainty of ± 0.1 cm or smaller is dubious because the ends of the object are rounded and it is hard to resolve ± 0.1 cm. Thus, you might want to record your measurement as (L ± δL) = (6 ± 1) cm, (L ± δL) = (6.0 ± 0.5) cm, or (L ± δL) = (6.0 ± 0.2) cm, since all three measurements would appear reasonable. For the purposes of discussion and uniformity in this laboratory manual, we will use the largest reasonable uncertainty. For our example, this is ± 1 cm. Figure 2: A measurement obtained by reading a scale. Acceptable measurements range from 6.0 ± 0.1 cm to 6.0 ± 0.2 cm, depending on the sharpness of your vision, the clarity of the scale, and the boundaries of the measured object. Examples of unacceptable measurements are 6 ± 2 cm and 6.00 ± 0.01 cm. 1.3 Percentage uncertainty of measurements When we speak of a measurement, we often want to know how reliable it is. We need some way of judging the relative worth of a measurement, and this is done by finding the percentage uncertainty of a measurement. We will refer to the percentage uncertainty of a measurement as the ratio between the measurement’s uncertainty and its measured value multiplied by 100%. You will often hear this kind of uncertainty or something closely related used with measurements – a meter is good to ± 3% of full scale, or ± 1% of the reading, or good to one part in a million. The percentage uncertainty of a measurement (Z ± δZ) is defined as δZ Z × 100%. Think about percentage uncertainty as a way of telling how much a measurement deviates from “perfection.” With this idea in mind, it makes sense that as the uncertainty 28 Purdue University Physics 152L Measurement Analysis 1 for a measurement decreases, the percentage uncertainty δZ Z × 100% decreases, and so the measurement deviates less from perfection. For example, a measurement of (2 ± 1) m has a percentage uncertainty of 50%, or one part in two. In contrast, a measurement of (2.00 ± 0.01) m has a percentage uncertainty of 0.5% (or 1 part in 200) and is therefore the more precise measurement. If there were some way to make this same measurement with zero uncertainty, the percentage uncertainty would equal 0% and there would be no deviation whatsoever from the measured value—we would have a “perfect” measurement. Unfortunately, this never happens in the real world. 1.4 Implied uncertainties When you read a physics textbook, you may notice that almost all the measurements stated are missing uncertainties. Does this mean that the author is able to measure things perfectly, without any uncertainty? Not at all! In fact, it is common practice in textbooks not to write uncertainties with measurements, even though they are actually there. In such cases, the uncertainties are implied. We treat these implied uncertainties the same way as we did when taking measurements in lab: — In a measurement with an implied uncertainty, the actual uncertainty is written as ± 1 in the smallest place value of the given measured value. For example, if you read g = 9.80146 m/s2 in a textbook, you know this measured value has an implied uncertainty of 0.00001 m/s2. To be more specific, you could then write (g ± δg) = (9.80146 ± 0.00001) m/s2. 1.5 Decimal points — don’t lose them If a decimal point gets lost, it can have disastrous consequences. One of the most common places where a decimal point gets lost is in front of a number. For example, writing .52 cm sometimes results in a reader missing the decimal point, and reading it as 52 cm — one hundred times larger! After all, a decimal point is only a simple small dot. However, writing 0.52 cm virtually eliminates the problem, and writing leading zeros for decimal numbers is standard scientific and engineering practice. 2 Agreement, Discrepancy, and Difference In the laboratory, you will not only be taking measurements, but also comparing them. You will compare your experimental measurements (i.e. the ones you find in lab) to some theoretical, predicted, or standard measurements (i.e. the type you calculate or look up in a textbook) as well as to experimental measurements you make during a second (or third...) data run. We need a method to determine how closely these measurements compare. To simplify this process, we adopt the following notion: two measurements, when compared, either agree within experimental uncertainty or they are discrepant (that is, they do 9.790 9.800 9.810 m/(s*s) g exp g std a: two values in experimental agreement 9.790 9.800 9.810 m/(s*s) g exp g std b: two discrepant values Purdue University Physics 152L Measurement Analysis 1 29 not agree). Before we illustrate how this classification is carried out, you should first recall that a measurement in the laboratory is not