Establishing Uncertainty


Every number used in the laboratory must be recorded with an uncertainty.


This appendix will discuss methods for obtaining that uncertainty. This section should be used in conjunction with the Error Analysis section, which contains the definitions, formulas and some additional examples. 


Error and uncertainty have different meanings.

  • Error is the difference between a value and its correct value or true value. The true value, of course, is not known.

·       Uncertainty is an estimate of the difference between a calculated or measured value and the true value.

An instrument measures values that are in error by a certain amount. Since the exact true value of a quantity is unknown, instrumental error is also unknown. The experimenter is forced to estimate and to judge the estimation. The measured value is an estimate for the true value and the uncertainty is an estimate for the error.


It will be important to understand what is an acceptable difference between two results. How much does one allow results to differ before the experiment is judged to have a serious flaw? The uncertainty is used to compare two values. It provides insight, when comparing two consecutive measurements, when comparing results with the theory, and when comparing two independent experiments.  Experimenters know that their uncertainty is a safety net. Because you have an uncertainty, your results cover a range of possible values. Large uncertainties make an experiment very defensible. The result cannot be called into question if the uncertainty is so large that all reasonable results are included. On the other hand, an extremely small uncertainty is a sign of a high quality experiment, the smaller the uncertainty the better the experiment.  This means that the optimal uncertainty has to balance these two opposing goals. The uncertainty should not be so small as to guarantee failure, nor so large that the experiment has no merit.  It is, of course, unethical to arbitrarily increase or decrease an uncertainty without justification. Knowing that the uncertainty you claim determines the correctness and quality of the result under all future scrutiny, many experimenters spend considerable effort searching for unknown sources of error (increasing uncertainty) and pushing the limits of a technique (minimizing uncertainty ).





There are different approaches for establishing uncertainty but in this lab the focus will be on common sense rather than rigorous mathematical analysis. With this laboratory course focusing on different aspects of the experimental process, some of the rigor that is required for a real experiment will be relaxed. Often a student can use a simple straightforward method for guessing an uncertainty.  Be sure to check with you lab instructor if you are not sure of your method.

  • The scale on the ruler can be read to about 0.5mm. The uncertainty can be estimated based on this limitation to be 0.5 mm.
  • A few   independent measurements (3 trials) can be used to calculate SD and SDM (see below). One of these two values could serve as the uncertainty. The discussion below should help you decide which one to use in any given situation.
  • Some instruments may show no indication of error in their measurement. In an actual experiment a separate measurement could be used to determine the uncertainty or the experimenter could consult the instrument manual. In this lab the student may conclude that the instrument contributes a negligible error.  This is true, for example for some voltage measurements.
  • Instrument uncertainty is sometimes given in a manual.
  • Tex book constants are typically good to 3 significant figures.
  • Your instructor may prefer to provide the uncertainty for some quantities.


Human Error, Hand waving arguments– Student is not allowed to introduce an uncertainty unless the student has developed a method to measure that uncertainty. There will be no hand waving arguments permitted.  If you believe that your experiment may be subject to errors that have not been included then you either develop a method to measure the uncertainty or you ignore it.


Absolutely no error can be introduced into any discussion unless some quantitative estimate can be made for its size and all estimates need to be justified.


The correct approach is to find a way to estimate these additional uncertainties, include them and reevaluate or to simply state that the experiment does not agree with theory within the uncertainty.


Not allowed:

·       The results are in agreement with predictions because in addition to the uncertainty measured there were some effects due to wind resistance.

·       We suspect that human error and an uneven table are the source of the difference between our result and the theoretical result.


·       By measuring for a longer time period we were able to see a loss in energy over many oscillations due to friction. As shown in the figure this resulted in a 2% change in the energy for one period. …

·       Comparing the measurements of different students we were able to see an average deviation of 3 mm, which we attribute to a differences in reaction times.  We therefore are factoring in a 3mm uncertainty in our analysis. …

·       This experiment includes all of the measurable uncertainties that we found. The result for g, however, differs significantly from the accepted value (4 times the uncertainty). Re-examining possible pitfalls and carefully re-measuring g did not change our result.



