Find the Error Between Two Solutions
Measurements and Error Analysis
"It is better to be roughly right than precisely wrong." — Alan Greenspan
The Uncertainty of Measurements
Some numerical statements are exact: Mary has 3 brothers, and 2 + 2 = 4. However, all measurements have some degree of uncertainty that may come from a variety of sources. The process of evaluating the uncertainty associated with a measurement result is often called uncertainty analysis or error analysis. The complete statement of a measured value should include an estimate of the level of confidence associated with the value. Properly reporting an experimental result along with its uncertainty allows other people to make judgments about the quality of the experiment, and it facilitates meaningful comparisons with other similar values or a theoretical prediction. Without an uncertainty estimate, it is impossible to answer the basic scientific question: "Does my result agree with a theoretical prediction or results from other experiments?" This question is fundamental for deciding if a scientific hypothesis is confirmed or refuted. When we make a measurement, we generally assume that some exact or true value exists based on how we define what is being measured. While we may never know this true value exactly, we attempt to find this ideal quantity to the best of our ability with the time and resources available. As we make measurements by different methods, or even when making multiple measurements using the same method, we may obtain slightly different results. So how do we report our findings for our best estimate of this elusive true value? The most common way to show the range of values that we believe includes the true value is:
( 1 )
measurement = (best estimate ± uncertainty) units
Let's take an example. Suppose you want to find the mass of a gold ring that you would like to sell to a friend. You do not want to jeopardize your friendship, so you want to get an accurate mass of the ring in order to charge a fair market price. You estimate the mass to be between 10 and 20 grams from how heavy it feels in your hand, but this is not a very precise estimate. After some searching, you find an electronic balance that gives a mass reading of 17.43 grams. While this measurement is much more precise than the original estimate, how do you know that it is accurate, and how confident are you that this measurement represents the true value of the ring's mass? Since the digital display of the balance is limited to 2 decimal places, you could report the mass as m = 17.43 ± 0.01 g. 17.44 ± 0.02 g. Accuracy is the closeness of agreement between a measured value and a true or accepted value. Measurement error is the amount of inaccuracy. Precision 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 among independent measurements of the same quantity; also the reliability or reproducibility of the result. The uncertainty estimate associated with a measurement should account for both the accuracy and precision of the measurement.
( 2 )
Relative Uncertainty = 
uncertainty measured quantity
Example: m = 75.5 ± 0.5 g = 0.006 = 0.7%.
( 3 )
Relative Error =
measured value − expected value expected value
If the expected value for m is 80.0 g, then the relative error is: = −0.056 = −5.6% Note: The minus sign indicates that the measured value is less than the expected value.
Types of Errors
Measurement errors may be classified as either random or systematic, depending on how the measurement was obtained (an instrument could cause a random error in one situation and a systematic error in another).
Random errors are statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement device. Random errors can be evaluated through statistical analysis and can be reduced by averaging over a large number of observations (see standard error).
Systematic errors are reproducible inaccuracies that are consistently in the same direction. These errors are difficult to detect and cannot be analyzed statistically. If a systematic error 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.
