Measurements and error analysis (2023)

"It is better to be right than exactly wrong." –Alan Greenspan

measurement uncertainty

Some numbers are correct: Maria has 3 siblings and 2 + 2 = 4. However, allmeasurementsthey have a degree of uncertainty that can come from a variety of sources. The process of evaluating the uncertainty associated with a measurement result is often calleduncertainty analysisosyntax error.A complete statement of a measured value must include an estimate of the confidence level associated with the value. Correctly reporting an experimental result, together with its uncertainty, allows others to make judgments about the quality of the experiment and facilitates meaningful comparisons with other similar values ​​or a theoretical prediction. Without an estimate of uncertainty, it is impossible to answer the fundamental scientific question: "Does my result agree with a theoretical prediction or with the results of other experiments?" This question is essential to decide whether to confirm or refute a scientific hypothesis.When we take a measurement, we generally assume that there is an exact or true value, depending on how we define what is being measured. Although we may never know exactly this actual value, we try to find this optimal amount to the best of our ability with the time and resources available. Because we take measurements using different methods, or even take multiple measurements using the same method, we may get slightly different results. So how do we relate our results to our best estimate of this elusive subject?real bravery🇧🇷 The most common way to display the range of values ​​that we think contains the true value is:


measure = (best estimate ± uncertainty) units

Let's take an example. Suppose you want to find the mass of a gold ring that you want to sell to a friend. You don't want to jeopardize your friendship, so you'll want to get an accurate mass of the ring to ask for a fair market price. They estimate the mass to be 10-20 grams based on the weight you feel in your hand, but that's not a very accurate estimate. After doing some research, you find an electronic balance that shows a mass of 17.43 grams. Although this measure is much moreprecisethat the original estimate as knownprecise, and how sure are you that this measurement represents the true value of the mass of the ring? Since the scale's digital display is limited to 2 decimal places, you can enter the mass as

metro= 17,43 ± 0,01 g.

Suppose you are using the same electronic balance and you get several different readings: 17.46 g, 17.42 g, 17.44 g, so the average mass appears to be in the range of

17,44 ± 0,02 g.

By now, you can be sure that you know the mass of this ring to the nearest hundredth of a gram, but how do you know that the actual value is definitely between 17.43 g and 17.45 g? Honestly, you opt for another scale that reads 17.22 g. This amount is well below the first balance amount range, and under normal circumstances, you may not care, but you want to be fair to your friend. So, what are you doing now? The answer lies in knowing something about the precision of each instrument.To answer these questions, we must first define the termsprecisionmiprecision:

precisionIt is the degree of agreement between a measured value and a true or accepted value. MeasurementErroris the amount of inaccuracy.

precisionit is a measure of how well a result can be determined (without reference to a theoretical or true value). It is the degree of consistency and agreement between independent measurements of the same size; also the reliability or reproducibility of the result.

ouncertaintyThe estimate associated with a measurement must take into account both the accuracy and the precision of the measurement.

Surveillance:unfortunately the conditionsErrormiuncertaintythey are often used interchangeably to describe vagueness and imprecision. This use is so common that it is impossible to avoid it entirely. When you come across these terms, make sure you understand whether they refer to accuracy or precision, or both.Note that to determine theprecisiongiven measure, we need to know the ideal and true value. Sometimes we have a "book" reading that we know and we assume that this is our "ideal" reading and use it to estimateprecisionour result. Other times we know a theoretical value calculated from basic principles, and this can also be taken as an "ideal" value. But physics is an empirical science, which means that the theory must be validated by experiment, and not the other way around. We can escape these difficulties and maintain a useful definition ofprecisionWe assume that we can count on the best even if we do not know its true valueacceptedValue against which we can compare our experimental value.For our gold ring example, there is no accepted reference value, and both measured values ​​have the same precision, so we have no reason to believe one over the other. We could look up the precision specifications provided by the manufacturer for each scale (the appendix at the end of this lab manual contains precision data for most of the instruments you will use), but the best way to judge the precision of a measurement is to compare it. with an acquaintancemeeting🇧🇷 For this situation, it may be possible to calibrate the scales with a standard mass that is accurate within a close, traceable tolerance to aprimary mass standardat the National Institute of Standards and Technology (NIST). Scale calibration should eliminate discrepancies between readings and provide: aprecisemeasure of massPrecision is often also given quantitatively.relativeofractional uncertainty:


Relative uncertainty =

Measurements and error analysis (1)

measured quantity

Measurements and error analysis (2)


metro= 75,5 ± 0,5 g

has a fractional uncertainty of:

0.5 grams
75.5 grams
=0,006= 0,7 %.