made up of one single value, but a whole range of values. With this in mind, we can say, Two measurements are in agreement if the two measurements share values in common; that is, their respective uncertainty ranges partially (or totally) overlap. Figure 3: Agreement and discrepancy of gravity measurements For example, a laboratory measurement of (gexp ± δgexp) = (9.801 ± 0.004) m/s2 is being compared to a scientific standard value of (gstd ± δgstd) = (9.8060 ± 0.0025) m/s2. As illustrated in Figure 3(a), we see that the ranges of the measurements partially overlap, and so we conclude that the two measurements agree. Remember that measurements are either in agreement or are discrepant. It then makes sense that, Two measurements are discrepant if the two measurements do not share values in common; that is, their respective uncertainty ranges do not overlap. Suppose as an example that a laboratory measurement (gexp ± δgexp) = (9.796 ± 0.004) m/s2 is being compared to the value of (gstd ± δgstd) = (9.8060 ± 0.0025) m/s2. From Figure 3 (b) we notice that the ranges of the measurements do not overlap at all, and so we say these measurements are discrepant. Precision & Accuracy precise, but not accurate accurate, but not precise (a) (b) 30 Purdue University Physics 152L Measurement Analysis 1 When two measurements being compared do not agree, we want to know by how much they do not agree. We call this quantity the discrepancy between measurements, and we use the following formula to compute it: The discrepancy Z between an experimental measurement (X ± δX) and a theoretical or standard measurement (Y ± δY ) is: Z = Xexperimental − Ystandard Ystandard × 100% As an example, take the two discrepant measurements (gexp ± δgexp) and (gstd ± δgstd) from the previous example. Since we found that these two measurements are discrepant, we can calculate the discrepancy Z between them as: Z = gexp − gstd gstd × 100% = 9.796 − 9.8060 9.8060 × 100% ≈ −0.10% Keep the following in mind when comparing measurements in the laboratory: 1. If you find that two measurements agree, state this in your report. Do NOT compute a discrepancy. 2. If you find that two measurements are discrepant, state this in your report and then go on to compute the discrepancy. 3 Precision and accuracy Figure 4: Precision and accuracy in target shooting. In everyday language, the words precision and accuracy are often interchangeable. In the sciences, however, the two terms have distinct meanings: 1 23 a: neither accurate nor precise 1 23 b: precise, but not accurate 1 23 c: both accurate and precise Purdue University Physics 152L Measurement Analysis 1 31 Precision describes the degree of certainty one has about a measurement. Accuracy describes how well measurements agree with a known, standard measurement. Let’s first examine the concept of precision. Figure 4(a) shows a precise target shooter, since all the shots are close to one another. Because all the shots are clustered about a single point, there is a high degree of certainty in where the shots have gone and so therefore the shots are precise. In Figure 5(b), the measurements on the ruler are all close to one another, and like the target shots, they are precise as well. Accuracy, on the other hand, describes how well something agrees with a standard. In Figure 4(b), the “standard” is the center of the target. All the shots are close to this center, and so we would say that the targets hooter is accurate. However, the shots are not close to one another, and so they are not precise. Here we see that the terms “precision” and “accuracy” are definitely not interchangeable; one does not imply the other. Nevertheless, it is possible for something to be both accurate and precise. In Figure 5(c), the measurements are accurate, since they are all close to the “standard” measurement of 1.5 cm. In addition, the measurements are precise, because they are all clustered about one another. Note that it is also possible for a measurement to be neither precise nor accurate. In Figure 5(a), the measurements are neither close to one another (and therefore not precise), nor are they close to the accepted value of 1.5 cm (and hence not accurate). Figure 5: Examples of precision and accuracy in length measurements. Here the hollow headed arrows indicate the ‘actual’ value of 1.5 cm. The solid arrows represent measurements. You may have noticed that we have already developed techniques to measure precision and accuracy. In Section 1.3, we compared the uncertainty of a measurement to its measured value to find the percentage uncertainty. The calculation of percentage uncertainty is actually a test to determine how certain you are about a measurement; in