When making careful measurements, our goal is to reduce as many sources of error as possible and to keep track of those errors that we can not eliminate. It is useful to know the types of errors that may occur, so that we may recognize them when they arise. Common sources of error in physics laboratory experiments:
Incomplete definition (may be systematic or random) — One reason that it is impossible to make exact measurements is that the measurement is not always clearly defined. For example, if two different people measure the length of the same string, they would probably get different results because each person may stretch the string with a different tension. The best way to minimize definition errors is to carefully consider and specify the conditions that could affect the measurement. Failure to account for a factor (usually systematic) — The most challenging part of designing an experiment is trying to control or account for all possible factors except the one independent variable that is being analyzed. For instance, you may inadvertently ignore air resistance when measuring free-fall acceleration, or you may fail to account for the effect of the Earth's magnetic field when measuring the field near a small magnet. The best way to account for these sources of error is to brainstorm with your peers about all the factors that could possibly affect your result. This brainstorm should be done before beginning the experiment in order to plan and account for the confounding factors before taking data. Sometimes a correction can be applied to a result after taking data to account for an error that was not detected earlier. Environmental factors (systematic or random) — Be aware of errors introduced by your immediate working environment. You may need to take account for or protect your experiment from vibrations, drafts, changes in temperature, and electronic noise or other effects from nearby apparatus. Instrument resolution (random) — All instruments have finite precision that limits the ability to resolve small measurement differences. For instance, a meter stick cannot be used to distinguish distances to a precision much better than about half of its smallest scale division (0.5 mm in this case). One of the best ways to obtain more precise measurements is to use a null difference method instead of measuring a quantity directly. Null or balance methods involve using instrumentation to measure the difference between two similar quantities, one of which is known very accurately and is adjustable. The adjustable reference quantity is varied until the difference is reduced to zero. The two quantities are then balanced and the magnitude of the unknown quantity can be found by comparison with a measurement standard. With this method, problems of source instability are eliminated, and the measuring instrument can be very sensitive and does not even need a scale. Calibration (systematic) — Whenever possible, the calibration of an instrument should be checked before taking data. If a calibration standard is not available, the accuracy of the instrument should be checked by comparing with another instrument that is at least as precise, or by consulting the technical data provided by the manufacturer. Calibration errors are usually linear (measured as a fraction of the full scale reading), so that larger values result in greater absolute errors. Zero offset (systematic) — When making a measurement with a micrometer caliper, electronic balance, or electrical meter, always check the zero reading first. Re-zero the instrument if possible, or at least measure and record the zero offset so that readings can be corrected later. It is also a good idea to check the zero reading throughout the experiment. Failure to zero a device will result in a constant error that is more significant for smaller measured values than for larger ones. Physical variations (random) — It is always wise to obtain multiple measurements over the widest range possible. Doing so often reveals variations that might otherwise go undetected. These variations may call for closer examination, or they may be combined to find an average value. Parallax (systematic or random) — This error can occur whenever there is some distance between the measuring scale and the indicator used to obtain a measurement. If the observer's eye is not squarely aligned with the pointer and scale, the reading may be too high or low (some analog meters have mirrors to help with this alignment). Instrument drift (systematic) — Most electronic instruments have readings that drift over time. The amount of drift is generally not a concern, but occasionally this source of error can be significant. Lag time and hysteresis (systematic) — Some measuring devices require time to reach equilibrium, and taking a measurement before the instrument is stable will result in a measurement that is too high or low. A common example is taking temperature readings with a thermometer that has not reached thermal equilibrium with its environment. A similar effect is hysteresis where the instrument readings lag behind and appear to have a "memory" effect, as data are taken sequentially 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 errors come from carelessness, poor technique, or bias on the part of the experimenter. The experimenter may measure incorrectly, or may use poor technique in taking a measurement, or may introduce a bias into measurements by expecting (and inadvertently forcing) the results to agree with the expected outcome.
Gross personal errors, sometimes called mistakes or blunders, should be avoided and corrected if discovered. As a rule, personal errors are excluded from the error analysis discussion because it is generally assumed that the experimental result was obtained by following correct procedures. The term human error should also be avoided in error analysis discussions because it is too general to be useful.
Estimating Experimental Uncertainty for a Single Measurement
Any measurement you make will have some uncertainty associated with it, no matter the precision of your measuring tool. 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 instrument, along with any other factors that might affect the ability of the experimenter to make the measurement.
For example, if you are trying to use a meter stick to measure the diameter of a tennis ball, the uncertainty might be ± 5 mm, ± 2 mm.
( 4 )
Measurement = (measured value ± standard uncertainty) unit of measurement
where the ± standard uncertainty indicates approximately a 68% confidence interval (see sections on Standard Deviation and Reporting Uncertainties). 6.7 ± 0.2 cm.
