Download Understanding Measurement Errors: Centrality, Systematic, and Random Errors and more Study notes Physics in PDF only on Docsity!
Notes on Measurement Errors
(Some material in this -handout excerpted from Stan Micklavzina's "Guidelines for Reporting Data" and from William Lichten's "Data arid Error Analysis," Allyn and Bacon)
Measurements are Central to Science. The laws of science are discovered through measurements. They are hypothesized from a related set of real-world measurements. They are verified and refined by means of critically designed measurements. Any law that has been contradicted by even a single measurement must be discarded immediately and be replaced by another. Measurements are the final authority in science. This paradigm has been an essential ingredient for the development of science in Europe starting about 500 years ago. The centrality of measurements remains unaltered in science today. As scientists, we must learn all we can about measurements.
Measurements are Approximate. Let's suppose you measured the length of your pencil with a ruler. It is incorrect for you to claim, "My new pencil is exactly 192 millimeters long." If you were to use a more exact measuring device you might say, "Oops! My pencil is 192. millimeters long." Your first measurement is good to the nearest millimeter; your second is good to the nearest 0.01 mm. We say that both values are inexact or approximate; both are subject to measurement uncertainties (or errors). The rest of this note discusses these uncertainties and how they affect our confidence in our own measurment results.
Mistakes Versus ErroIS. The word "error" has a special non-colloquial meaning in science. Error is different from mistake. Mistakes, such as measuring a 32-em-Iong object to be 42 em, can be avoided. As we shall see, errors cannot be avoided, even by the most careful measurements. lienee, errors quantify the degree of confidence we have in the associated measurements.
Precision Versus Accuracy: Random and Systematic ErroIS. Let's go back to the example of the pencil. Suppose everyone in the class uses the same ruler, measures the pencil to the nearest millimeter, and all agree it is 192 mm long. All say that it couldn't be either 191 or 193 mm long. We say that the class has measured the length of the pencil to a precision of 1 mm. Precision is the reliability or repeatability of a measurement. Suppose that the instructor now points out, "You all have made the same mistake. You lined up one end of 1
the pencil and one end of the ruler together. The end of the ruler is worn badly; it doesn't
begin at zero. Try to remeasure the pencil by putting it in the middle of the ruler. Then find
the position of both ends." (see Table 1 below.) "Subtract one value from the other to find the
length." Now the class finds that the pencil is 187 mm long! How can this be? Both
measurements are equally precise. The second one is more accurate than the first, because a
systematic error (caused by the worn end of the ruler) is no longer there. A systematk
error is an effect that changes all measurements by the same amount or by the same
percentage. The class's experience with the ruler is a mirror of the history of science.
Systematic errors have often crept unsuspectedly into measurements. The only way to
eliminate systematic errors is to look carefully for them and to understand well the nature of
the experiment or measurement.
TABLE 1 Measure8ent of the Length of a Pencil.
Random Errors: We Can Not Avoid Them. Let's return to the example of the class. measurement of the length of a pencil; when measuring to the nearest millimeter, everyone got the same value. Let's try to push the precision further and ask each person to measure to the nearest tenth of a millimeter. Now disagreements appear. We find different values: 186.7, 187.0, 187.3 mm, as shown. Is someone making a mistake? No, even the most careful and skillful person will come up with values that vary by one- or two-tenths of a millimeter. Now we are at the limit of measurement by use of the naked eye and rulers. The unavoidable change in successive measurements, due to small irregularities in the ruler, difficulty in estimating precisely, and the like, is called a random error, or error for short.
Your Best Estimate. Thus far, you have been careful not to make any mistakes, you have
avoided all systematic errors, and you have narrowed your uncertainty to the random error of
measurement. What's next? Common sense tells you to take the average of several
measurements, called the arithmetic mean or mean. The algebraic expression for the
Left End (cm) Right End (em) Length (em) Deviation^ from Mean 10.16 28.83 18.67^ -0. 15.87 34.57^ 18.70^ 0. 20.22 38.95 18.73 +0.