other words, how precise the measurement is. In Section 2, we learned how to compare a measurement to a standard 32 Purdue University Physics 152L Measurement Analysis 1 or accepted value by calculating a percent discrepancy. This comparison told you how close your measurement was to this standard measurement, and so finding percent discrepancy is really a test for accuracy. It turns out that in the laboratory, precision is much easier to achieve than accuracy. Precision can be achieved by careful techniques and handiwork, but accuracy requires excellence in experimental design and measurement analysis. During this laboratory course, you will examine both accuracy and precision in your measurements and suggest methods of improving both. 4 Propagation of uncertainty (worst case) In the laboratory, we will need to combine measurements using addition, subtraction, multiplication, and division. However, measurements are composed of two parts—a measured value and an uncertainty—and so any algebraic combination must account for both. Performing these operations on the measured values is easily accomplished; handling uncertainties poses the challenge. We make use of the propagation of uncertainty to combine measurements with the assumption that as measurements are combined, uncertainty increases—hence the uncertainty propagates through the calculation. Here we show how to combine two measurements and their uncertainties. Often in lab you will have to keep using the propagation formulae over and over, building up more and more uncertainty as you combine three, four or five set of numbers. 1. When adding two measurements, the uncertainty in the final measurement is the sum of the uncertainties in the original measurements: (A ± δA)+(B ± δB)=(A + B) ± (δA + δB) (1) As an example, let us calculate the combined length (L ± δL) of two tables whose lengths are (L1 ± δL1) = (3.04 ± 0.04) m and (L2 ± δL2) = (10.30 ± 0.01) m. Using this addition rule, we find that (L ± δL) = (3.04 ± 0.04) m + (10.30 ± 0.01) m = (13.34 ± 0.05) m 2. When subtracting two measurements, the uncertainty in the final measurement is again equal to the sum of the uncertainties in the original measurements: (A ± δA) − (B ± δB)=(A − B) ± (δA + δB) (2) For example, the difference in length between the two tables mentioned above is (L2 ± δL2) − (L1 ± δL1) = (10.30 ± 0.01) m − (3.04 ± 0.04) m = [(10.30 − 3.04) ± (0.01 + 0.04)] m = (7.26 ± 0.05) m Purdue University Physics 152L Measurement Analysis 1 33 Be careful not to subtract uncertainties when subtracting measurements— uncertainty ALWAYS gets worse as more measurements are combined. 3. When multiplying two measurements, the uncertainty in the final measurement is found by summing the percentage uncertainties of the original measurements and then multiplying that sum by the product of the measured values: (A ± δA) × (B ± δB)=(AB) " 1 ± à δA A + δB B !# (3) A quick derivation of this multiplication rule is given below. First, assume that the measured values are large compared to the uncertainties; that is, A À δA and B À δB. Then, using the distributive law of multiplication: (A ± δA) × (B ± δB) = AB + A(±δB) + B(±δA)+(±δA)(±δB) ∼= AB ± (A δB + B δA) (4) Since the uncertainties are small compared to the measured values, the product of two small uncertainties is an even smaller number, and so we discard the product (±δA)(±δB). With further simplification, we find: AB + A(±δB) + B(±δA) = AB + B(±δA) + A(±δB) = AB " 1 ± à δA A + δB B !# It should be noted that the above equation is mathematically undefined if either A or B is zero. In this case equation 4 is used to obtain the uncertainty since it is valid for all values of A and B. Now let us use the multiplication rule to determine the area of a rectangular sheet with length (l ±δl) = (1.50±0.02) m and width (w ±δw) = (20±1) cm = (0.20 ± 0.01) m. The area (A ± δA) is then (A ± δA)=(l ± δl) × (w ± δw)=(lw) " 1 ± à δl l + δw w !# = (1.50 × 0.20) · 1 ± µ0.02 1.50 + 0.01 0.20¶¸ m2 = 0.300[1 ± (0.0133 + 0.0500)] m2 = 0.300[1 ± 0.0633] m2 = (0.300 ± 0.0190)m2 ≈ (0.30 ± 0.02) m2 34 Purdue University Physics 152L Measurement Analysis 1 Notice that the final values for uncertainty in the above calculation were determined by multiplying the product (lw) outside the bracket by the sum of the two percentage uncertainties (δl/l + δw/w) inside the bracket. Always remember this crucial step! Also, notice how the final measurement for the area was rounded. This rounding was performed by following the rules of significant figures, which are explained in detail later in Section 5. Recall our discussion of percentage uncertainty in Section 1.3. It is here that we see the benefits of using such a quantity; specifically, we can use it to tell right away which of the two original measurements contributed most to the final area uncertainty. In the above example, we see that the percentage uncertainty of the width measurement (δw/w)×100% is 5%, which is larger than the percentage uncertainty (δl/l)×100% ≈ 1.3% of the length measurement. Hence, the width measurement contributed most to the final area uncertainty, and so if we wanted to improve the precision of our area measurement, we should concentrate on reducing width uncertainty δw (since it would have a greater effect on the total uncertainty) by changing our method for measuring width. 