You will be expected to include all the important sources of errors but you cannot merely state that something might be a source of error. You need to provide a justifiable guess as to how large it is. If you decide wind resistance might have influenced your measurement then, in order to mention it, you must think of a way to figure out how large an influence it is. If you cannot find all the errors in your experiment, you may have to conclude that your experiment failed to demonstrate the principle the lab was exploring.  It is okay to have a failed experiment as long as another experimenter using the same equipment would get the same result.  There are some sources of error that are to complex or subtle to be discovered in an introductory lab. There may be instruments that are not calibrated and cannot be tested by the student.  Materials and components may be flawed in ways that are undetectable.  It is advisable to do a dry run and perform calculations immediately to see that things are going as planned but sometimes even well designed experiments fail. If your experiment is unsuccessful and there is no obvious flaw then you should receive a good grade.  Naturally your instructor will try and see why you failed. Your report may therefore require a more complete description and may be more difficult to write.


Unknown – There will be times when estimating the uncertainty is not critical to the particular laboratory procedure.  The student should still include the uncertainty as unknown. You should check with your instructor to see if this appropriate.


Constants – Values from the textbook are usually given to 3 significant figures. You can use this rule for most of the constants used in the lab.  The value of g, 9.80 m/s2,  should be assigned an uncertainty of 0.01 m/s2.




These are called statistics. They are functions of the data points that can be used by the experimentalist as an estimate for a quantity of interest. Usually the mean is a good estimate of the quantity measured. Most students already understand that averages can be superior to a single measurement. The underlying assumption is that the measurements are random. Some data points are greater and some less than the true value. Averaging tends to cancel these fluctuations.  Be sure that you are comfortable with the notion that the mean of these data qualifies as a good estimator for the result.


The standard deviation will be used as one estimate of uncertainty.  The interpretation of the SD is a more subtle point. Since the SD is the average deviation of the measurements from their central value, one expects the SD could be used to estimate the uncertainty in a typical measurement. If 10 measurements of the same mass are on average 6 gm from their central value then assigning an uncertainty of 6 gm to each measurement is reasonable. The SD is then assigned as the error for each of the 10 measurements.  Those close and those far from the mean are given the same value for their uncertainty (6 gm). 


To summarize:


MEAN is an estimate for the true value of the quantity being measured.


STANDARD DEVIATION is an estimate for the error in any ONE of the measurements averaged.


The averaging process that provides the mean value often reduces the actual error. As mentioned above the mean is superior to a single measurement because of the cancellation due to the averaging.  One can go further and state that averages are better when more values are used. Let N be the number of individual measurements. As N increases the average value improves as an estimate of the true value. If this point doesn’t seem obvious accept it for now and we will explore it later.  This leads to the conclusion that the mean is closer to the true value than the standard deviation may suggest and that the uncertainty of the mean should depend on N. In fact another statistic, the standard deviation of the mean, SDM, is usually used to estimate the error if the mean is used as the estimate of the true value rather than SD the uncertainty in one of the individual measurements.


STANDARD DEVIATION OF THE MEAN is an estimate of the error associated with using the mean as an estimate for the true value.


A discussion of the mean, SD and SDM must include the limitations of these statistics as estimators.  It is probably apparent that one cannot improve a measurement by simply recording and averaging more and more data.  The limits arise due to a second type of error. These errors are called systematic errors. They are different from random errors because they influence each measurement in the same manner. A ruler that is too short is an example of a systematic error and such a ruler will measure all values to be short. Averaging cannot correct for this error.


SYSTEMATIC errors cannot be reduced by averaging and they limit the extent to which averaging data can be used to reduce experimental error.


When an experimenter judges, based on an evaluation of the experiment, that systematic errors could be a significant then the SD should be used so that one doesn’t underestimate the error.


