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Computing Uncertainties in Laboratory Data and Result

This section considers the error and uncertainty in experimental measurements and calculated results. First, here are some fundamental things you should realize about uncertainty:

Every measurement has an uncertainty associated with it, unless it is an exact, counted integer, such as the number of trials performed.

Every calculated result also has an uncertainty, related to the uncertainty in the measured data used to calculate it. This uncertainty should be reported either as an explicit ± value or as an implicit uncertainty, by using the appropriate number of significant figures.

• The numerical value of a "plus or minus" (±) uncertainty value tells you the range of the result. For example a result reported as 1.23 ± 0.05 means that the experimenter has some degree of confidence that the true value falls in between 1.18 and 1.28.

• When significant figures are used as an implicit way of indicating uncertainty, the last digit is considered uncertain. For example, a result reported as 1.23 implies a minimum uncertainty of ±0.01 and a range of 1.22 to 1.24.

• For the purposes of General Chemistry lab, uncertainty values should only have one significant figure. It generally doesn't make sense to state an uncertainty any more precisely.

To consider error and uncertainty in more detail, we begin with definitions of accuracy and precision. Then we will consider the types of errors possible in raw data, estimating the precision of raw data, and three different methods to determine the uncertainty in calculated results.

Accuracy and Precision

The accuracy of a set of observations is the difference between the average of the measured values and the true value of the observed quantity. The precision of a set of measurements is a measure of the range of values found, that is, of the reproducibility of the measurements.

The relationship of accuracy and precision may be illustrated by the familiar example of firing a rifle at a target where the black dots below represent hits on the target:

You can see that good precision does not necessarily imply good accuracy. However, if an instrument is well calibrated, the precision or reproducibility of the result is a good measure of its accuracy.

Types of Error

The error of an observation is the difference between the observation and the actual or true value of the quantity observed. Returning to our target analogy, error is how far away a given shot is from the bull's eye. Since the true value, or bull's eye position, is not generally known, the exact error is also unknowable. Errors are often classified into two types: systematic and random. Systematic errors may be caused by fundamental flaws in either the equipment, the observer, or the use of the equipment. For example, a balance may always read 0.001 g too light because it was zeroed incorrectly. In a similar vein, an experimenter may consistently overshoot the endpoint of a titration because she is wearing tinted glasses and cannot see the first color change of the indicator. Systematic errors can result in high precision, but poor accuracy, and usually do not average out, even if the observations are repeated many times. Furthermore, they are frequently difficult to discover.

Random errors vary in a completely nonreproducible way from measurement to measurement. However, random errors can be treated statistically, making it possible to relate the precision of a calculated result to the precision with which each of the experimental variables (weight, volume, etc.) is known.

Take, for example, the simple task (on the face of it) of measuring the distance between these two parallel vertical lines:

                    |                     |

You would probably pull out a ruler, align one end with one bar, and read off the distance to the next. And you might think that the errors arose from only two sources,

(1) Instrumental error
(How "well calibrated" is the ruler? How thin and how closely spaced are the ruler's graduations?)

(2) Uncertainties in the thing being measured
(How thin are the lines? Is the paper subject to temperature and humidity changes?)

But a third source of error exists, related to how any measuring device is used. In this case, the main mistake was trying to align one end of the ruler with one mark. One should put the ruler down at random (but as perpendicular to the marks as you can, unless you can measure the ruler's angle as well), note where each mark hits the ruler, and then subtract the two readings. This should be repeated again and again, and average the differences. This eliminates the systematic error (i.e., the error that occurs in each measurement as a result of the measuring process itself) that aligning one end with one mark introduces.

This same idea—taking a difference in two readings, neither of which is pre-judged—holds in many of the operations you will do in this course. Here are two examples:

A. The lab manual says, "Fill one buret with..."

B. "Accurately weigh about 0.2 g..."

and here are two common mistakes associated with each:

A. You fill the buret to the top mark and record 0.00 mL as your starting volume. Note that burets read 0.00 mL when "full" and 10.00 mL when "empty", to indicate the volume of solution delivered.

