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PHYS 1007 Fall 2021

C. Uncertainties and Significant Figures A measurement is an attempt to determine the best estimate of the true value of a given quantity. Every time that a measurement is performed, one could obtain slightly different results. In science, to account for that variation, numerical results are always expressed as:

(𝒎 ± 𝝈𝒎 ) 𝒖𝒏𝒊𝒕𝒔

where 𝑚 is the measured value and 𝜎𝑚 is its associated uncertainty. Uncertainties are due to the limits on the precision that can be achieved in an experiment and no faults or mistakes are implied on the part of the experimenter or apparatus. From statistical theory, the uncertainty has probabilistic connotations: we say that there is a 68% chance that the true value lies within ±𝜎𝑚 of the measured value. In other words, if we were to repeat the same measurement, at least 68% of the time we would obtain a value in the [𝑚 − 𝜎𝑚 , 𝑚 + 𝜎𝑚 ] interval. Experimental uncertainties can be classified as systematic uncertainties and random uncertainties: •

Systematic uncertainties refer to the effects of procedure, apparatus or method, which equally affect all measurements of a particular quantity. Some examples of systematic uncertainties are: - An incorrect instrument calibration that would produce a bias in one direction. For example, a manufacturer supplied meter-stick that is subdivided into 1000 x 1 mm divisions but is actually 91.5 cm long. - Experimental conditions. For example, using the instrument under other conditions than those for which it was calibrated. - Assumptions in the theory. For example, not accounting for the friction of the surfaces in an inclined plane experiment. Systematic uncertainties must be identified and corrected at the earliest practical opportunity in the measurement process.

•

Random uncertainties arise from the fluctuations in results of repeated measurements of the same quantity that vary in an unpredictable manner. Some examples of random uncertainties are: - Noise in the data. For example, electrical noise in a circuit causing fluctuations in the measured voltages. - Intrinsic variations in the definition of the measured quantity. For example, measuring the length of a rectangular table with rough edges. - Fluctuations in the environment. For example, temperature and pressure variations in the laboratory. Random uncertainties may be analyzed statistically.

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PHYS 1007 Fall 2021

Accuracy and Precision If an experiment has low systematic uncertainty, it is said to be accurate. Accuracy describes the closeness of agreement between the measured quantity and its true value. A measurement, 𝑚, is said to have high accuracy if its relative difference with respect to the true value, 𝑥𝑡𝑟𝑢𝑒 , i.e.

|

𝑚−𝑥𝑡𝑟𝑢𝑒 𝑥𝑡𝑟𝑢𝑒

|, is small.

If the experiment has low random uncertainty, it is said to be precise. Precision describes the unpredictable variability of repeated measurements of a quantity. A measurement is said to have 𝜎 high precision if its relative uncertainty, i.e. | 𝑚 ⁄𝑚 |, is small.

In an experiment, one attempts for high precision. However, low precision measurements are also useful. If you are told that the distance to Carleton Place is 30 ± 10 km, the precision of such value is poor, but the overall measurement is good enough to allow you to decide whether to walk, take your bike or drive. The precision associated with any measurement also gives the experimenter an indication of what effort is needed to significantly improve the experiment.

C.1

Instrumental Uncertainty

When estimating the uncertainty of a single measurement, one is mainly limited by the precision and accuracy of the measuring instrument used. Under optimum conditions, say when the scale is very sharply scribed and the reference mark or pointer has a sharp edge, sometimes it is easy to estimate to a tenth of a division with acceptable reliability. Under some conditions, such as when there is a large gap between the division mark and the reference mark, it may not be possible to estimate to better than a half of a division, even one division. It is up to the experimenter to decide how reliably he or she can read the scale and consequently how large an instrumental uncertainty to quote.

C.2

Statistical Uncertainty

If multiple measurements are made of the same quantity, one can determine the uncertainty in the measurement using statistical procedures. In most measurements, there are a number of factors contributing to fluctuations in the observed quantity. The measured quantity, itself, can change from one observation to another or the pointer on a meter may give slightly different readings for the same input. Added to these, the observer's estimates of the pointer position may produce other variations, which further increase the fluctuations. In most situations, most of the sources of variations add up in such a manner that the end effect is indistinguishable from that of a single random source. This means that a result of one reading is independent of the rest. In that case, it turns out that the distribution of results follows the so-called

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PHYS 1007 Fall 2021

Gaussian or Normal distribution. Many other distributions (binomial, Poisson, etc.) for large number of events also tend to a Gauss distribution when added together. Thus, we can consider the total uncertainty as the effect of a single Gaussian source of random uncertainty. You will recognize the Gaussian distribution from its shape (see Figure 1) and by its functional form, which can be written as: 𝑓(𝑥) =