Example: Diameter of tennis ball =
Estimating Uncertainty in Repeated Measurements
Suppose you time the period of oscillation of a pendulum using a digital instrument (that you assume is measuring accurately) and find: T = 0.44 seconds. This single measurement of the period suggests a precision of ±0.005 s, but this instrument precision may not give a complete sense of the uncertainty. If you repeat the measurement several times and examine the variation among the measured values, you can get a better idea of the uncertainty in the period. For example, here are the results of 5 measurements, in seconds: 0.46, 0.44, 0.45, 0.44, 0.41.
( 5 )
Average (mean) =
x 1 + x 2 + 
N
For this situation, the best estimate of the period is the average, or mean.
Whenever possible, repeat a measurement several times and average the results. This average is generally the best estimate of the "true" value (unless the data set is skewed by one or more outliers which should be examined to determine if they are bad data points that should be omitted from the average or valid measurements that require further investigation). Generally, the more repetitions you make of a measurement, the better this estimate will be, but be careful to avoid wasting time taking more measurements than is necessary for the precision required.
Consider, as another example, the measurement of the width of a piece of paper using a meter stick. Being careful to keep the meter stick parallel to the edge of the paper (to avoid a systematic error which would cause the measured value to be consistently higher than the correct value), the width of the paper is measured at a number of points on the sheet, and the values obtained are entered in a data table. Note that the last digit is only a rough estimate, since it is difficult to read a meter stick to the nearest tenth of a millimeter (0.01 cm). 
( 6 )
Average =
= = 31.19 cm sum of observed widths no. of observations
This average is the best available estimate of the width of the piece of paper, but it is certainly not exact. We would have to average an infinite number of measurements to approach the true mean value, and even then, we are not guaranteed that the mean value is accurate because there is still some systematic error from the measuring tool, which can never be calibrated perfectly. So how do we express the uncertainty in our average value? One way to express the variation among the measurements is to use the average deviation. This statistic tells us on average (with 50% confidence) how much the individual measurements vary from the mean.
( 7 )
d =
|x 1 − x | + |x 2 − x | + N
However, the standard deviation is the most common way to characterize the spread of a data set. The standard deviation is always slightly greater than the average deviation, and is used because of its association with the normal distribution that is frequently encountered in statistical analyses.
Standard Deviation
To calculate the standard deviation for a sample of N measurements:
-
1
Sum all the measurements and divide by N to get the average, or mean. -
2
Now, subtract this average from each of the N measurements to obtain N "deviations". -
3
Square each of these N deviations and add them all up. -
4
Divide this result by( N − 1)
and take the square root.
We can write out the formula for the standard deviation as follows. Let the N measurements be called x 1, x 2, ..., xN . Let the average of the N values be called x . δ x i = x i − x , for i = 1, 2,
In our previous example, the average width x d = 0.086 cm. s = x ± 2 s,
The average deviation is:
= 0.12 cm. 
(0.14)2 + (0.04)2 + (0.07)2 + (0.17)2 + (0.01)2 5 − 1
Figure 1
Standard Deviation of the Mean (Standard Error)
When we report the average value of N measurements, the uncertainty we should associate with this average value is the standard deviation of the mean, often called the standard error (SE).
( 9 )
σ x =
s
N
The standard error is smaller than the standard deviation by a factor of 1/ Average paper width = 31.19 ± 0.05 cm.
. N
. 5
Anomalous Data
The first step you should take in analyzing data (and even while taking data) is to examine the data set as a whole to look for patterns and outliers. Anomalous data points that lie outside the general trend of the data may suggest an interesting phenomenon that could lead to a new discovery, or they may simply be the result of a mistake or random fluctuations. In any case, an outlier requires closer examination to determine the cause of the unexpected result. Extreme data should never be "thrown out" without clear justification and explanation, because you may be discarding the most significant part of the investigation! However, if you can clearly justify omitting an inconsistent data point, then you should exclude the outlier from your analysis so that the average value is not skewed from the "true" mean.