Precision is often also given quantitatively.relative error:


relative error =

measured value - expected value
expected value

If the expected value ofmetrois 80.0 g, so the relative error is:

75,5 − 80,0
=−0,056 = −5,6%

Surveillance:The minus sign indicates that the measured value isnot lessthan the expected value.

When analyzing experimental data, it is important to understand the difference between precision and accuracy.precisionindicates the quality of the measurement, with no guarantee that the measurement is "correct".precisionon the other hand, it assumes an ideal value and tells you how far your answer is from this ideal "correct" answer. These concepts are directly relatedCoincidentallymisystematicMeasurement error.

types of errors

Measurement errors can be classified asCoincidentallyosystematic, depending on how the measurement was obtained (an instrument can introduce random error in one situation and systematic error in another).

random bugsare statistical fluctuations (in any direction) in the measured data due to precision limitations of the measuring device. Random errors can be assessed by statistical analysis and reduced by averaging a large number of observations (see standard errors).

systematic errorsthey are reproducible inaccuracies that constantly go in the same direction. These errors are difficult to detect and cannot be statistically analyzed. If bias is identified when calibrating against a standard, applying a correction or correction factor to compensate for the effect can reduce the bias. Unlike random errors, systematic errors cannot be detected or reduced by increasing the number of observations.

Our goal is to reduce as many sources of error as possible through careful measurement and to keep an eye on errors that we cannot eliminate. It is useful to know the types of errors that can occur so that we can detect them when they do occur. Common sources of error in experiments in the physics laboratory:

incomplete definition(can be systematic or random) - One of the reasons for the impossibility of making exact measurements is that the measurement is not always clearly defined. For example, if two different people were to measure the length of the same rope, they would probably get different results because each person can stretch the rope with a different tension. The best way to minimize definition errors is to carefully consider and specify the conditions that may affect the measurement.Error when considering a factor(usually systematic) – The hardest part of designing an experiment is trying to control or account for all possible factors other than the only independent variable being tested. For example, you may inadvertently ignore air resistance when measuring acceleration in free fall, or you may not take into account the effect of the Earth's magnetic field when measuring the field near a small magnet. The best way to explain these sources of error is to discuss with your colleagues all the factors that could affect your result. This brainstorming should be doneIn frontStart the experiment to plan and account for confounding factors before collecting data. sometimes acorrectioncan be applied to a resultafter thisTaking into account the data of an error that was not previously detected.environmental factors(systematic or random) - Watch for errors caused by your immediate working environment. You may need to accommodate or shield your experiment from vibrations, drafts, temperature changes, and electronic noise or other effects from nearby equipment.device resolution(random): All instruments have finite precision, which limits the ability to resolve small measurement differences. For example, a metric ruler cannot be used to distinguish distances with a precision much better than half its smallest division (0.5mm in this case). One of the best ways to get more accurate measurements is to use azero differencemethod instead of measuring a lot directly.NulloEquilibriumThe methods involve the use of instruments to measure the difference between two similar quantities, one of which is known with great precision and is adjustable. The adjustable reference quantity is varied until the difference is reduced to zero. The two quantities are then balanced, and the size of the unknown quantity can be determined by comparison with a standard of measurement. With this method, source instability problems are eliminated and the meter can be very sensitive and does not even need a scale.calibration(systematic) — Whenever possible, the calibration of an instrument should be verified before data is collected. If a calibration standard is not available, the accuracy of the device should be verified by comparing it to another device that is at least as accurate, or to the specifications provided by the manufacturer. Calibration errors are generally linear (measured as a fraction of full scale), so larger values ​​result in larger absolute change(systematically) — Whenever you take a measurement with a gauge micrometer, electronic balance, or electric meter, always check the zero reading first. If possible, zero the instrument or at least measure and record the zero offset so that the readings can be corrected later. It's also a good idea to check for null during your experiment. Failure to zero a device will result in a constant error that is more significant at smaller readings than at larger values.physical variations(random) — It is always a good idea to take multiple measurements over as large an area as possible. This often reveals deviations that would otherwise go undetected. These variances may require closer examination, or they may be combined to create an average.Parallax(systematic or random) – This error can occur when there is some distance between the measurement scale and the indicator used to obtain a measurement. If the viewer's eye is not aligned with the pointer and scale, the reading may be too high or too low (some analog gauges have mirrors to help with this alignment).instrument drift(systematic) – Most electronic instruments have readings that change over time. The amount of deviation is not usually an issue, but on occasion this source of error can be significant.make an appointmentmihysteresis(Systematic): Some meters take time to reach equilibrium and taking a reading before the instrument is stable will result in a higher or lower reading. A common example is temperature measurements with a thermometer that has not reached thermal equilibrium with the environment. A similar effect ishysteresiswhere instrument readings lag and appear to have a "memory effect" as data is acquired sequentially by moving up or down through a range of values. Hysteresis is most commonly associated with materials that become magnetized when a changing magnetic field is applied.personal mistakesthey come from carelessness, poor technique, or bias on the part of the experimenter. The experimenter may measure incorrectly or use poor technique when making a measurement, or introduce bias into the measurements by expecting (and inadvertently forcing) the results to match the expected result.