1

𝜎𝑆𝐷 √2𝜋

𝑒

−

(𝑥−𝑥𝑜 )2 2𝜎𝑆𝐷 2

(C.1)

The function is normalized (∫−∞ 𝑓(𝑥)𝑑𝑥 = 1), so if one measures 𝑥 something must occur. The +∞

probability that 𝑥 will have values between 𝑥1 and 𝑥2 is 𝑝(𝑥1 < 𝑥 < 𝑥2 ) = ∫𝑥 2 𝑓(𝑥)𝑑𝑥. This last 𝑥

1

integral only can be calculated through numerical integration. The expected, or average value of 𝑥 is: ∫

+∞

−∞

𝑥𝑓(𝑥)𝑑𝑥 = 𝑥𝑜

(C.2)

2 and the standard deviation (𝜎𝑆𝐷 ) is the square root of the variance (𝜎𝑆𝐷 ): 2 =∫ 𝜎𝑆𝐷

+∞

−∞

(𝑥 − 𝑥𝑜 )2 𝑓(𝑥)𝑑𝑥

(C.3)

We suppose that the quantity being measured, e.g. 𝑥, is well defined, that is, it has an actual definite value. For example, one could measure the distance, 𝑑 , covered by a falling object, starting from rest, in exactly one second. Different measurements will give different 𝑥𝑖 , where the subscript 𝑖, is just a label or tag we use to identify the specific measurement Average, Standard deviation and Standard deviation of the mean

Figure 1: The Gaussian or bell curve distribution. If a single measurement is repeated, the result will be in the interval [xo-σSD, xo+σSD] with probability 0.68, and between [xo-2σSD, xo+2σSD] with the probability of 0.95.

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PHYS 1007 Fall 2021

Now, assume that 𝑁 measurements were performed with results {𝑥1 , 𝑥2 , … 𝑥𝑁 }, the best description of the real value of 𝑥 is the average value (𝒙𝒂𝒗 ): 𝑥𝑎𝑣

𝑁

1 = 𝑁 ∑ 𝑥𝑖 𝑖=1

(C.4)

The sample standard deviation (𝝈𝑺𝑫 ) can tell what result to expect if another single measurement is added to the previously made 𝑁 measurements. This 𝑁 + 1 measurement will be inside (𝑥𝑎𝑣 − 𝜎𝑆𝐷 , 𝑥𝑎𝑣 + 𝜎𝑆𝐷 ) with a 0.68 probability and inside (𝑥𝑎𝑣 − 2𝜎𝑆𝐷 , 𝑥𝑎𝑣 + 2𝜎𝑆𝐷 ) with a 0.95 probability. It is also an indication of how spread out is the data. 𝜎𝑆𝐷

𝑁

1 =√ ∑( 𝑥𝑖 − 𝑥𝑎𝑣 )2 𝑁−1 𝑖=1

(C.5)

A particular (𝑥𝑖 − 𝑥𝑎𝑣 ) may be positive or negative, but by squaring them, one deals with positive quantities only. If all 𝑥𝑖 = 𝑥𝑎𝑣 then 𝜎𝑆𝐷 = 0. On the other hand, if results are scattered away from the average, 𝜎𝑆𝐷 will become larger.

Figure 2: Gauss distribution diagrams for different σ.

The standard deviation of the mean (𝝈𝒎𝒆𝒂𝒏) is used if one intends to repeat the whole experiment, i.e. to repeat all 𝑁 measurements. Then the average value of that new experiment, 𝑛𝑒𝑤 𝑥𝑎𝑣 , will belong to the interval (𝑥𝑎𝑣 − 𝜎𝑚𝑒𝑎𝑛 , 𝑥𝑎𝑣 + 𝜎𝑚𝑒𝑎𝑛 ) with the probability 0.68, where: 𝜎𝑚𝑒𝑎𝑛 =

𝜎𝑆𝐷

√𝑁

(C.6)

If one can increase 𝑁, (number of measurements), it usually means a better sample mean and smaller standard deviation of the mean.

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PHYS 1007 Fall 2021

It can be shown that if the same experiment with 𝑁 measurements were repeated many times, all the new values of 𝑥𝑎𝑣 would follow a Gaussian distribution with a width equal to the standard deviation of the mean.