Fractional Uncertainty Revisited
When a reported value is determined by taking the average of a set of independent readings, the fractional uncertainty is given by the ratio of the uncertainty divided by the average value. For this example,
( 10 )
Fractional uncertainty = = = 0.0016 ≈ 0.2%
Note that the fractional uncertainty is dimensionless but is often reported as a percentage or in parts per million (ppm) to emphasize the fractional nature of the value. A scientist might also make the statement that this measurement "is good to about 1 part in 500" or "precise to about 0.2%". The fractional uncertainty is also important because it is used in propagating uncertainty in calculations using the result of a measurement, as discussed in the next section.
Propagation of Uncertainty
Suppose we want to determine a quantity f, which depends on x and maybe several other variables y, z, etc. We want to know the error in f if we measure x, y, ... with errors σ x , σ y , ... Examples:
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f = xy (Area of a rectangle)
( 12 )
f = p cos θ ( x -component of momentum)
( 13 )
f = x / t (velocity)
For a single-variable function f(x), the deviation in f can be related to the deviation in x using calculus:
( 14 )
δ f = 
Thus, taking the square and the average:
( 15 )
δ f 2 =
δ x 2 2
and using the definition of σ , we get:
( 16 )
σ f = 
Examples: (a) f =
x
( 17 )
=
1 2
x
( 18 )
σ f =
, or = σ x 2
x
(b) f = x 2
(c) f = cos θ
( 22 )
σ f = |sin θ | σ θ , or = |tan θ | σ θ Note : in this situation, σ θ must be in radians.
In the case where f depends on two or more variables, the derivation above can be repeated with minor modification. For two variables, f(x, y), we have:
The partial derivative means differentiating f with respect to x holding the other variables fixed. Taking the square and the average, we get the law of propagation of uncertainty:
If the measurements of x and y are uncorrelated, then δ x δ y = 0,
Examples: (a) f = x + y
( 27 )
∴ σ f =
σ x 2 + σ y 2
When adding (or subtracting) independent measurements, the absolute uncertainty of the sum (or difference) is the root sum of squares (RSS) of the individual absolute uncertainties. When adding correlated measurements, the uncertainty in the result is simply the sum of the absolute uncertainties, which is always a larger uncertainty estimate than adding 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.
(b) f = xy
( 29 )
∴ σ f =
y 2 σ x 2 + x 2 σ y 2
Dividing the previous equation by f = xy, we get:
(c) f = x / y
Dividing the previous equation by f = x / y ,
When multiplying (or dividing) independent measurements, the relative uncertainty of the product (quotient) is the RSS of the individual relative uncertainties. When multiplying correlated measurements, the uncertainty in the result is just the sum of the relative uncertainties, which is always a larger uncertainty estimate than adding in quadrature (RSS). Multiplying or dividing by a constant does not change the relative uncertainty of the calculated value.
Note that the relative uncertainty in f, as shown in (b) and (c) above, has the same form for multiplication and division: the relative uncertainty in a product or quotient depends on the relative uncertainty of each individual term. Example: Find uncertainty in v, where v = at
( 34 )
= = =
= 0.031 or 3.1% 
(0.010)2 + (0.029)2
Notice that the relative uncertainty in t (2.9%) is significantly greater than the relative uncertainty for a (1.0%), and therefore the relative uncertainty in v is essentially the same as for t (about 3%). Graphically, the RSS is like the Pythagorean theorem:
Figure 2
The total uncertainty is the length of the hypotenuse of a right triangle with legs the length of each uncertainty component.
Timesaving approximation: "A chain is only as strong as its weakest link."
If one of the uncertainty terms is more than 3 times greater than the other terms, the root-squares formula can be skipped, and the combined uncertainty is simply the largest uncertainty. This shortcut can save a lot of time without losing any accuracy in the estimate of the overall uncertainty.