B. You take forever at the balance adding a bit and taking away a bit until the balance indicates 0.2000 g.

The correct procedures are these:

A. Add enough solution so that the buret is nearly full, but then simply read the starting value to whatever precision the buret allows and record that value. It will be subtracted from your final buret reading to yield the most unbiased measurement of the delivered volume.

B. The key terms are "accurately weigh" and "about 0.2 g". The first specifies precision (0.1 mg, usually) and the second specifies a broad target. Together they mean that any mass within 10% or ±0.02 g of 0.2 g will probably do, as long as it is known accurately. Therefore you tare the weighing container (beaker, weighing paper, etc.), zero the balance, and add a small amount of the solid and determine its mass. Taring involves subtraction of the weight of the vessel from the weight of the sample and vessel to determine the weight of the sample. The analytical balance does this by electronically resetting the digital readout of the weight of the vessel to 0.0000. Solid is then added until the total mass is in the desired range, 0.2 ± 0.02 g or 0.18 to 0.22 g. You record the sample weight to the 0.1 mg, for example 0.1968 g. Now have an "accurately known" sample of "about 0.2 g".

A final type of experimental error is called erratic error or a blunder. These errors are the result of a mistake in the procedure, either by the experimenter or by an instrument. An example would be misreading the numbers or miscounting the scale divisions on a buret or instrument display. An instrument might produce a blunder if a poor electrical connection causes the display to read an occasional incorrect value. If you are aware of a mistake at the time of the procedure, the experimental result should be discounted and the experiment repeated correctly. If the mistake is not noticed, blunders can be difficult to trace and can give rise to much larger error than random errors. A widely errant result, a result that doesn't fall within a propagated uncertainty, or a larger than expected statistical uncertainty in a calculated result are all signs of a blunder. If a result differs widely from a known value, or has low accuracy, a blunder may be the cause. If a result differs widely from the results of other experiments you have performed, or has low precision, a blunder may also be to blame. The best way to detect erratic error or blunders is to repeat all measurements at least once and to compare to known values, if they are available. There are rigorous statistical tests to determine when a result or datum can be discarded because of wide discrepancy with other data in the set, but they are beyond the scope of this guide.

Precision of Instrument Readings and Other Raw Data

The first step in determining the uncertainty in calculated results is to estimate the precision of the raw data used in the calculation. Consider three weighings on a balance of the type in your laboratory:

        1st weighing of object: 6.3302 g
        2nd weighing of object: 6.3301 g
        3rd weighing of object: 6.3303 g

The average, or mean, weight of the object

In this example, the precision or reproducibility of the measurement is ± 0.0001 g. All three measurements may be included in the statement that the object has a mass of 6.3302 ± 0.0001 g. The balance allows direct reading to four decimal places, and since the precision is roughly 0.0001 g, or an uncertainty of ± 1 in the last digit, the balance has the necessary sensitivity for this measurement.

In the above example, we have little knowledge of the accuracy of the stated mass, 6.3302 ± 0.0001 g. The accuracy of the weighing depends on the accuracy of the internal calibration weights in the balance as well as on other instrumental calibration factors. The stated accuracy of our analytical balances is ± 0.0001 g and this is checked every time the balance is put in the calibration mode.

The precision of two other pieces of apparatus that you will often use is somewhat less obvious from a consideration of the scale markings on these instruments. The 10 milliliter burets used are marked (graduated) in steps of 0.05 mL. Thus you might suspect that readings from a buret will be precise to ± 0.05 mL. Actually since the scale markings are quite widely spaced, the space between 0.05 mL marks can be mentally divided into five equal spaces and the buret reading estimated to the nearest 0.01 mL. In fact, since the estimation depends on personal factors ("calibrated eyeballs"), the precision of a buret reading by the average student is probably on the order of ± 0.02 mL. Nevertheless, buret readings estimated to the nearest 0.01 mL will be recorded as raw data in your notebook. Similarly, readings of your Celsius (centigrade) scale thermometer can be estimated to the nearest 0.1 °C even though the scale divisions are in full degrees.