Serious personal mistakes, sometimes also calledErroroError, should be avoided and corrected when discovered. Personal errors are usually excluded from the discussion of error analysis, since it is generally assumed that the experimental result was obtained by doing the right thing.Also, the term human error should be avoided in discussions of error analysis, as it is too general to be useful..

Estimation of experimental uncertainty for a single measurement

Every measurement you make comes with some degree of uncertainty, no matter how precise your measurement tool is. So how do you determine and report this uncertainty?

The uncertainty of a single measurement is limited by the precision and accuracy of the measuring device, along with other factors that may affect the experimenter's ability to make the measurement.

For example, if you are trying to measure the diameter of a tennis ball with a metric ruler, the uncertainty might be higher.

± 5 mm,

but if you use calipers the uncertainty could be reduced to maybe

± 2 mm.

The limiting factor with the metric rule is parallax, while the second case is limited by the ambiguity in the definition of the tennis ball's diameter (it's inaccurate!). In both cases, the uncertainty is greater than the smallest graduation marked on the measuring tool (probably 1 mm and 0.05 mm respectively). Unfortunately, there is no general rule for determining uncertainty in all measurements. The experimenter is the one who can best estimate and quantify the uncertainty of a measurement based on all the possible factors that affect the result. Therefore, the person making the measurement is obliged to make the best possible assessment of the uncertainty and to express the uncertainty in a way that clearly explains what the uncertainty represents:


measurement = (measured value ± standard uncertainty) unit of measurement

where the ±standard uncertaintygives a confidence interval of approximately 68% (see sections on Standard Deviation and Statement of Uncertainties).
Example: diameter of tennis ball =

6,7 ± 0,2 cm.

Estimation of uncertainty in repeated measurements

Suppose you measure the period of a pendulum with a digital instrument (which you assume to measure accurately) and find:T= 0.44 seconds. This single period measurement suggests an accuracy of ±0.005 s, but the accuracy of this instrument may not provide a complete sense of uncertainty. By repeating the measurement several times and examining the scatter between readings, you can get a better picture of uncertainty over time. For example, here are the results of 5 measurements in seconds: 0.46, 0.44, 0.45, 0.44, 0.41.


Means (Means) =

X1+X2+Measurements and error analysis (3)+Xnorte

For this situation, the best estimate of the period is theAverage, omi.

If possible, repeat a measurement several times and average the results. This average is usually the best estimate of the "true" value (unless the data set is skewed by one or moredeviatethat should be examined to determine if they are incorrect data points that should be omitted from the average or valid measurements that require further investigation). In general, the more times you repeat a measurement, the better the estimate. However, be careful not to waste time taking more measurements than are necessary for the required accuracy.