4. When dividing two measurements, the uncertainty in the final measurement is found by summing the percentage uncertainties of the original measurements and then multiplying that sum by the quotient of the measured values: (A ± δA) (B ± δB) = µ A B ¶ " 1 ± à δA A + δB B !# (5) As an example, let’s calculate the average speed of a runner who travels a distance of (100.0 ± 0.2) m in (9.85 ± 0.12) s using the equation v = D/t, where ¯v is the average speed, D is the distance traveled, and t is the time it takes to travel that distance. v¯ = D ± δD t ± δt = µD t ¶ " 1 ± à δD D + δt t !# = µ100.0 m 9.85 s ¶ ·1 ± µ 0.2 100.0 + 0.12 9.85¶¸ = 10.15 [1 ± (0.002000 + 0.01218) ] m/s = 10.15 [1 ± (0.01418) ] m/s = (10.15 ± 0.1439) m/s ≈ (10.2 ± 0.1) m/s In this particular example the final uncertainty results mainly from the uncertainty in the measurement of t, which is seen by comparing the percentage uncertainties of the time and distance measurements, (δt/t) ≈ 1.22% and (δD/D) ≈ 0.20%, respectively. Therefore, to reduce the uncertainty in (v ± δv), we would want to look first at changing the way t is measured. 5. Special cases—inversion and multiplication by a constant: Purdue University Physics 152L Measurement Analysis 1 35 (a) If you have a quantity X ± δX, you can invert it and apply the original percentage uncertainty: 1 X ± δX = µ 1 X ¶ " 1 ± δX X # (b) To multiply by a constant, k × (Y ± δY )=[kY ± kδY ] It is important to realize that these formulas and techniques allow you to perform the four basic arithmetic operations. You can (and will) combine them by repetition for the sum of three measurements, or the cube of a measurement. Normally it is impossible to use these simple rules for more complicated operations such as a square root or a logarithm, but the trigonometric functions sin θ, cos θ, and tan θ are exceptions. Because these functions are defined as the ratios between lengths, we can use the quotient rule to evaluate them. For example, in a right triangle with opposite side (x±δx) and hypotenuse (h±δh), sin θ = (x±δx) (h±δh) . Similarly, any expression that can be broken down into arithmetic steps may be evaluated with these formulas; for example, (x ± δx)2 = (x ± δx)(x ± δx). 6. Finding the uncertainty of a square root The method for obtaining the square root of a measurement. uses some algebra coupled with the multiplication rule. Let (A ± δA) and (B ± δB) be two measurements. Further, assume that the square root of (A ± δA) is equal to the measurement (B ± δB). Then, q (A ± δA)=(B ± δB) (6) Squaring both sides, we obtain (A ± δA)=(B ± δB) 2 Using the multiplication rule on (B ± δB)2, we find (A ± δA)=(B ± δB) 2 = B2 " 1 ± à 2δB B !# = (B2 ± 2BδB) Thus, (A ± δA)=(B2 ± 2BδB) which means B = √ A and δB = δA 2B = δA 2 √ A. 36 Purdue University Physics 152L Measurement Analysis 1 Making this substitution into Equation 6, we arrive at the final result q (A ± δA) = Ã√ A ± δA 2 √ A ! (7) This technique for finding the uncertainty in a square root will be required in E4 — E6. 7. Another example: a case involving a triple product. The formula for the volume of a rod with a circular cross–section (πr2) and length l is given by V = πr2l. Given initial measurements (r ±δr) and (l±δl), derive an expression for (V ±δV ). Note that π has no uncertainty. Using the derivation of the worst case multiplication propagation rule (Equation 4) as a guide, we start with (V ± δV ) = π(r ± δr)(r ± δr)(l ± δl) and expand the terms involving r on the left hand side. (V ± δV ) = π[r2 + r(±δr) + r(±δr)+(±δr)(±δr)](l ± δl) Discarding the term involving the product of measurement uncertainties (δr)(δr), since it is small compared to the other terms, we obtain (V ± δV ) = π[r2 + 2r(±δr)](l ± δl) Next we multiply out the final product on the left. (V ± δV ) = π[r2 l + r2 (±δl)+2rl(±δr)+2r(±δr)(±δl)] Again we discard terms involving products of measurement uncertainties such as (δr)(δl) to obtain (V ± δV ) = π[r2 l + r2 (±δl)+2rl(±δr)] Finally, we can factor out r2l to obtain (V ± δV ) = πr2 l(1 ± δl l ± 2 δr r ) 