Every measurement that you make in the lab should be accompanied by a reasonable estimate of its precision or uncertainty.

Absolute and Relative Uncertainty

Precision can be expressed in two different ways. Absolute precision refers to the actual uncertainty in a quantity. For the example of the three weighings, with an average of 6.3302 ± 0.0001 g, the absolute uncertainty is 0.0001 g.

Relative uncertainty expresses the uncertainty as a fraction of the quantity of interest. Other ways of expressing relative uncertainty are in per cent, parts per thousand, and parts per million. For our example of an object weighing 6.3302 ± 0.0001 g, the relative uncertainty is 0.0001 g/6.3302 g which is equal to 2
x 10–5. This relative uncertainty can also be expressed as 2 x 10–3 percent, or 2 parts in 100,000, or 20 parts per million. Relative uncertainty is a good way to obtain a qualitative idea of the precision of your data and results.

In general, results of observations should be reported in such a way that the last digit given is the only one whose value is uncertain due to random errors. The digits that constitute the result, excluding leading zeros, are then termed significant figure. Appendix A of your textbook contains a thorough description of how to use significant figures in calculations. A brief description is included in the examples, below

Error Propagation and Precision in Calculations

The remainder of this guide is a series of examples to help you assign an uncertainty to your experimental data and calculated results. There are three different ways of calculating or estimating the uncertainty in calculated results. The following diagram describes these ways and when they are useful.

To illustrate each of these methods, consider the example of calculating the molarity of a solution of NaOH, standardized by titration of KHP.

First the calculated results

A 0.2181 g sample of KHP was titrated with 8.98 mL of NaOH. What is the molarity of the NaOH?

First we convert the grams of KHP to moles. We need this because we know that 1 mole of KHP reacts with 1 mole of NaOH, and we want the moles of NaOH in the volume used:

Now we can calculate the molarity of the NaOH, from the moles NaOH and the mL used:

Now for the significant figure analysis

You already know the rules for significant figures, but here is a quick and dirty summary.

Multiplication and division: The result has the same number of significant figures as the smallest of the number of significant figures for any value used in the calculation.

Addition and subtraction: The result will have a last significant digit in the same place as the left-most of the last significant digits of all the numbers used in the calculation.

The mass of KHP has four significant figures, so the moles of KHP should also have four significant figures and should be reported as 1.068
x 10–3 moles. Note that you should use a molecular mass to four or more significant figures in this calculation, to take full advantage of your mass measurement's accuracy. The moles of NaOH then has four significant figures and the volume measurement has three. So the final result should be reported to three significant figures, or 0.119 M. If this was your experiment, the results would mean that you have determined the concentration to be, at best, 0.119 ± 0.001 M or between 0.118 and 0.120 M. The accuracy of the volume measurement is the limiting factor in the uncertainty of the result, because it has the least number of significant figures.

Now for the error propagation

To propagate uncertainty through a calculation, we will use the following rules. These rules are similar to those for combining significant figures. The number of significant figures, used in the significant figure rules for multiplication and division, is related to the relative uncertainty. The left-most significant figure, used to determine the result's significant figures for addition and subtraction, is related to the absolute uncertainty.