As another example, consider measuring the width of a piece of paper with a metric ruler. Taking care to keep the ruler parallel to the edge of the paper (to avoid a systematic error that would cause the measured value to be constantly greater than the correct value), the width of the paper is measured at different points on the sheet and the obtained values ​​are inserted into a data table. Please note that the last digit is only a rough estimate, as it is difficult to read a metric ruler to the nearest tenth of a millimeter (0.01 cm).Measurements and error analysis (4)


average =

Sum of observed latitudes
Do not do. of the observations

(Video) Measurement and Error Lab

This average is the best available estimate of the width of the paper, but it is certainly not exact. We would have to average an infinite number of measurements to get close to the true mean, and even then we have no guarantee that the mean is correct.precisebecause there is stillnonesystematic error of the measurement tool that can never be calibratedPerfect🇧🇷 So how do we express uncertainty in our environment?One way to express the variation between measurements is to usemean deviation🇧🇷 This statistic tells us on average (with 50% confidence) how much individual readings deviate from the mean.



|X1X| + |X2X| +Measurements and error analysis (5)+ |XnorteX|

However theStandard deviationIt is the most common way to characterize the variability of a data set. EITHERStandard deviationis always a little bigger than thatmean deviation, and is used because of its association withnormal distributionwhich is often found in statistical analyses.

Standard deviation

To calculate the standard deviation of a sample ofnorteMeasurements:

  • 1

    Add all the measurements and dividenorteto get thatAverage, omi.
  • 2

    Now subtract thisAveragedecadenorteget actionnorte"deviations".
  • 3

    Squareeach one of thesenorte deviationsand add them all.
  • 4

    Share this result

    (norte− 1)

    and take the square root.

We can write the standard deviation formula as follows. leave it alonenortethe measures are calledX1,X2, ...,Xnorte🇧🇷 Leave the stockingnorteare called values


Any deviation is given by

dXUE=XUEX, ProUE= 1, 2,Measurements and error analysis (6),norte.

oStandard deviationEs:



Measurements and error analysis (7)
(dX12+dX22+Measurements and error analysis (8)+dXnorte2)
(norte− 1)
Measurements and error analysis (9)
Measurements and error analysis (10)
(norte− 1)

In our example above, the average width


measures 31.19cm The deviations are:Measurements and error analysis (11)oAveragedeviation is:

d= 0,086 cm.

omeetingdeviation is:


Measurements and error analysis (12)
(0,14)2+ (0,04)2+ (0,07)2+ (0,17)2+ (0,01)2
5 - 1
= 0,12 cm.

The meaning of the standard deviation is as follows: If you take another measurement using the same metric ruler, you can reasonably expect (with 68% confidence) that the new measurement will be within 0.12 cm of the estimated mean of 31, 19 feet cm. In fact, it makes sense to use the standard deviation as the associated uncertainty.not marriednew measure. But the uncertainty aboutAveragethe value is theStandard deviation Average, whatever isnot lessthan the standard deviation (see the next section).Consider an example where 100 measurements of a set were made. The mean or mean was 10.5 and the standard deviation wass= 1.83. The following figure is ahistogramof 100 measurements, showing how often a certain range of values ​​was measured. For example, 20 readings ranged from 9.5 to 10.5, and most of the readings werefencefor the mean of 10.5. standard deviationsfor this set of measurements is approximately how far from the meanmostof dropped readings. With a large enough sample, about 68% of the readings will be within one standard deviation of the mean, and 95% of the readings will be within the range.

X± 2 s,

and almost all (99.7%) of the readings are within 3 standard deviations of the mean. The smooth curve superimposed on the histogram is thegaussianoonormalDistribution predicted by theory for measurements with random errors. As more measurements are taken, the histogram more closely follows the Gaussian bell curve, but the standard deviation of the distribution remains about the same.

Measurements and error analysis (13)

illustration 1

Standard deviation of the mean (standard error)

If we report the average value ofnortemeasurements, the uncertainty that we have to assign to this average value is thestandard deviation of the mean, often calleddefault error(SE).



Measurements and error analysis (14)norte

odefault errorit's smaller than thatStandard deviationby a factor of


Measurements and error analysis (15)norte

This reflects the fact that we expect the mean uncertainty to decrease as we use a larger number of measurements,norte🇧🇷 In the example above, we found the standard error of 0.05 cm by dividing the standard deviation of 0.12

Measurements and error analysis (16)5

The final result should be given as follows:

Average paper width = 31.19 ± 0.05 cm.

abnormal date

The first step you should take when analyzing data (and even collecting data) is to examine the data set as a whole to look for patterns anddeviate.anomalousdata points that lieForThe general trend of the data may indicate an interesting phenomenon that may lead to a new discovery, or it may simply be the result of error or random fluctuations. In either case, an outlier should be investigated further to determine the cause of the unexpected result. Extreme data should never be "thrown away" without a clear justification and explanation, as the most important part of the research may be lost! However, if you can clearly justify omitting an inconsistent data point, you should exclude the outlier from your analysis so that the average is notwrongthe "true" means.