is reported to be 0.999Solution. The worn end mof long; the rulerthe second causes isthe reported same error. to be 10 No cmmatter long. what the theleng~n other^ ot^ thehand,^ object, the uniform^ it^ will shrinkage^ appear^ toof^ bethe^ 1 mm meter^ longer stick^ than causes^ its truethe samevalue.^ On fractional or percentage error. We first note that the meter stick is actually 999 mm long. The 999-mm-long object would appear to be 1 m long and the error in
measuringThe percentage the lengtherror would-bewould be 0.1%.1-mm.. TheFor fractionalthe short object,error would the wornbe (1end mm causes I 999mm). an
error of 1 mm, a fractional error of (1 mm I 100 mm = 0.01), and a percentage
error of 1%. The shrinkage of a 10 cm length of the ruler is only one-tenth of the I 10 shrinkagecm) = 0.001, of 1 m.the sameThus asthe for error the islong 0.1 object. mm. TheThe fractional percentage errorerror is (^) is(0.1 mm again 0.1%. The uniform shrinkage or expansion of an meter stick or any other scale causes the same fractional or percentage error.
Si~cant Figures. RULE: Experimental values - both measurements and results calculated from these measurements - should always be reported with only one uncertain digit.
As you have seen in previous discussions, the right-most digit of any experimental
measurement contains an error. "Significant figures" are those digits which have significance
- which have meaning. If you had measured the length of this page with a mm-ruler and
given the result as 279.33 mm, you would be incorrect. The smallest division is a mm, and
you can estimate one more digit corresponding to a place between two divisions. You have
absolutely no information on hundredths of a mm. That is, in this measurement, the tenths
place is somewhat uncertain but the hundredths place is completely uncertain - it has no
experimental significance. So your reported value should be 279.3 but not 279.33, or 279.
Things were simple before calculators. If you carried out your calculations with a slide rule (an antiquated hand calculator known only to persons born before 1955) you would be limited to three figures. If you did them by hand, you would be only too happy to round off. But, because at the touch of a button you have eight or nine digits displayed before your eyes, you will have difficulty with this simple rule. The calculator makes life difficult because you must decide which of these digits are uncertain, and you must round off all but one uncertain digit. The rules for rounding off are as follows:
Examine the digits to be discarded.
If the first digit is larger than a 5, round up.
If the first digit is less than a 5, round down.
If the first digit is a 5 followed by other digits, at least one of which
is not zero, round up.
If the first digit is 5 followed only by zeros, or by no other digits at
all, round up or down to make the last digit retained even. [For
example, 2.55 is rounded to 2.6, while 2.45 is rounded to 2.4.]
This business of "significant figures" is the simplest of the error analysis, which you will try to
master throughout your scientific career. For the moment, let's look at a few examples to
establish useful rules about significant figures. To keep track of uncertain digits, they will be
emboldened and underlined in these examples. Suppose you have the object made up of
three separate component parts - a ball, a cylinder and a plate. You measure the masses by
using a precision analytical balance for the ball and the cylinder and a triple beam balance for
the plate. The masses are .282 gm, 79.545 gm and 422.23 gm. The total mass is
The result as written is incorrect - it has two uncertain digits. To conform to the rule, the
mass must be rounded up, and be written as 502.06 gm. Note that this isn't at all profound!
What we are saying in this example is that if you can only measure the mass of one part of
the object to a precision of hundredths of a gram, you cannot possibly know the mass of the
composite object to a precision of thousandths of a gram. The general rule for addition,
which works as well for subtraction is: When calculating an experimental result by
adding or subtracting experimental data, round off so only one uncertain digit
remains in the result.
Now suppose we want to find the area of a rectangle. The length, measured with a meter 5
- - -^ ----
significant figures. The question is, "Is there a simple way to track the error propagation?" The answer is yes. Again, the simple rules we will develop here mirror more rigorous results which you will learn later.
Let us calculate the area of a square, whose side is measured to be L = 6.71 :t 0.02 em. Let's
write this as
L=6.71x[1:t~], where the bracketed quantity is a mathematical construct, not a measured quantity. Here, 8 must be equal to ( 0.02 / 6.71 ). We get, upon squaring, A=L2= (6.71)2x[1:t~]x[1:t~], where we note that the first and the second 8 are completely correlated. In fact they are one and same. Hence, A=45.0X[1:t2~ +O(~2)].