Addition and subtraction: Uncertainty in results depends on the absolute uncertainty of the numbers used in the calculation. For the R = a + b or R = a – b,

the absolute uncertainty in R is calculated

The result would be reported as R ± σR

Example: For our KHP example, we first need to estimate the uncertainty in each measured value. The uncertainty in the mass measurement is ± 0.0001 g, at best. For the volume measurement, the uncertainty is estimated based on the ability to read a buret. For a 10 mL buret, with graduation marks every 0.05 mL, a single reading might have an uncertainty of ± 0.01 or 0.02 mL. In a titration, two volume readings are subtracted to calculate the volume added. Using the rules for addition and subtraction and the conservative uncertainty estimate of ± 0.02 mL for each reading, the uncertainty of the subtracted result can be calculated

The calculated volume for the titration would be reported as 8.98 ± 0.03 mL. This means that the true value of the volume is determined by the experiment to be in the range between 8.95 and 9.01 mL

Multiplication and division: Uncertainty in results depends upon the relative uncertainty of the data used as input in the calculation. Relative uncertainty is the uncertainty divided by the number it refers to. For result R, with uncertainty σR the relative uncertainty is σR/R.

This error propagation rule may be clearer if we look at some equations. We will let R represent a calculated result, and a and b will represent measured quantities used to calculate R. The symbol σR stands for the uncertainty in R. For the result R = a
x b or R = a/b, the relative uncertainty in R is


where σa and σb are the uncertainties in a and b, respectively. The absolute uncertainty, σR, can be calculated from this result and R. The result would then be reported as R ± σR.

Example: We can now apply the multiplication and division rule to the first step of our two-step molarity calculation:

This can be rearranged and the calculated number of moles substituted to give

σmoles= (5
x 10–4) (1.0679 x 10–3 mol) = 5 x 10–7 mol

The calculated result would be reported as (1.0679 ± 0.0005)
x 10–3 moles. One thing to notice about this result is that the relative uncertainty in the molecular mass of KHP is insignificant compared to that of the mass measurement. In fact, we could leave it out and would get the same uncertainty. Also notice that the uncertainty is given to only one significant figure. It doesn't make sense to specify the uncertainty in a result with a higher degree of precision than this. Finally, the error propagation result indicates a greater accuracy than the significant figures rules did.

Now we can apply the same methods to the calculation of the molarity of the NaOH solution.

This can be rearranged and the calculated molarity substituted to give

σM = (3 x 10–3) (0.11892 M) = 4 × 10–4 M

The final result would be reported as 0.1189 ± 0.0004 M. Again, the error propagation, using relative errors, shows which uncertainty contributes the most to the uncertainty in the result. The relative uncertainty in the volume is greater than that of the moles, which depends on the mass measurement, just like we saw in the significant figures analysis. To reduce the uncertainty, you would need to measure the volume more accurately, not the mass. Again, the uncertainty is less than that predicted by significant figures. The reason for this, in this particular example, is that the relative uncertainty in the volume, 0.03/8.98 = 0.003, or three parts per thousand, is closer to that predicted by a four significant figure result (one part per thousand) than that of the three significant figures used (one part in 100). Significant figures are a more approximate method of estimating the uncertainty than error propagation. A strict following of the significant figure rules resulted in a loss of precision, in this case.

Finally, the statistical way of looking at uncertainty

This method is most useful when repeated measurements are made, since it considers the spread in a group of values, about their mean. This analysis can be applied to the group of calculated results. If you have a set of N calculated results, R, you can average them to determine the mean, using the following equation


Where the Ri are the individual results. The standard deviation of a set of results is a measure of how close the individual results are to the mean. Your calculator probably has a key that will calculate this for you, if you enter a series of values to average. The standard deviation is given the symbol s and can be calculated as follows:


The standard error of the mean is a measure of the uncertainty of the mean and depends on the number of results. This is given by


Notice that the more measurements that are averaged, the smaller the standard error will be. Finally, an uncertainty can be calculated as a confidence interval. For a 95% confidence interval, there will be a 95% probability that the true value lies within the range of the calculated confidence interval, if there are no systematic errors. Confidence intervals are calculated with the help of a statistical device called the Student's t. These are tabulated values that relate the standard error of a mean to a confidence interval. Values of the t statistic depend on the number of measurements and confidence interval desired. Your textbook has a table of t values in Appendix A, and some values are included at the end of this section. The confidence interval is defined as the range of values calculated using the following equation


where t is the value of the t statistic for the number of measurements averaged and the confidence interval desired.