The uncertainty of the fracture is resumed

When a reported value is determined by averaging a series of independent readings, the fractional uncertainty is the ratio of the uncertainty divided by the average value. For this example


Bruchunsicherheit =

= 0,0016 ≈ 0,2 %

Note that fractional uncertainty is dimensionless, but is usually expressed as a percentage or in parts per million (ppm) to emphasize the fractional nature of the value. A scientist might also claim that this measurement is "good to about 1 in 500" or "accurate to about 0.2%."Fractional uncertainty is also important because it is used inPropagandaUncertainty in calculations using the result of a measurement, as discussed in the next section.

propagation of uncertainty

Suppose we want to determine a lotF, which dependsXand maybe several other variablesj,z, etc. We want to know the error inFwhen we measureX,j, ... with mistakespagX,pagj, ...Examples:


F=xy(area of ​​a rectangle)


F=pagwhyUE(X- moment component)



For a function of a single variableF(X), the deviation inFmay be related to the discrepancyXUse calculation:



Measurements and error analysis (17)


Measurements and error analysis (18)


So let's take the square and the mean:



Measurements and error analysis (19)

Measurements and error analysis (20)2

(Video) 4. What's Significant in Laboratory Measurement? Error Analysis Lecture

and with the definition ofpag, Have:



Measurements and error analysis (21)


Measurements and error analysis (22)




Measurements and error analysis (23)X


Measurements and error analysis (24)X



Measurements and error analysis (25)X
, o




= 2X


= 2


F= becauseUE


= −senUE


pagF🇺🇸 SinUE|pagUE, o

= | Sun tanningUE|pagUE

surveillance: in this situation,pagUEmust be specified in radians.

cae whereFdepends on two or more variables, the above derivation can be repeated with minor modifications. For two variables it applies:F(X,j), have:



Measurements and error analysis (26)


Measurements and error analysis (27)


Measurements and error analysis (28)


Measurements and error analysis (29)


A partial derivative


means to differentiateFupXKeep track of the other variables. Taking the square and the mean, we getLaw of Propagation of Uncertainty:



Measurements and error analysis (30)

Measurements and error analysis (31)2

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Measurements and error analysis (33)2

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Measurements and error analysis (37)

dX dj

If the measurementsXmijesnot correlated, after

dX dj= 0,

and we get:



Measurements and error analysis (38)

Measurements and error analysis (39)


Measurements and error analysis (40)

2 pagX2+

Measurements and error analysis (41)


Measurements and error analysis (42)

2 pagj2




(Video) Units and Measurements 06 || Error Analysis - Part 1 JEE/NEET

= 1,
= 1



Measurements and error analysis (43)pagX2+pagj2

When adding (or subtracting)Independentmeasures thatabsolute uncertaintythe sum (or difference) is the individual's root sum of squares (RSS)absolute uncertainties🇧🇷 By addingcorrelatedmeasurements, the uncertainty in the result is simply the sum of the absolute uncertainties, which is always an uncertainty estimate greater than the sum in quadrature (RSS). Adding or subtracting a constant does not change the absolute uncertainty of the calculated value as long as the constant is an exact value.







Measurements and error analysis (44)j2pagX2+X2pagj2

Divide the above equation byF=xy, Have:


Measurements and error analysis (45)

Measurements and error analysis (46)


Measurements and error analysis (47)


Measurements and error analysis (48)


Measurements and error analysis (49)





= −



Measurements and error analysis (50)

Measurements and error analysis (51)


Measurements and error analysis (52)


Measurements and error analysis (53)


Measurements and error analysis (54)


Divide the above equation by




Measurements and error analysis (55)

Measurements and error analysis (56)


Measurements and error analysis (57)


Measurements and error analysis (58)


Measurements and error analysis (59)


When multiplying (or dividing) independent measurements, therelative uncertaintyof the product (quotient) is the RSS of the individualrelative uncertainties🇧🇷 By multiplyingcorrelatedmeasurements, the uncertainty in the result is just the sum of the relative uncertainties, which is always an uncertainty estimate greater than quadrature addition (RSS). Multiplying or dividing by a constant does not change the relative uncertainty of the calculated value.

