Measurement Analysis: Uncertainties, Propagation of Error, Least Square Fitting, and Graphical Analysis

1. Measurements

1.1 Uncertainty in Measurements In an ideal world, measurements are always perfect. Wooden boards can be cut to exactly two meters in length and three meters in width. An aluminum block can exactly have a mass of four kilograms. All measurements will have exact values and hence, calculations involving measurements will be textbook-style simple. Unfortunately, experiments are done in a real world, not an ideal world. In the real world case measurements are never perfect. Measuring devices have limitations such that there will always be imprecisions and inaccuracies in measurements. The imperfection inherent in all experimental measurements is termed an uncertainty. In the Physics 152 laboratory, uncertainties must always be considered every time a measurement is taken. The notation for stating a measurement together with its uncertainty is: (best-estimate ± uncertainty) proper-units.

Fig. 1 Measurement and uncertainty: g = (9.801 ± 0.002) m/s2 Consider the measurement g = (9.801 ± 0.002) m/s2 indicate by the arrow in Figure 1. This measurement can be interpreted as a value that can lie between (9.801 + 0.002) m/s2 and (9.801 - 0.002) m/s2, or as an interval 9.799 m/s2 < g < 9.803 m/s2 . You can therefore see that an experimental measurement is not an exact value but a range of possible values. This range is set by the uncertainty of the measurement. Below are two more examples of measurements:

V=(4.000 ± 0.002) m3 G=(6.67 ± 0.01) x 10-11 Nm2/kg2 The rule used in writing these measurements is:

When stating a measurement, the best estimate and its uncertainty must always have the same number of digits after the decimal point.

We will have to do algebra with uncertainties so we need to define how we write a measurement in terms of variables. The convention is to use the Greek delta to indicate the uncertainty:

)( XX δ± How do we determine these uncertainties in the laboratory?

9.794 9.796 9.798 9.800 9.802 9.804 9.806 9.808 m/s2

9.801 m/s2

1.2 Uncertainties in Experimental Measurements

Since experimental measurements are written as a pair of numbers, a complete measurement must yield both a best-estimate and an uncertainty. The best estimate may be determined by simply reading a scale or digital readout but determining the uncertainty can be more challenging. The convention we will use is that the uncertainty in a measurement will be the smallest quantity that can be resolved on the measuring instrument used. For example, if a digital scale displays 1.35 g, then the measurement will be stated as (1.35 ± 0.01) g since that measuring instrument measures in increments of 0.01 g. For analog measuring devices like a meter stick, again the uncertainty is the smallest division marked on the device. For example, meter sticks commonly have labels every centimeter and 10 divisions per centimeter. Therefore, the smallest measurement that can be made by a meter stick is one millimeter. And so, the uncertainty for distance measurements made by a meter stick is also one millimeter, e.g. (0.353 ± 0.001) m. You certainly could estimate between the millimeter divisions, but that estimate would be highly dependent on the particular experimenter. Our convention is meant to make measurements that are independent of who performs the experiment. It is commonly called taking the largest reasonable uncertainty.

1.3 Percentage Uncertainties of Measurements It is often important (and instructive) to determine the quality of a given measurement. To determine how that measurement compares to other measurements of different quantities. In order to achieve this, the concept of percentage uncertainties will be introduced. The percentages will allow us to compare apples and oranges. For example, we will want to decide whether it is the length measurement or the mass measurement that is causing the value we find for the acceleration of gravity to have a large uncertainty. This concept will be used heavily in the Analysis section of your lab reports. The percentage uncertainty of a given measurement is defined as the ratio between that measurement's uncertainty and its best estimate and then multiplied by 100%. For example, say the measurement is

MM δ± . Then the percentage uncertainty is :

Percentage uncertainty = %100×M

Mδ

1.4 Implied Uncertainties In physics textbooks, you may have noticed that most measurements are stated without any uncertainties. This does not mean that the authors where able to measure things exactly. Unless a textbook is about Error Analysis, it is usually the practice not to include the uncertainties on measurements. The uncertainties in this case are termed implied. For measurements involving implied uncertainties, the actual uncertainty is defined by the least significant decimal place of the stated measurement. For example, if a certain textbook states that the acceleration due to gravity is g = 9.80146 m/s2, then the implied uncertainty is 0.00001 m/s2. So we then write this measurement as g ± δg = (9.80146 ± 0.00001) m/s2.