8. Other uncertainty propagation techniques. The worst case uncertainty propagation assumes that all measurement uncertainties conspire to give the worst possible uncertainty in your final result. Fortunately this does not usually happen in nature, and there are techniques to take this into account, the simplest being the addition of uncertainties in quadrature and taking the square root of the sum. However, these techniques are more complex and inconsistent with the mathematical requirements for PHYS 152, and we have avoided them. A good start in learning about these more sophisticated techniques is to read the references listed at the end of this chapter. Purdue University Physics 152L Measurement Analysis 1 37 5 Rounding measurements The previous sections contain the bulk of what you need to take and analyze measurements in the laboratory. Now it is time to discuss the finer details of measurement analysis. The subtleties we are about to present cause an inordinate amount of confusion in the laboratory. Getting caught up in details is a frustrating experience, and the following guidelines should help alleviate these problems. An often-asked question is, “How should I round my measurements in the laboratory?” The answer is that you must watch significant figures in calculations and then be sure the number of decimal places of a measured value and its uncertainty agree. Before we give an example, we should explore these two ideas in some detail. 5.1 Treating significant figures The simplest definition for a significant figure is a digit (0 - 9) that actually represents some quantity. Zeros that are used to locate a decimal point are not considered significant figures. Any measured value, then, has a specific number of significant figures. See Table 1 for examples. There are two major rules for handling significant figures in calculations. One applies for addition and subtraction, the other for multiplication and division. 1. When adding or subtracting quantities, the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum (or difference). Examples: 51.4 − 1.67 = 49.7 7146 − 12.8 = 7133 20.8 + 18.72 + 0.851 = 40.4 2. When multiplying or dividing quantities, the number of significant figures in the final answer is the same as the number of significant figures in the least accurate of the quantities being multiplied (or divided). Examples: 2.6 × 31.7 = 82 not 82.42 5.3 ÷ 748 = 0.0071 not 0.007085561 5.2 Measured values and uncertainties: Number of decimal places As mentioned earlier in Section 1.1, we learned that for any measurement (X ± δX), the number of decimal places of the measured value X must equal those of the corresponding uncertainty δX. Below are some examples of correctly written measurements. Notice how the number of decimal places of the measured value and its corresponding uncertainty agree. (L ± δL) = (3.004 ± 0.002) m (m ± δm) = (41.2 ± 0.4) kg 38 Purdue University Physics 152L Measurement Analysis 1 Measured value Number of significant figures 123 3 1.23 3 1.230 4 0.00123 3 0.001230 4 Table 1: Examples of significant figures 5.3 Rounding Suppose we are asked to find the area (A ± δA) of a rectangle with length (l ± δl) = (2.708 ± 0.005) m and width (w ± δw) = (1.05 ± 0.01) m. Before propagating the uncertainties by using the multiplication rule, we should first figure out how many significant figures our final measured value A must have. In this case, A = lw, and since l has four significant figures and w has three significant figures, A is limited to three significant figures. Remember this result; we will come back to it in a few steps. We may now use the multiplication rule to calculate the area: (A ± δA)=(l ± δl) × (w ± δw) = (lw) " 1 ± à δl l + δw w !# = (2.708 × 1.05) · 1 ± µ0.005 2.708 + 0.01 1.05¶¸ m2 = (2.843) [1 ± (0.001846 + 0.009524)] m2 = 2.843 (1 ± 0.011370) m2 = (2.843 ± 0.03232) m2 Notice that in the intermediate step directly above, we allowed each number one extra significant figure beyond what we know our final measured value will have; that is, we know the final value will have three significant figures, but we have written each of these intermediate numbers with four significant figures. Carrying the extra significant figure ensures that we will not introduce round-off error. We are just two steps away from writing our final measurement. Step one is recalling the result we found earlier—that our final measured value must have three significant figures. Thus, we will round 2.843 m2 to 2.84 m2. Once this step is accomplished, we round our uncertainty to match the number of decimal places in the measured value. In this case, we round 0.03233 m2 to 0.03 m2. Finally, we can write (A ± δA) = (2.84 ± 0.03) m2