Example: To apply this statistical method of error analysis to our KHP example, we need more than one result to average. Let's consider the following table of results.

Trial [NaOH]
1 0.1180 M
2 0.1176
3 0.1159
4 0.1192

The first step is to calculate the mean value of the molarity, using Equation 3. Substituting the four values above gives

Next, we will use Equation 4 to calculate the standard deviation of these four values:

Using Equation 5 with N = 4, the standard error in the mean is

Finally, we can calculate a 95% confidence interval for these results. The table gives a t-statistic for a 95% confidence interval and 4 results as 3.18. The 95% confidence interval is calculated with Equation 6:

The final molarity would be reported as the 95% confidence interval. In this example that would be written 0.118 ± 0.002 (95%, N = 4). The values in parentheses indicate the confidence interval and the number of measurements. This confidence interval result means that, with 95% probability, the true value of the concentration is between 0.116 and 0.120 M.

The results of the three methods of estimating uncertainty are summarized below:

Significant Figures: 0.119 M (±0.001 implied by 3 significant figures)
True value lies between 0.118 and 0.120M
Error Propagation: 0.1189 ± 0.0004 M
True value lies between 0.1185 and 0.1194 M
Statistical Method: 0.118 ± 0.002 M (95%, N = 4)
True values lies between 0.116 and 0.120 M

The significant figures and statistical method results are the same molarity, within the experimental uncertainty. The Error Propagation and Significant Figures results are in agreement, within the calculated uncertainties, but the Error Propagation and Statistical Method results do not agree, within the uncertainty calculated from Error Propagation. Notice that the ± value for the statistical analysis is twice that predicted by significant figures and five times that predicted by the error propagation. This is because the spread in the four values indicates that the actual uncertainty in this group of results is greater than that predicted for an individual result, using just the inherent uncertainties of the volume and mass measurements. This could be the result of a blunder in one or more of the four experiments. If these were your data and you wanted to reduce the uncertainty, you would need to do more titrations, both to increase N and to (we hope) increase your precision and reduce the spread in the values, given by s. This will be reflected in a smaller standard error and confidence interval.

These examples illustrate three different methods of finding the uncertainty due to random errors in the molarity of an NaOH solution. Although three different uncertainties were obtained, all are valid ways of estimating the uncertainty in the calculated result. The method of uncertainty analysis you choose to use will depend upon how accurate an uncertainty estimate you require and what sort of data and results you are dealing with. The most important thing to remember is that all data and results have uncertainty and should be reported with either an explicit ? uncertainty value or with uncertainty implied by the appropriate number of significant figures.

Student's t statistics

Confidence Intervals
Number of observations 90% 95% 99%
2 6.31 12.7 63.7
3 2.92 4.30 9.92
4 2.35 3.18 5.84
5 2.13 2.78 4.60
6 2.02 2.57 4.03
7 1.94 2.43 3.71
8 1.90 2.36 3.50
9 1.86 2.31 3.36
10 1.83 2.26 3.25
11 1.81 2.23 3.17
12 1.80 2.20 3.11
13 1.78 2.18 3.06
14 1.77 2.16 3.01
15 1.76 2.14 2.98
1.64 1.96 2.58

"Student" is W. S. Gossett, who was an employee of Guinness Breweries and who first published these values under the pseudonym "A. Student" in 1908.

For more information about uncertainty

Zumdahl, Chemical Principles, Appendix A.

Oxtoby and Nachtrieb, Principles of Modern Chemistry, Appendix A.

Daniel C. Harris, Quantitative Chemical Analysis, 4th ed., Freeman, 1995.

David Shoemaker, Carl Garland, and Joseph Nibler, Experiments in Physical Chemistry, 5th ed. McGraw-Hill, 1989.

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