Note that the relative uncertainty inF, as shown in (b) and (c) above, has the same form for multiplication and division: the relative uncertainty in a product or quotient depends onrelativeUncertainty of each term.Example: find the uncertainty inv, From where


coma= 9,8 ± 0,1 m/s2,t= 1,2 ± 0,1 s


Measurements and error analysis (60)

Measurements and error analysis (61)


Measurements and error analysis (62)


Measurements and error analysis (63)


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Measurements and error analysis (68)


Measurements and error analysis (69)

Measurements and error analysis (70)(0,010)2+ (0,029)2
= 0.031 o 3.1%

Note that the relative uncertainty int(2.9%) is significantly greater than the relative uncertainty fora(1.0%) and therefore the relative uncertainty invis essentially the same as fort(about 3%).Graphically, RSS is like the Pythagorean theorem:

Measurements and error analysis (71)

Figure 2

The total uncertainty is the length of the hypotenuse of a right triangle with legs of the length of each uncertainty component.

Time saving approach:"A chain is only as strong as its weakest link."
If one of the uncertainty terms is more than three times greater than the other terms, the square root formula can be ignored and the combined uncertainty is simply the largest uncertainty. This shortcut can save a lot of time without sacrificing accuracy when estimating total uncertainty.

The upper-lower bound of the uncertainty propagation method

An alternative and sometimes simpler procedure to the tedious procedurePropagation of the Law of Uncertaintyand theUpper-Lower Bound Methodspread of uncertainty. This alternative method does not produce astandard uncertaintyEstimate (with a 68% confidence interval), but returns aappropriateUncertainty estimation for practically any situation. The basic idea of ​​this method is to use the uncertainty ranges of each variable to calculate the maximum and minimum values ​​of the function. You can also think of this technique as an examination of best and worst case scenarios. Suppose you measure an angle like this:UE= 25° ± 1° and you had to findF= becauseUE, after:


Fmaximum= cos(26°) = 0,8988


FMinimum= cos(24°) = 0,9135


F= 0,906 ±0,007

Measurements and error analysis (72)

where 0.007 is half the difference betweenFmaximummiFMinimum

Measurements and error analysis (73)

(Video) Percent Error Made Easy!

Note that even thoughUEwas measured with only 2 significant digits,Fis known by 3figuren. Use ofPropagation of the law of uncertainty:

pagF🇺🇸 SinUE|pagUE= (0,423)(Pi/180) =0,0074

(same result as above).

The uncertainty estimate of the upper-lower bound method is generally larger than the standard uncertainty estimate found in uncertainty law propagation, but both methods provide a reasonable estimate of the uncertainty in a computed value.

The upper-lower bound method is particularly useful when the functional relationship is unclear or incomplete. A practical application is forecasting the expected range in an expense budget. In this case, some costs may be fixed while others may be uncertain, and the range of these uncertain terms can be used to predict upper and lower bounds on total costs.

significant algharisms

The number of significant digits in a value can be defined as all digits between and including the first non-zero digit from the left to the last digit. For example, 0.44 has two significant digits and the number 66.770 has 5 significant digits. Zeros are significant except when used to locate the decimal point, such as in the number 0.00030, which has 2 significant digits. Zeros may or may not be significant for numbers like 1200 when it is not clear whether two, three, or four significant digits are shown. To avoid this ambiguity, these numbers should be expressed in scientific notation (for example, 1.20 × 103clearly shows three significant digits).When using a calculator, the display often shows many digits, only a few of which areimportant(it makes sense in another sense). For example, if you want to estimate the area of ​​a circular playing field, you can set the radius to 9 meters and use the formula:A=Pir2🇧🇷 When calculating this area, the calculator can report a value of 254.4690049 m2🇧🇷 It would be extremely misleading to give this number as the area of ​​the field, as it would suggest that you know the area with an absurd degree of precision, down to a fraction of a square millimeter! Since only the radius of one significant digit is known, the final answer must also contain only one significant digit: area = 3 × 102metro2.From this example, we can see that the number of significant numbers reported for a value implies a certain level of precision. In fact, the number of significant numbers suggests a rough estimate of the relative uncertainty:

The number of significant digits implies an approximate relative uncertainty:
1 significant number indicates a relative uncertainty of approximately 10% to 100%
2 significant figures indicate a relative uncertainty of approximately 1% to 10%
3 significant figures indicate a relative uncertainty of approximately 0.1% to 1%

To better understand this relationship, consider a value with 2 significant digits, such as 99, indicating an uncertainty of ±1 or a relative uncertainty of ±1/99 = ±1%. (In fact, some people might argue that the implied uncertainty in 99 is ±0.5 since the range of values ​​that would round to 99 is 98.5 to 99.4. But since the uncertainty here is just an estimate It doesn't make much sense to argue about the factor of 2.) The smallest number with 2 significant digits, 10, also indicates an uncertainty of ±1, which in this case corresponds to a relative uncertainty of ±1/10 = ±10 %. Ranges for other significant digit numbers can be similarly justified.

Use significant numbers to easily propagate uncertainty

By following a few simple rules, significant numbers can be used to find the proper precision for a calculated result for the four most basic mathematical functions, without using complicated uncertainty propagation formulas.

In multiplication and division, the number of reliably known significant digits in a product or quotient is the same as the smallest number of significant digits in any of the original numbers.


48369.42=48 × 103
(2 significant digits)
(5 significant digits)
(2 significant digits)

For addition and subtraction, the result must be rounded to the last decimal place entered for the less precise number.



If a calculated number is to be used in subsequent calculations, it's a good idea to keep an extra digit to reduce rounding errors that can accumulate. The final answer should then be rounded according to the guidelines above.

Uncertainty, significant numbers, and rounding

For the same reason that it is dishonest to state a result with more significant numbers than are known with certainty, the value of uncertainty should also not be stated with undue precision.For example, it would be inappropriate for a student to report a result such as:


measured density = 8.93 ± 0.475328 g/cm3INCORRECT!

The measurement uncertainty cannot be known exactly! For most experimental work, the confidence in the uncertainty estimate is not much better than about ±50% because of all the different sources of error, none of which can be known exactly. Therefore, uncertainty values ​​should be given to a single significant digit (or perhaps 2 significant digits if the first digit is 1).

Because experimental uncertainties are inherently imprecise, they should be rounded to one, or at most, to two significant digits.

To give you an idea of ​​how confident you can be in the standard deviation, the following table provides the relative uncertainty associated with the standard deviation for various sample sizes. Note that it would take over 10,000 measurements to justify this level of precision to report an uncertainty value with 3 significant digits!Measurements and error analysis (74)*The relative uncertainty results from the approximation formula:

Measurements and error analysis (75)2(norte− 1)

When making an explicit uncertainty estimate, the uncertainty term indicates how many significant digits should be given in the measurement (not the other way around!). For example, the uncertainty in the above density measurement is about 0.5 g/cm3, so this tells us that the digit in the tenths place is uncertain and should be entered last. The other digits in the hundreds digit and beyond are irrelevant and should not be reported:

measured density = 8.9 ± 0.5 g/cm3.


An experimental value should be rounded to be consistent with the magnitude of its uncertainty. This generally means that the last significant digit in any reported value must be reported to the same decimal place as the uncertainty.

In most cases, this practice of rounding an experimental result to be consistent with the estimate of uncertainty produces the same number of significant digits as the rules discussed above for simple propagation of uncertainty for addition, subtraction, multiplication, and divisions.

Caution:When conducting an experiment, it is important to keep this in mind.precision is expensive(both in terms of time and material). Don't waste time trying to get an accurate result when only a rough estimate is required. The cost increases exponentially with the precision required, so the potential benefit of that precision must be weighed against the additional cost.