The result of an experiment is then reported as the sample mean, 𝑥𝑎𝑣 , and the associated uncertainty is 𝜎𝑚𝑒𝑎𝑛 . The usual way of writing this is: 𝑥 = (𝑥𝑎𝑣 ± 𝜎𝑚𝑒𝑎𝑛 ) units

e.g.: 𝑔 = (9.78 ± 0.03)

C.3

𝑚

(C.7)

𝑠2

Propagation of Uncertainty

A quantity may often be calculated using a functional relationship involving measured quantities and their measured uncertainties. If a quantity 𝑓 is determined from input variables 𝑥, 𝑦, … , 𝑧 through a functional relationship 𝑓(𝑥, 𝑦, … , 𝑧), then the uncertainties 𝜎𝑥 , 𝜎𝑦 , … , 𝜎𝑧 will propagate through the calculation to an uncertainty 𝜎𝑓 .

Assuming that the uncertainties on the different variables are not correlated (random) and that the variables themselves are independent, one can determine the propagated uncertainty 𝜎𝑓 from the Pythagorean theorem; taking the square root of the sum of the uncertainties squared. In the case of a more complex calculation with several variables, the dependence of each variable has to be taken into account using partial derivatives, as follows:

where

𝜕𝑓

𝜕𝑥

𝜎𝑓 = √(

𝜕𝑓

𝜕𝑥

)

2

𝜎𝑥2

𝜕𝑓 2 2 + ( ) 𝜎𝑦 + ⋯ 𝜕𝑦

(C.8)

denotes the partial derivative of the function 𝑓 with respect to 𝑥 . When the partial

derivative is taken with respect to one variable, the other variables are treated as constants. Keep in mind that the partial derivative (and ordinary derivative, as well) tells you how much the function changes with respect to the said variable. Multiplying the derivative by the corresponding uncertainty gives a measure of the contribution of the uncertainty on the function. Example 1: Sum or difference of two variables with constant coefficients (𝒂, 𝒃). For some commonly recurring expressions calculated uncertainties are: 𝑓(𝑥, 𝑦) = 𝑎𝑥 ± 𝑏𝑦

𝜎𝑓 = √𝑎2 𝜎𝑥2 + 𝑏 2 𝜎𝑦2

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PHYS 1007 Fall 2021

Example 2: Products and quotients

𝑓(𝑥, 𝑦, 𝑧) = 𝑥 𝑚 𝑦 𝑛 𝑧 𝑝 𝜎𝑥

2

𝜎𝑦 2 𝜎𝑧 ) ) + 𝑛2 ( ) + 𝑝2 ( 𝑧 𝑥 𝑦 The formula above is applicable to both products and quotients since it is valid for both positive and negative powers 𝑚, 𝑛 and 𝑝. 𝜎𝑓 = 𝑓(𝑥, 𝑦, 𝑧)√𝑚2 (

C.4

2

Comparing Results

It is expected that there will be discrepancies between the results of different measurements of the same quantity. It is important to know whether these results are in agreement or not. If they don’t agree, then one or both of the results are wrong and they contain unknown systematic uncertainties. Suppose that 𝑥1 and 𝑥2 , with uncertainties 𝜎𝑥1 and 𝜎𝑥2 respectively, are the results of two measurements of the same quantity. We evaluate the consistency of two measurements using the t-test: 𝑡=

|𝑥1 − 𝑥2 |

√𝜎𝑥21 + 𝜎𝑥22

(C.9)

This ratio can serve as a measure of how reliable the results are. Assuming that the measured quantities follow a Gaussian distribution, their difference (discrepancy) will also have the same distribution. A difference of one 𝜎 is expected to occur 32% of the time, of 2𝜎 (or greater) about 5% of the time, and of 3𝜎 (or greater) about 0.3% of the time. In science, a new discovery usually requires a 5𝜎 discrepancy from a normal measurement. In this course the choice for a reasonable uncertainty is 2𝜎, thus, 𝑡 ≤2 𝑡 >2

𝑥1 and 𝑥2 are consistent (there is a 95% probability that 𝑥1 and 𝑥2 represent the same quantity) 𝑥1 and 𝑥2 are inconsistent

(C.10) (C.11)

If one of the two measurement lacks information about its uncertainty or if a comparison with a given value is required, it is acceptable to assume that 𝜎𝑥2 = 0.

NOTE: 𝜎𝑥1 and 𝜎𝑥2 are NOT the standard deviations, they are the uncertainties associated with the two

values being compared. The origin of these uncertainties (whether propagated, statistical or instrumental) are irrelevant for the purpose of the consistency test.