2. Agreements and Discrepancies One of the main purposes in performing an experiment is to acquire measurements so that these could be compared to other measurements. There are two types of measurement comparison. One is to compare the measurement with a standard or predicted number. The other is when a set of measurements is gathered and then compared among themselves. For both cases, we need a convention for deciding if the measurements agree with other measurements of the same quantity. We also need methods to describe numerically how close these measurements are. Let us adopt a convention on how to specify agreement:

Two measurements agree if they share common values; that is, their uncertainty ranges overlap.

The overlapping of uncertainty ranges can either be total, in which case the measurements have the same best estimates and uncertainties, or partial, in which case only some values are common to both. This is illustrated by the three different comparisons of measurements indicated above. The best estimate is given by the line and the gray box indicates the range. When two measurements differ, it is necessary to determine the magnitude of their disagreement. For this case, we calculate a quantity termed discrepancy. Below is the definition: The discrepancy Z between an experimental measurement ( XX δ± ) and another measurement (usually a theoretical or standard measurement) ( YY δ± ) is

%100×−

=Y

YXZ

Notice that when computing the discrepancy, we use only best-estimate values, not the uncertainties, in the calculation. If you are comparing two different experimental measurements that you take, then use the first measurement as the standard and calculate how discrepant the second measurement is compared to the first. Keep the following in mind when comparing measurements in your experiments:

• If a key finding agrees with an accepted standard in one of your experiments be sure to state that agreement in the abstract, analysis and conclusion of your lab report. You may also report the discrepancy if you believe it to be significant.

• Conversely, if it disagrees you must calculate the discrepancy and note it in the abstract, analysis and conclusion of your lab report.

Agree Disagree Agree

2.1 Precision and Accuracy In everyday language, the words precision and accuracy are often interchangeable. In the sciences, however, the two terms have distinct meanings: Precision measures the repeatability of your measurements. It describes how certain you are that the next measurement will be close to the previous measurements. A numerical measure of the degree of certainty one has about a measurement the percentage uncertainty. The smaller the percentage the higher the precision of the experiment. Accuracy describes how well measurements agree with a known, standard measurement. A numerical measure of this accuracy is the discrepancy we defined in the previous section.

3. Worst Case Propagation of Uncertainty In the laboratory, we will need to combine measurements using addition, subtraction, multiplication, and division. However, measurements are composed of two parts --- the best estimate and an uncertainty --- and so any algebraic combination must account for both. Performing these operations on the best estimates is done in the usual fashion; handling uncertainties poses the challenge. We make use of a method of propagating uncertainties to combine measurements that incorporates the assumption that as measurements are combined, uncertainty always increases. Here we show how to combine two measurements and their uncertainties. Often in the 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 sets of numbers. In Physics152L we use a worst case uncertainty propagation methof, which assumes that all measurement uncertainties conspire to give the worst possible uncertainty in the 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 therefore avoided them. Nevertheless, you need to be aware of these techniques, because they provide straightforward ways of dealing with mathematical operations more complicated than addition, subtraction, multiplication, and division. A good start in learning about these more sophisticated techniques is to read the references listed at the end of this appendix. The following are the rules for algebra with uncertainties. Addition: The uncertainty in the final measurement is the sum of the uncertainties in the original measurements,

|)||(|)()()( BABABBAA δδδδ +±+=±+± Subtraction: The uncertainty in the final measurement is again the sum of the uncertainties in the original measurements,

|)||(|)()()( BABABBAA δδδδ +±−=±−±

Multiplication: 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,

+±=±×±

B

B

A

AABBBAA

δδδδ 1)()()(

This can be derived easily with the assumption that the uncertainties are much smaller than the best estimates. In that case, when we multiply out the product on the left hand side of the equation we can discard the term δAδB as being very small. Rearranging the remaining terms gives the result on the right hand side of the equation. Note if we were to multiply three measurements together the result would be

++±=±×±×±

C

C

B

B

A

AABCCCBBAA

δδδδδδ 1)()()()(

and so on as more terms are multiplied. It should be noted that the above equations are mathematically undefined if either A or B is zero. In this case the assumption that the uncertainties are smaller than the best estimates are not valid and so we must keep all the terms to calculate the uncertainties. Division: 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:

+±

=

±

±

B

B

A

A

B

A

BB

AA δδδδ

1)(

)(

This can be derived using a binomial expansion of the denominator using the assumption that the uncertainties are much smaller than the best estimates. As an example, let's calculate the average speed of a runner who travels a distance D ± δD = (100.0 ± 0.2) m in t ± δt = (9.85 ± 0.12),

( )[ ]

( )

( ) sm

sm

sm

s

m

t

t

D

D

t

D

tt

DDv

/1.02.10

/1439.015.10

/01218.000200.0115.10

85.9

12.0

0.100

2.01

85.9

0.100

1)()(

±=

±=

+±=

+±

=

+±

=

±

±=

δδδδ

In this particular example the final uncertainty stems 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. This suggests that if we want to improve our uncertainty for the average velocity we would first look at improving the way time is measured, i.e. it would make sense to look into buying a better stopwatch before buying a better tape measure.

Notice that we kept more decimal places in the intermediate steps and rounded the answer to the correct number of significant figures only at the end. We will shortly discuss the rules for significant figures and rounding. Other algebraic operations:

Inversion:

±

=

± B

B

BBB

δδ

11

)(

1

Multiplication by a constant: k AkkAAA δδ ±=± )()(

Square Root: A

AAAA

2

δδ ±=±

4. Rounding measurements The previous sections contain most of what you need to acquire and analyze measurements in the laboratory. This section will now be concerned with the finishing details of correctly stating the measurements. There are two major concepts:

• Significant Figures – the number of digits in a measurement that are known to have significance or meaning. Since we know all measurements have limitations, there is a decimal place in each measurement that represents the highest accuracy possible for this measurement. For example, if your scale only reads to 0.1g it makes no sense to report a best estimate of 433.33333g. We should only report what we know, so the correct way to write this best estimate is 433.3g. This best estimate has 4 significant figures.

• Rounding should be accomplished so that the best estimate and its uncertainty agree on the number of decimal places. It makes no sense to have either have a different number of decimal places. For example, measurements written as (433.3333±0.1)g or (433±0.1)g both give conflicting messages about the accuracy of your measurements.

4.1 Significant Figures Significant figures are all the digits in a physical quantity that have meaning or agree with the accuracy of the measurement of those physical quantities. Zeros that are used to locate a decimal point are not considered as significant figures. Any measured value, then, has a specific number of significant figures. See Table 1 for examples.

Table 1. Examples of significant figures. Example 4 shows an ambiguous situation in which the number of significant figures could be 1 or 2. Examples 5 and 6 show how to resolve this ambiguity.

Example Measured Value Sig. Figs. 1 1 1 2 11. 2 3 11.1 3 4 10 1 or 2? 5 10. 2 6 1.0 x 101 2 7 1.10 x 10-3 3 8 0.001 1 9 1.001 4

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. A few examples are

51.4 - 50.63 = 0.8; 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 should be the same as the number of significant figures of the least precise of the quantities that is being multiplied or divided. A couple examples are

2.6 x 31.7 = 82 not 82.42 5.3 / 748=0.0071 not 0.007085561

Again, take note that when adding or subtracting two numbers, the number of decimal places should be considered. However, when multiplying or dividing two numbers, the number of significant figures should be considered. As mentioned above, the number of decimal places in the best estimate of a measurement should agree with the number of decimal places in its uncertainty. However, this means that the number of significant figures in a best estimate and in an uncertainty will not ordinarily agree.

4.2 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. We should first figure out how many significant figures our final best estimate A must have. In this case, A=lw, and since l has four significant figures and w has three significant figures, A should be limited to only three significant figures. Remember this result; we will need it later. We may now use the multiplication rule to calculate the uncertainties of the area:

+±=

+±=±×±

05.1

01.0

708.2

005.01)05.1)(708.2(1)()()(

w

w

l

llwwwll

δδδδ

( )[ ]2

2

m)03232.0843.2(

m009524.0001846.01843.2

±=

+±=

Notice that in the intermediate step above, we allowed each number one extra significant figure beyond what we know our final best estimate 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 this extra non-significant figure ensures that we will not introduce a round-off error. We are just two steps away from writing our final measurement. First we round the best estimate 2.843 m2 to 2.84 m2 and then we round the 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.