SD can be used as an overall estimate of the error (uncertainty) when the student suspects that there are systematic limits.


An actual experiment must explore the extent of both types of error and develop methods to evaluate both of these errors.  The introductory physics labs do not always require this level of thoroughness.


More on Measurement Differences and Uncertainty


An expression can be constructed for the likelihood of obtaining a certain result given the true value TV and an uncertainty σ. This expression is often a gaussian function. For illustration let us assume that you know the length of a field is exactly 50m (TV) and when measuring the length of the field the average uncertainty is 10m (σ).  The gaussian function would be


A plot of this function that describes this situation is shown below. The lines show those x values that are one σ (40, 60) away from TV (50).

















































































































The likelihood of a measurement falling somewhere in this region is 68.3% (1 σ, 40 to 60). If we increase the range of values (2 σ, 30 to 70) there is a 95.4% probability that a measurement will fall somewhere within this range. If we increase the range of values (3 σ, 20 to 80) there is a 99.7% probability that a measurement will fall somewhere within this range. The probability of getting a measured value outside these ranges is 31.7%, 4.6% and 0.3%, respectively.  You can conclude that finding two measured values of the same quantity that are 1 σ apart in not that unlikely but finding two measured values 3 σ apart is very unlikely and probably indicates one of the measurements is bad.

When do my measurements agree with another experiment or with a theoretical value ?


The graph above shows 6 measurements that were performed and compared to a theoretical prediction of g (circles, 9.8 m/s-s). The first thing to note is that both the theoretical (circles) and the experimental (squares) results have an associated uncertainty. Also note that measurement A, B, and C have a theoretical uncertainty that is very small compared to the experimental uncertainty. For measurements E, F, and G the experimenter is using a prediction for g that has a comparable uncertainty.  A rigorous evaluation of the agreement or disagreement involves probability statements. However, our first goal is to get a sense of what the measurements mean and in this lab this will be the only requirement. Here are three rules of thumb one may use. The experiments are labeled as 1, 2 and 3 sigma. Sigma denotes the uncertainty chosen by the experimenter and is reflected in the size of error bars drawn.


1.      Measurements and/or theoretical results that agree to within one sigma are in agreement.

a.      Shown as case A and D.

2.      Measurements and/or theoretical results that agree to within 2 sigma are in agreement but suggest there may be problems. The experimenter needs to review his/her data and methods. The experimenter might duplicate the experiment.

a.      Shown as case B and E.

3.      Measurements that agree only at the 3 sigma level are in disagreement.

a.      Shown as case C and F.


Comment: In looking at cases D, E, F the experimenter needs to establish his point of view. The experimenter can ask if the two data points are compatible by using the two data points to establish a new estimate for the true value.  The average would be an appropriate choice. The individual measurements in example E would lie approximately 1 sigma away from this average.  So one could choose as the 1-sigma point. For our case we are simplifying the analysis by simply comparing the experimenter’s value to another value using only the uncertainty in the experimenter’s measurement.



When you OBTAIN your FINAL result and your FINAL UNCERTAINTY be sure to state THESE resultS with the correct number of significant figures.

The Error Analysis Section of this lab (front of manual) has a detailed description of significant figures with numerous examples. In general, numbers presented in spreadsheet tables do not need to given with the correct number of significant figures. Intermediate results do not need to be given with the correct number of significant figures. Summary tables that are providing results for a lab section or a final result should be listed with the correct number of significant figures. If you are unsure ask your instructor.



