Uncertainty, Error, and Precision in Quantitative Measurements an Introduction Much of the work in any chemistry laboratory involves the measurement of numerical quantities. A quantitative measurement tells us three things: 1. Numerical quantity, 2. Appropriate units, 3. Uncertainty of the measurement. The first two of these are fairly easy to understand but the last one, uncertainty, needs some explanation. There is uncertainty associated with physical measurements, arising not only from the care with which you take the measurement, but also from the care with which the measuring device is calibrated. If you have done all that you can to minimize error in taking the measurement, your recorded values will reflect the uncertainty of the measuring tool. This is usually the smallest numerical value that can be estimated with the measuring device. For example, measure the length of the following line segment: Is the length between 4.0 and 4.5 cm? Yes, it looks that way. But is the length 4.3 cm? Is it 4.4 cm? So, what we can say is that the actual length is around 4.4 cm, but it might be closer to 4.3 cm, or it might be closer to 4.5 cm. In other words, we think the length is 4.4 cm but we might be off by 0.1 cm in either direction. We would record this measurement in this way: 4.4 ± 0.1 cm We use significant figures to imply the precision of a measurement without having to state the uncertainty explicitly. This works okay and the measurement above ends up being recorded just as 4.4 cm and the uncertainty of ±0.1 cm is implied. Note that when using this method, it is very important that you record all significant digits. If you measured a mass and found it to be 2.0000 ± 0.0001g, it would be wrong to record the mass as: 2 g (Wrong!) Instead you must include all significant figures, even if they happen to be trailing zeros: 2.0000 g (Right!) The uncertainty that we have been discussing so far is always associated with individual physical measurements. The value that you are trying to find when you make such a measurement has a true value that is unknown and is fundamentally unknowable. Because there is (unavoidable) uncertainty in measurements, the values you get when taking a series of measurements will tend to scatter around the true value. The difference between the true value and any given measured value is called the error in the measurement. Experimental error has a very specific meaning and does not necessarily imply a mistake or blunder. If you know about a mistake or blunder, you can, at least in principle, fix the problem and eliminate the mistake. Some experimental error is intrinsic. While it can be minimized, it cannot be eliminated. A perfectly executed experiment with no
mistakes or blunders still has experimental error. Experimental error falls into two categories: determinate and indeterminate. Indeterminate or random errors in a series of repeated experiments cause data to be scattered around a mean value. Random errors are caused by many uncontrollable parameters existing in any scientific measurement and cannot be predicted. Making repeated measurements of the same quantity can reduce random error in an experimentally measured quantity. The standard deviation of a set of repeated measurements shows how closely the data is clustered around the mean; i.e., it is a measure of your precision. The standard deviation of the mean of a set of repeated measurements (which is equal to the standard deviation divided by the square of the number of measurements made) shows how close the average value of your measurements is to the true value; i.e., it is a measure of your accuracy. Errors that can be attributed to definite identifiable causes are known as determinate or systematic errors. In a series of repeated experiments, systematic errors will cause experimental values to differ from the expected value in a potentially predictable, non-random, way. They arise from such things as your own experimental technique (doing the same thing incorrectly every time), problems in the experimental method and instrumental error. Systematic errors can be tracked down and corrected for by standardizing the equipment being used. The two terms accuracy and precision should be carefully differentiated in a scientific context. An accurate result closely agrees with the actual or true value for that quantity. Precision describes how well repeated measurements agree with each other. A good experiment is both precise and accurate. As an example, Table One is a listing of allowed systematic error for volumetric glassware. There is an uncertainty associated with volume measurement that is dependent on the glassware you use. Note that these tolerances are valid only if the glassware is scrupulously clean. Table One: Tolerances for Volumetric Glassware by the National Bureau of Standards Capacity/mL Vol Flask Vol Pipette 0 to 2 0.006 2 to 5 0.01 5 to 10 0.02 10 to 25 0.03 0.03 25 to 50 0.05 0.05 50 to 100 0.08 0.08 100 to 200 0.1 0.1 200 to 500 0.15 500 to 1000 0.3 The numerical value of an error in a measurement is the absolute uncertainty. The quality of a measurement can be gauged by the relative uncertainty which is a dimensionless quantity given by the ratio of the absolute uncertainty to the absolute value of the measurement. For repeated measurements, the relative uncertainty will be the ratio of the standard deviation and the mean of the measurements. Relative uncertainty is sometimes converted into a percentage uncertainty
Statistical Analysis Statistics is a mathematical science whose laws are very useful in studying experimental data because they take into account the random nature of measurement. The small number of measurements we take in lab can be considered to be a fraction of an infinite number of identical experiments (yielding a statistical range of results) that could be made given an infinite amount of time. The relationship of a limited set of data to the true data can be estimated using statistics. There is not enough time this semester to cover even a significant fraction of this field, so we will only point out a number of important concepts. As discussed previously, all individual measurements are subject to uncertainty and the uncertainty should be reported. The uncertainty tells what you think the magnitude of error is in each of your individual measurements. Whenever you make three or more quantitative measurements of some observable, you should report a standard deviation. When reporting your value, follow these guidelines: 1. Calculate the standard deviation (mean = 10.145, s = 0.467) 2. Round standard deviation to one significant digit (mean = 10.145, s = 0.5) 3. Round mean so that it has the same number of digits after the decimal point as does the standard deviation (mean = 10.1, s = 0.5) 4. Report mean and standard deviation as mean ± one standard deviation (10.1 ± 0.5) Standard deviation can be used to compare two values or to compare an experimentally determined value with a literature value. A property of normal distribution is that 95.5% of the values in a series of measurements fall within two standard deviations of the mean. If you take another measurement, you can expect with 95.5% certainty that it will fall within two standard deviations of the mean. If, instead, it falls (say) seven standard deviations away, then the probability is very low that you were still measuring the same thing. Calculation of the mean and standard deviation: Calculation of the mean and standard deviation is a small part of the very large field of statistics. The mean of a series of measurements is equal to the sum of the individual measurements divided by the total number of measurements (N): Once the mean has been calculated, the standard deviation (s) is determined by the following equation: When in the statistics mode, modern calculators can quickly calculate the average and standard deviation. Because you will frequently need to report these values, it is important that you learn to do the calculations using the calculator rather than going through these tedious calculations by hand.