5.0 Statistical Considerations When a particular measurement is repeated several times and seemingly random differences occur on each measurement (maybe due to resolution limitations), we can apply probabilistic methods to analyze the uncertainties. To do this, it is common to assume that the distribution of measurements follow a Gaussian or Normal distribution.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

number of standard deviations

pro

bab

ility

den

sity

Gaussian or Normal distribution function First, we will define several statistical quantities. The statistical mean value is exactly equivalent to the quantity we knew in high school as the simple average for a set of repeated measurements. For example, a set of N different measurements represented by X1, X2, … Xi, … , XN, has a mean given by

∑==

N

iiX

NX

1

1

As an example, suppose we try to measure the speed of sound and made five repetitions of the measurement. The following are the data: 341 m/s, 344 m/s, 338 m/s, 340 m/s and 343 m/s. The sum of these five values is 1706 m/s and dividing by five we get 341.2 m/s. Rounding to the correct three significant figures gives the mean value of 341 m/s for this set of data. The second important statistical quantity is the standard deviation. It gives a measure of how wide the distribution of measurements is or how far they are spread out around the mean value. The standard deviation is usually represented by the lowercase s appended with an appropriate subscript, i.e. sX. It is written in terms of the mean and the set of measurements as follows,

2

1

2

1

22 1)(

1XX

NXX

Ns

N

ii

N

iix −

∑=∑ −===

sx

mean

Consider again the speed of sound data we collected in the laboratory. Having already found the mean value of 341m/s for this data, we now wish to calculate the standard deviation. Evaluating the above equation we

find 2vs = 4.56 (m/s)2 or sv = 2.14 m/s.

Finally, if we assume that the measurements follow a Gaussian random distribution, then the uncertainty in the best estimate or the mean value X is given by the standard error of the mean. This standard error of the mean is usually denoted by xσ or sometimes by SEM and is given by:

N

sSEM x

x == σ

Note that the SEM can be easily reduced increasing the number of measurements (i.e. a larger value of N). The more the data gathered, the less is the uncertainty in the measurement. Finally we write the best estimate and uncertainty from this set of data as

xX σ±

Returning to the measurement of the speed of sound, we find that 96.05/ == vv sσ m/s. So finally, the

measured speed of sound from our collection of measurements is (341 ± 1) m/s. Notice that we rounded the uncertainty to agree with the number of decimal places in the best estimate.

6.0 Comparing Data to a Line It is common that we will have data that forms a straight line. For example, if a cart is moving with constant positive velocity the plot of the position vs. time will be a line with a positive slope (the slope of the line is the velocity). We like to present data in the form of a straight line because your eye can easily detect a deviation from a straight line. It is also very easy to do a least squares fit of the best straight line through a set of data. Once we have this fit we can extract various physical quantities and determine the level of uncertainty for those quantities. So even if our data has a quadratic dependence, i.e. y=ax2 , we tend to plot y vs x2. The slope of the resulting line is then the constant a.

6.1 Least Square Fit Algorithm The idea of the algorithm is to find the straight line that minimizes the squared distance between a collection of data points and that straight line. It is not a difficult derivation but it requires partial differentiation and so is beyond the scope of this course. However, the algorithm itself is straightforward to implement in an Excel spreadsheet. The slope (m) and the y-intercept (b) for the best fit line (y=mx+b) are:

( ) ( )( )∆

∑∑−∑= i ii ii ii yxyxN

m

( )( ) ( )( )∆

∑∑−∑∑= i iii ii ii i yxxyx

b2

the constant ∆ is defined for convenience to be ( ) ( )22 ∑−∑=∆ i ii i xxN .

The uncertainties for the least square fit quantities are given in terms of standard deviation of the y values from the best fit line.

( )∑ −−= i iiy mxbyN

22 1σ

Knowing this the values for the uncertainties in m and b can be found:

∆

∑=

∆=

i iyb

ym

xN 222

22 ;

σσ

σσ

Therefore, the least squares fit gives us the following best estimates and uncertainties for the slope and y-intercept.

( ) ( )bm bm σσ ±± ;

Note that in Physics 152L we use the common rule of reporting only one significant digit in the uncertainty given by the least squares curve fit. So that if you find m=1.567 and σm=0.027 we report the slope to be ( )03.057.1 ± .