What is a measurement?
A measurement tells you about a property of something you are investigating, giving it a number and a unit. Measurements are always made using an instrument of some kind. Rulers, stopclocks, chemical balances and thermometers are all measuring instruments.

Some processes seem to be measuring, but are not, e.g. comparing two lengths of string to see which one is longer. Tests that lead to a simple yes/no or pass/fail result do not always involve measuring.

The quality of measurements
Evaluating the quality of measurements is an essential step on the way to sensible conclusions. Scientists use a special vocabulary that helps them think clearly about their data. Key terms that describe the quality of measurements are:

§  Validity 

§  Accuracy 

§  Precision (repeatability or reproducibility) 

§  Measurement uncertainty  

Validity: A measurement is ‘valid’ if it measures what it is supposed to be measuring. What is measured must also be relevant to the question being investigated.

If a factor is uncontrolled, the measurements may not be valid. For example, if you were investigating the heating effect of a current (P = I²R ) by increasing the current, the resistance of the wire may change as it is heated by the current to different temperatures. This would skew the results.

Correct conclusions can only be drawn from valid data.

Accuracy: This describes how closely a measurement comes to the true value of a physical quantity. The ‘true’ value of a measurement is the value that would be obtained by a perfect measurement, i.e. in an ideal world. As the true value is not known, accuracy is a qualitative term only.

Many measured quantities have a range of values rather than one ‘true’ value. For example, a collection of resistors all marked 1 kΩ will have a range of values, but the mean value should be 1 kΩ. You can have more confidence in a number of measurements of a sample rather than an individual measurement. The variation enables you to identify a mean, a range and the distribution of values across the range.

Precision: The closeness of agreement between replicate measurements on the same or similar objects under specified conditions.

Repeatability or reproducibility (precision): The extent to which a measurement replicated under the same conditions gives a consistent result. Repeatability refers to data collected by the same operator, in the same lab, over a short timescale. Reproducibility refers to data collected by different operators, in different laboratories. You can have more confidence in conclusions and explanations if they are based on consistent data.

Measurement uncertainty: The uncertainty of a measurement is the doubt that exists about its value. For any 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 centimeter’.

The doubt about a measurement has two aspects: 

§  the width of the margin, or ‘interval’. This is the range of values one expects the true value to lie within. (Note this is not necessarily the range of values one might obtain when taking measurements of the value, which may include outliers.) 

§  Confidence level’, i.e. how sure the experimenter is that the true value lies within that margin. Discussion of confidence levels is generally appropriate only in advanced level science courses. 

Uncertainty in measurements can be reduced by using an instrument that has a scale with smaller scale divisions. For example, if you use a ruler with a centimeter scale then the uncertainty in a measured length is likely to be ‘give or take a centimeter’. A ruler with a millimeter scale would reduce the uncertainty in length to ‘give or take a millimeter’.

Measurement errors
It is important not to confuse the terms ‘error’ and ‘uncertainty’. Error refers to the difference between a measured value and the true value of a physical quantity being measured. Whenever possible we try to correct for any known errors: for example, by applying corrections from calibration certificates. But any error whose value we do not know is a source of uncertainty.

Measurement errors can arise from two sources: 

§  a random component, where repeating the measurement gives an unpredictably different result;

§  a systematic component, where the same influence affects the result for each of the repeated measurements.  

Every time a measurement is taken under what seem to be the same conditions, random effects can influence the measured value. A series of measurements therefore produces a scatter of values about a mean value. The influence of variable factors may change with each measurement, changing the mean value. Increasing the number of observations generally reduces the uncertainty in the mean value.

Systematic errors (measurements that are either consistently too large, or too small) can result from: 

§  poor technique (e.g. carelessness with parallax when sighting onto a scale); 

§  zero error of an instrument (e.g. a ruler that has been shortened by wear at the zero end, or a newton meter that reads a value when nothing is hung from it); 

§  poor calibration of an instrument (e.g. every volt is measured too large). 

Whenever possible, a good experimenter will try and correct for systematic errors, thus improving accuracy. For example, if it is known that a balance always reads 2 g greater than the true reading it is perfectly possible to compensate for that error by simply subtracting 2 g from all readings taken.

Sometimes you can only find a systematic error by measuring the same value by a different method.

Errors that are not recognized contribute to measurement uncertainty.