When a number of measured quantities are combined to get a result the uncertainty associated with the result is a function of the uncertainties of measured quantities used to calculate the result. Examples:

  • distances and times to find g,
  • temperatures and masses to measure the heat capacity


This type of analysis can be quite complicated. The correct way to add independent uncertainties is to add uncertainties in quadrature.  The formulas in the Error Analysis section at the front of this manual follow this rule and therefore one sees that many of the formulas sum squares and then take the square root.  The following analysis will use a simpler less rigorous approximation. This approximation will overestimate the uncertainties. (always double check with your instructor as to the level of rigor expected in an analysis):

§       Absolute refers to actual uncertainty. Fractional or percent involves the ratio of the uncertainty to another quantity. The absolute uncertainty divided by the measured quantity times 100 is often used for the percent uncertainty.

§       When quantities add or subtract, add absolute uncertainties.

§       When quantities are multiplied or divided, the percent uncertainty in the result is the sum of the percent uncertainties in the quantities used in the calculation.

§       When a quantity A is raised to the power j, B=Aj. The percent uncertainty is j times the percent uncertainty in A.   (% uncertainty B) =  j  (% uncertainty A)

§       For formulas that consist of several different operations, combine the uncertainties as you perform the calculation.  A spreadsheet is ideal for this type of calculation.

§       One can find uncertainties by plugging in values +/- the uncertainty into a formula and see how the result changes. This can be misleading when some of your values should be combined as smaller values with others as larger values to get the largest fluctuation.  Students can typically ignore this effect.

§       More complicated formulas will require more complicated relationship. Discuss these with your instructor.


Example of Calculating Uncertainties


Equations 4 and 6 shown below are extracted from a lab designed to measure Lf and Lv.


            mi(Lf + Cw(Tf-0)) = mwCw(To - Tf) + mcCc(To - Tf)                 (4)


            msLv + msCw(Tbp-Tf) = mwCw(Tf - To) + mcCc(Tf - To)                        (6)


How do we find the uncertainty, for example, in equation 6 for L assuming that we measure the following quantities:








fractional uncert.

mass of water






Dmw/ mw


final temperature




D Tf


D Tf / Tf


initial temperature




D To


D To / To


boiling point of water




D Tbp


D Tbp / Tbp


mass of the steam added






D ms /ms


mass of the container








known sp. heat of water



cal/gm oC





known sp. heat of copper



cal/gm oC







The quantities Cw and Cc are assumed to be known exactly.  The experimenter measures several quantities and determines the uncertainties in each quantity measured. There are various techniques for finding the uncertainty including statistical analysis and consulting instrument specifications.


Let us solve equation 6 for Lv.


msLv  = mwCw(Tf - To) + mcCc(Tf - To) - msCw(Tbp-Tf)



Substituting the values above we find Lv = 543. 


There are three terms added together






To determine the uncertainty we first note that each term includes a difference in temperatures.

The uncertainty associated with these differences (add absolute uncertainties) is





fractional uncertainty

Tf - To


0.3 + 0.3 = 0.6


Tbf - Tf


0.3 + 0.2 = 0.5



Once the temperature difference and its uncertainty have been determined, each term becomes a product (or quotient) of numbers. Therefore we add fractional uncertainties to get the uncertainty in each term. For example, to determine the uncertainty for term 1 we sum the fractional uncertainty in the mass of water, the fractional uncertainty for specific heat, the fractional uncertainty for the temperature difference (above table) and the fractional uncertainty for the mass of steam.




fractional uncertainty




0.0003 + 0 + 0.0271 + 0.0090 = 0.0365




0.0012 + 0 + 0.0271 + 0.0090 = 0.0373




0 + 0.0092 = 0.0092



Since you add the terms together we must sum up absolute uncertainty of each term in the sum.


DLv = 21.536 +0.297 + 0.5 = 22.3


DLv /Lv =.041    or 4.1%


Final result

Lv = 543  ± 22        cal/gm









List of all directly measured quantities:

Name of quantity



Method for estimating error


























Some measured quantities are indirect and must be calculated from a set of direct measurements.


Indirect measurements     Quantity:

Formula or relationship used to calculate this quantity:

List of Direct quantities

absolute  uncertainty

Percent uncertainty





















FINAL RESULTS - USE THE CORRECT number of significant figures.