The Upper-Lower Bound Method of Uncertainty Propagation
An alternative, and sometimes simpler procedure, to the tedious propagation of uncertainty law is the upper-lower bound method of uncertainty propagation. This alternative method does not yield a standard uncertainty estimate (with a 68% confidence interval), but it does give a reasonable estimate of the uncertainty 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 procedure as examining the best and worst case scenarios. For example, suppose you measure an angle to be: θ = 25° ± 1° and you needed to find f = cos θ , then:
( 35 )
f max = cos(26°) = 0.8988
( 36 )
f min = cos(24°) = 0.9135
( 37 )
∴ f = 0.906 ± 0.007 
Note that even though θ was only measured to 2 significant figures, f is known to 3 figures. By using the propagation of uncertainty law: σ f = |sin θ | σ θ = (0.423)( π /180) = 0.0074
The uncertainty estimate from the upper-lower bound method is generally larger than the standard uncertainty estimate found from the propagation of uncertainty law, but both methods will give a reasonable estimate of the uncertainty in a calculated value.
The upper-lower bound method is especially useful when the functional relationship is not clear or is incomplete. One practical application is forecasting the expected range in an expense budget. In this case, some expenses may be fixed, while others may be uncertain, and the range of these uncertain terms could be used to predict the upper and lower bounds on the total expense.
Significant Figures
The number of significant figures in a value can be defined as all the digits between and including the first non-zero digit from the left, through the last digit. For instance, 0.44 has two significant figures, and the number 66.770 has 5 significant figures. Zeroes are significant except when used to locate the decimal point, as in the number 0.00030, which has 2 significant figures. Zeroes may or may not be significant for numbers like 1200, where it is not clear whether two, three, or four significant figures are indicated. To avoid this ambiguity, such numbers should be expressed in scientific notation to (e.g. 1.20 × 103 clearly indicates three significant figures). When using a calculator, the display will often show many digits, only some of which are meaningful (significant in a different sense). For example, if you want to estimate the area of a circular playing field, you might pace off the radius to be 9 meters and use the formula: A = π r 2. When you compute this area, the calculator might report a value of 254.4690049 m2. It would be extremely misleading to report this number as the area of the field, because it would suggest that you know the area to an absurd degree of precision—to within a fraction of a square millimeter! Since the radius is only known to one significant figure, the final answer should also contain only one significant figure: Area = 3 × 102 m2. From this example, we can see that the number of significant figures reported for a value implies a certain degree of precision. In fact, the number of significant figures suggests a rough estimate of the relative uncertainty: The number of significant figures implies an approximate relative uncertainty:
1 significant figure suggests a relative uncertainty of about 10% to 100%
2 significant figures suggest a relative uncertainty of about 1% to 10%
3 significant figures suggest a relative uncertainty of about 0.1% to 1%
Use of Significant Figures for Simple Propagation of Uncertainty
By following a few simple rules, significant figures can be used to find the appropriate precision for a calculated result for the four most basic math functions, all without the use of complicated formulas for propagating uncertainties.
For multiplication and division, the number of significant figures that are reliably known in a product or quotient is the same as the smallest number of significant figures in any of the original numbers.
Example:
6.6 × 7328.7 48369.42 = 48 × 103
(2 significant figures) (5 significant figures) (2 significant figures)
For addition and subtraction, the result should be rounded off to the last decimal place reported for the least precise number.
Examples:
223.64 5560.5 + 54 + 0.008 278 5560.5
Uncertainty, Significant Figures, and Rounding
For the same reason that it is dishonest to report a result with more significant figures than are reliably known, the uncertainty value should also not be reported with excessive precision. For example, it would be unreasonable for a student to report a result like:
( 38 )
measured density = 8.93 ± 0.475328 g/cm3 WRONG!