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PHYS 1007 Fall 2021

C.5

Averaging Results

When combining results, great care must be taken to calculate the right average. If numbers with different uncertainties are combined, it is logical that those with the greater uncertainties should have less effect, or less weight; this is why the weighted average is the best way to combine results if they are consistent. Consistent Results From statistical theory, it can be shown that the appropriate weight factor, when averaging consistent numbers, is 1/𝜎 2 . Then, two consistent measurements 𝑥1 ± 𝜎𝑥1 and 𝑥2 ± 𝜎𝑥2 can be combined into a weighted average with its corresponding uncertainty as follows 𝑥𝑎𝑣

𝑥1 𝜎𝑥22 + 𝑥2 𝜎𝑥21 = 𝜎𝑥21 + 𝜎𝑥22

𝜎𝑥𝑎𝑣 =

Consistent results

𝜎𝑥1 𝜎𝑥2

√𝜎𝑥21 + 𝜎𝑥22

(C.12)

(C.13)

Inconsistent Results allowed one, (

∆

> 2 ), then the above procedure is not justified. The fact that a number of

When a set of data is inconsistent, i.e. when the discrepancies between them are greater than the 𝜎∆

measurements of a quantity are not consistent indicates that there are some unknown systematic uncertaintys in some or all of the measurements.

Depending on how much information is available about the set of results, two approaches can be taken. The first is to study the methods of measurement carefully and decide which of the experiments are from a consistent subgroup, and seem most reliable, i.e., use well understood methods which have minimal systematic uncertainties. This subgroup can then be handled as a consistent set using the above method. The remaining data are discarded. If there is no way of deciding which of the data are more reliable, a second approach must be taken. The actual value sought is more likely to be somewhere within the cluster than outside, so the arithmetic average is used. The only statement that can be made about the actual value is that it is likely to be in the region bounded by the more extreme data points and their uncertainties. For two inconsistent results, 𝑥1 ± 𝜎𝑥1 and 𝑥2 ± 𝜎𝑥2 , one can adopt the following expressions as an average and corresponding undecertainty:

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PHYS 1007 Fall 2021

Inconsistent results

C.6

𝜎𝑥𝑎𝑣 =

𝑥𝑎𝑣 =

(𝑥1 + 2 𝑥2 )

|𝑥1 − 𝑥2 | + (𝜎𝑥1 + 𝜎𝑥2 ) 2

(C.14) (C.15)

Significant Figures and Rounding Numbers

In science, the numbers entered into calculations have a precision bounded by the limits of measuring instruments and measuring uncertainties. Measurements in an undergraduate laboratory are seldom known to better than one percent, or at best a few parts per thousand. This means that only the first three or four most significant digits of these numbers have any meaning. The question is how to determine the number of digits to use for measured quantities and for quoting the result of an experiment.

Significant figures •

All non-zero digits are significant.

•

The leftmost nonzero digit is the most significant figure; o If there is no decimal point the last nonzero digit is the least significant figure; (Example: 1004500 has 5 significant figures) o If there is a decimal point, the rightmost digit is the least significant figure, zero included; (Example: 0.00230 has 3 significant figures)

•

In scientific notation the number of digits written is the number of significant figures. (Example: 100 would have 1 significant figure, but 1.00×102 has 3 significant figures)

Rounding Results 1. Round the uncertainty, 𝜎𝑚 , to 1 significant figure.

2. Round the value, 𝑚, to match the same number of decimal places as the uncertainty.

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PHYS 1007 Fall 2021

Small numerical uncertainties introduced by rounding numbers can accumulate if several calculations are required. Consequently, a few extra significant digits, say three or four, should be kept in all intermediate results through all calculations. Numbers are then rounded to the correct number of significant figures for presentation in tables and in the final results.

Table 1: Example of steps to round results

Before rounding Rounding uncertainty Matching the value Final Result

𝑥 = 4.58931 mm

𝑥 = 4.589 mm

𝜎𝑥 = 0.003212 mm 𝜎𝑥 = 0.003 mm

𝑥 = (4.589 ± 0.003 ) mm

Table 2: More example of significant figures for propagated uncertainties

Before Rounding

After Rounding

(1734 ± 0.92) g

(1734.0 ± 0.9) g

(19.3471 ± 0.131 )mm

(19.4 ± 0.1) mm

(0.3426 ± 0.2) cm

(0.3 ± 0.2 ) cm

(0.237810954 ± 0.04593) V

(0.24 ± 0.05) V

(53.33 × 10−9 ± 2.315 × 10 −10) m

(5.33 ± 0.02) × 10−8 m***

***When using scientific notation, the factor of ten multiplier should be the same for the measurement and the uncertainty.

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