Combine and report uncertainties

In 1993, the International Organization for Standardization (ISO) published the first official version of theGuide to express measurement uncertainties.Prior to this time, uncertainty estimates were evaluated and reported according to different conventions depending on the measurement context or scientific discipline. Here are some key points from this 100-page guide, which can be found in modified form atTuned NIST website.When reporting a measurement, the measured value should be reported along with an estimate of the totalcombined standard uncertainty


of value. The total uncertainty is found by combining the uncertainty components based on the two types of uncertainty analysis:
  • Type A evaluation of standard uncertainty- Uncertainty evaluation method through statistical analysis of a set of observations. This method mainly involvesCoincidentallyError.
  • Type B evaluation of standard uncertainty- Method of determination of uncertainty by means other than the statistical analysis of series of observations. This method includessystematicErrors and any other uncertainties that the experimenter considers important.
The individual components of uncertaintyUdsUEmust be combined withLaw of Propagation of Uncertainty, commonly known as the "Root Sum of Squares" or "RSS" method. Once this is done, the pooled standard uncertainty should equal the standard deviation of the result, so that this uncertainty value corresponds to a 68% confidence interval. If a wider confidence interval is desired, the uncertainty can be multiplied by acoverage factor(usuallyk= 2 or 3) to provide an uncertainty range that is assumed to contain the true value with 95% confidence (eg.k= 2) o 99.7% (p. ej.k= 3). If an expansion factor is used, its meaning must be clearly explained so as not to confuse the reader when interpreting the meaning of the uncertainty value.You should be aware that the uncertainty notation ± can be used to indicate different confidence intervals depending on the scientific discipline or context. For example, an opinion poll may report that the results of aerror range±3%, which means that readers can be 95% (not 68%) confident that the results reported are accurate to within 3 percentage points. also a makertoleranceThe classification usually assumes a confidence level of 95% or 99%.

Conclusion: "When do the measurements match up?"

We now have the resources to answer the fundamental scientific question posed at the beginning of this discussion of error analysis: "Does my result agree with a theoretical prediction or with the results of other experiments?"In general, a measured result agrees with a theoretical prediction if the prediction is within the range of experimental uncertainty. The same applies if two measured values ​​are availablestandard uncertaintyIntervals that overlap, so it says in the measuresconsistent(They agree). If the uncertainty bands do not overlap, one speaks of measurementsdeviate(they disagree). However, you should recognize that these overlapping criteria may provide two opposing answers depending on the assessment and the level of confidence in the uncertainty. It would be unethical to arbitrarily inflate the uncertainty range just to make a measurement match an expected value. A better approach would be to discuss the size of the difference between the measured and expected values ​​in the context of uncertainty and try to find the source of the discrepancy if the difference is really significant. To verify your own data, you must use themeasurement comparison toolavailable inLabor website.Here are some examples of using this chart analysis tool:

Measurements and error analysis (76)

figure 3

A= 1,2 ± 0,4

B= 1,8 ± 0,4

These measuresto acceptinside his insecurities, even though thePercentage differencebetween its central values ​​is 40%.But with mean uncertainty ± 0.2, same measurementsI disagreesince their uncertainties do not overlap. Further investigation would be necessary to determine the cause of the discrepancy. Perhaps the uncertainties were underestimated, a systematic error may have been overlooked, or there may have been a real difference between these values.

Measurements and error analysis (77)

Figure 4

An alternative method of determining agreement between values ​​is to calculate the difference between the values ​​divided by their pooled standard uncertainty. This ratio gives the number of standard deviations that separate the two values. If this ratio is less than 1.0, it is reasonable to assume that the values ​​agree. If the ratio is greater than 2.0, it is very unlikely (less than 5% chance) that the values ​​are equal.Example above with

Uds= 0,4:

|1,2 − 1,8|
= 1,1.

That's why,AmiByou probably agree.Example above with

Uds= 0,2:

|1,2 − 1,8|
= 2,1.

Therefore, it is unlikely thatAmiBto accept.


Baird, DCExperimentation: An Introduction to Measurement Theory and Design of Experiments, 3rd Editionthird🇧🇷 editionLehrlingshalle: Englewood Cliffs, 1995.Bevington, Phillip y Robinson, D.Data Reduction and Error Analysis for the Physical Sciences, 2North Dakota🇧🇷 editionMcGraw-Hill : New York , 1991 .THEY ARE LIKE THIS.Guide to express measurement uncertainties.International Organization for Standardization (ISO) and International Committee for Weights and Measures (CIPM): Switzerland, 1993.Lights, William.Data analysis and errors., 2North Dakota🇧🇷 editionApprentice Hall: Upper Saddle River, NJ, 1999.NIST.Fundamentals of the expression of measurement uncertainty., Juan.Introduction to error analysis, 2North Dakota🇧🇷 editionUniversity Academic Books: Sausalito, 1997.

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