This should remind you a lot of the beginning of the differential calculus. In fact the error analysis does deal with negligibly small quantities in the same way the calculus does. In the following discussion, our measurement errors will be referred to as standard errors and are denoted by the variable u with appropriate suffixes. You will learn the exact relationship between our measurement errors and standard errors later.
Propagation of ErroIS: Single Measurement. Let us assume that we made a measurement of an x and ask how the standard error Uxpropagates as different functions of x are computed. It is convenient to consider not only the error itself, but also the relative (fractional) error (ux / x). In error theory, we always consider the fractional error to be small compared to 1;
i.e., (ux/ x) < < 1.. (Large fractional errors are very unusual in physics. laboratories.) All
expressions that follow are based on this assumption.
Consider an arbitrary function z(x). analytical geometry, we obtain, o z = d z ox 0 = d z 0. Ox dx Z dx >. Now, suppose that z(x) of the form,
We wish to know Uz. Applying the concept learned in the
z=axn then Manipulating further, oz=naxn-1ox(;)=zn( ;)
ox
7
This specially simple relationship between two relative errors exists only for this particular
functional form. It is useful nevertheless, because so many error tracking operations in
practice involve this type of functions (e.g. unit conversions). For other functional forms, one
must go back to the general expression listed above.
Suppose you have a tiny postal scale that only weighs to 1 ounce, and you want to mail a ream (500 sheets) of paper. How much does it weigh? You cleverly take a packet 1 ounce. of fiveUnder sheets the assumptionof paper, putthat themall together,sheets of andpaper find have the thetotal same weight,weight to bethe weight of a ream is 100 oz, or 6 lb and 4 oz. You put on enough postage for a 7-lbestimate package. of the Willweight. your packageSuppose makeyour weighingit? That haddepends an error on theof error0.1 oz.of yourThen the ream would have an error of 100 x 0.1 oz = 10 oz. At most, (^) rour^ package^ would weight 6 lb 14 oz, and you are safe. In this example you mu tiplied your measured quantity and its error by 100. You can readily imagine the reverse, in which you have only a large set of scales, weigh a ream of paper in pounds, and find the weight of a five-sheet letter by division. In this case, .you would divide both your result and error by 100. To summarize: When a measured quantity is. multiplied (divided) by a constant, the absolute error is likewise multiplied (divided) by the same constant.
Propagation of Errors. More Than One Measurement. When we started the discussion of error propagation, we considered an example of calculating the area of a square. In that example, we took one single measurement, and squared the data. Because 8's were completely correlated, the error doubled. What happens if we add, subtract, multiply or divide two independently measured quantities? The answer is not simple. But it is clear that those 8's are not correlated. In fact there is a probability that they may partially cancel each other out. You will learn in Physics 353 the problem of "drunkard's walk" which involves the treatment of uncorrelated errors. We will borrow the result from that treatment.
When two or more independent measurements are combined, there are four simple rules to
remember. They are;
Rule #1: When two measurements are combined by addition or subtraction, use absolute enurs, and use the recipe - Given z=x~y with x, ax' y, Gy' then
az =Ja; + a; (A cocktail party phrase.. In addition of measured quantities, absolute
errors are added in quadrature.)
We measured ~ = 0.20(1), a = 0.52(2) radians, t = 2.0(1) and g = 980.5(2) cm /S2. Solutiona~sp~ays usingo~4.u~~~o~1 four rules listed.cm. Le~ us keepFirst, one themore predicteddigit than distance. what the Myrules calculator about the significant figures say; 6.34let us firstcalculate% errors. x 102 cm. They are: 5% for ~, 4% for To understand the propagation a, 5% for t, and of errors, 0.02% for g. Clearly, we need not worry about the error in g. (Rule #4) Error in sin ex=a.cosex=0. Error in eosex =a. sinex =0. 01"" 1.1% ErrorError inin JJcosexparan =5%=0.017 (from 5.3% II) 0. Error in t2 =0 .1.2'2 =10% Total error =v'102+5.32=11% 72 Thus, the final answer is L=(6.3:f:O.7)x102 em You may try to use the the general formula;
(
OL
)
2. ot =[g(sJ.nex -Jlcos ex) t]
(
OL )
2 [
t ]
2 011 = - g2 cosex
(~;r=[ gg2 (Cosex+IISinex)r Complete several lines of algebra and obtain the final answer.