The uncertainty in the measurement cannot possibly be known so precisely! In most experimental work, the confidence in the uncertainty estimate is not much better than about ±50% because of all the various sources of error, none of which can be known exactly. Therefore, uncertainty values should be stated to only one significant figure (or perhaps 2 sig. figs. if the first digit is a 1). Because experimental uncertainties are inherently imprecise, they should be rounded to one, or at most two, significant figures. = measured density = 8.9 ± 0.5 g/cm3.
*The relative uncertainty is given by the approximate formula:
1
2(N − 1)
An experimental value should be rounded to be consistent with the magnitude of its uncertainty. This generally means that the last significant figure in any reported value should be in the same decimal place as the uncertainty.
In most instances, this practice of rounding an experimental result to be consistent with the uncertainty estimate gives the same number of significant figures as the rules discussed earlier for simple propagation of uncertainties for adding, subtracting, multiplying, and dividing.
Caution: When conducting an experiment, it is important to keep in mind that precision is expensive (both in terms of time and material resources). Do not waste your time trying to obtain a precise result when only a rough estimate is required. The cost increases exponentially with the amount of precision required, so the potential benefit of this precision must be weighed against the extra cost.
Combining and Reporting Uncertainties
In 1993, the International Standards Organization (ISO) published the first official worldwide Guide to the Expression of Uncertainty in Measurement. Before this time, uncertainty estimates were evaluated and reported according to different conventions depending on the context of the measurement or the scientific discipline. Here are a few key points from this 100-page guide, which can be found in modified form on the NIST website. When reporting a measurement, the measured value should be reported along with an estimate of the total combined standard uncertainty U c
Conclusion: "When do measurements agree with each other?"
We now have the resources to answer the fundamental scientific question that was asked at the beginning of this error analysis discussion: "Does my result agree with a theoretical prediction or results from other experiments?" Generally speaking, a measured result agrees with a theoretical prediction if the prediction lies within the range of experimental uncertainty. Similarly, if two measured values have standard uncertainty ranges that overlap, then the measurements are said to be consistent (they agree). If the uncertainty ranges do not overlap, then the measurements are said to be discrepant (they do not agree). However, you should recognize that these overlap criteria can give two opposite answers depending on the evaluation and confidence level of the uncertainty. It would be unethical to arbitrarily inflate the uncertainty range just to make a measurement agree with an expected value. A better procedure would be to discuss the size of the difference between the measured and expected values within the context of the uncertainty, and try to discover the source of the discrepancy if the difference is truly significant. To examine your own data, you are encouraged to use the Measurement Comparison tool available on the lab website. Here are some examples using this graphical analysis tool:
Figure 3
A = 1.2 ± 0.4 B = 1.8 ± 0.4
Figure 4
An alternative method for determining agreement between values is to calculate the difference between the values divided by their combined standard uncertainty. This ratio gives the number of standard deviations separating the two values. If this ratio is less than 1.0, then it is reasonable to conclude that the values agree. If the ratio is more than 2.0, then it is highly unlikely (less than about 5% probability) that the values are the same. Example from above with u = 0.4: = 1.1. u = 0.2: = 2.1.
References
Baird, D.C. Experimentation: An Introduction to Measurement Theory and Experiment Design, 3rd. ed. Prentice Hall: Englewood Cliffs, 1995. Bevington, Phillip and Robinson, D. Data Reduction and Error Analysis for the Physical Sciences, 2nd. ed. McGraw-Hill: New York, 1991. ISO. Guide to the Expression of Uncertainty in Measurement. International Organization for Standardization (ISO) and the International Committee on Weights and Measures (CIPM): Switzerland, 1993. Lichten, William. Data and Error Analysis., 2nd. ed. Prentice Hall: Upper Saddle River, NJ, 1999. NIST. Essentials of Expressing Measurement Uncertainty. http://physics.nist.gov/cuu/Uncertainty/ Taylor, John. An Introduction to Error Analysis, 2nd. ed. University Science Books: Sausalito, 1997.
Find the Error Between Two Solutions
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