Data rejection Useful Analysis Techniques When you have a set of data with one data point that seems very different to all the rest, there must be a scientific basis for deciding whether to reject that point. You should not just throw it away without justification. If you have a series of repeated measurements of the same property, then the Q-test is a useful statistical way to determine whether one of your points in that series of measurements can be disregarded. The Q-test is a simple statistical test to determine if a data point that is very different from the other data points in a set can be rejected. Only one data point may be discarded using the Q-test. Q = outlier - value closest to the outlier / highest value - lowest value Table of Q critical values (90% confidence) N Q C 3 0.94 4 0.76 5 0.64 6 0.56 7 0.51 8 0.47 9 0.44 10 0.41 If Q is larger than Q C the outlier can be discarded with 90% confidence. The Q test cannot be used to disregard a point from a set of data when you've only measured each point once. In this case, if you have no basis for rejecting a data point other than some gut level feeling that dropping a particular data point will make all your data look so much better, then this is what you should do: Do the analysis using all your data. Show that the results using all your data are very different than what you would expect. Point out how bad that one data point looks (e.g., the plot is linear except for this one point). Speculate about at least one specific experimental reason that could explain how that data point could have been in error. Redo the analysis dropping that data point. Make self-congratulatory statements about how that fixed all your calculations. Estimation of uncertainties While performing all experiments, you are required to estimate the uncertainty in each type of measurement that you make, based on how that measurement was made. This will enable you to use appropriate significant figures when reporting your results. Uncertainty is in the same units as the value, so quote the units after both the value and the uncertainty. If you quote your value in scientific notation, do the same for the uncertainty.
Most experimental measurements involve reading a scale or a digital readout. When using a scale, determine how well you can read that scale. Depending how far apart each marking is, you may be able to estimate a 1/10 or a 1/2 of the smallest scale. That is your uncertainty. For example, if you feel that you can read a ruler to the nearest millimeter, your uncertainty will be ± 0.5 mm. You may be able to estimate a large pressure gauge marked in 1 Torr increments to the nearest 0.2 Torr, resulting in an uncertainty of ± 0.1 Torr. A digital display may seem to give you many significant figures. This can be misleading. Repeating a measurement several times and calculating the random error will help you decide how many of the decimal places are actually significant. Your data analysis will often involve numbers that you did not personally measure; e.g., molecular weights, densities, etc. If you use a numerical value without an uncertainty, and you are not told that the uncertainty is negligible, you should estimate the uncertainty to be of the order of magnitude of the smallest significant figure. For example, if you were given a boiling point for your chemical of 334.8 K, you would estimate the error to be ± 0.1K. The appropriate uncertainty in fundamental constants and unit conversions depend on how many significant figures you use. In table two, some of the more common fundamental constants you may need with their associated relative uncertainties are provided. If you choose to round off one of these constants, then you will have to use the uncertainty dictated by the number of significant figures you have chosen to use; e.g., the unit conversion for Temperature from Celsius to Kelvin (T/K =t/ C+273.15) is exact if all five digits are used (the uncertainty in 273.15 is zero), but if you choose to simply use 273, then the uncertainty you should use for this number will be ±1. Note the uncertainties quoted in this table are relative; to find the absolute uncertainty, you will have to multiply the relative uncertainty by the value of the constant. You may round the resulting absolute uncertainty to a single significant figure. Table Two: Fundamental Physical Constants with associated relative uncertainties Relative Quantity Symbol Value Units uncertainty Speed of light in vacuum c 299792458 m/s exact Planck Constant h 6.62606876(52)E-34 J s 7.8E-08 elementary charge e 1.602176462(63)E-19 C 3.9E-08 electron mass me 9.10938188(72)E-31 kg 7.9E-08 proton mass mp 1.6726158(13)E-27 kg 7.9E-08 Faraday Constant F 96485.3415(39) C/mol 4.0E-08 Gas Constant R 8.314472(15) J/(mol*K) 1.7E-06 Once you have your data, you will perform a series of calculations using those experimental quantities to come to some final meaningful scientific quantity. The way in which your initial experimental uncertainties are dealt with throughout the calculations is called error propagation. The next section will deal with this topic in a minimal way as an introduction. If you take upper division science labs that require calculus, you may need to do a more complete analysis of the propagation of error.
Error Propagation You may have been shown several ways to treat error propagation in other classes. There is no single "right" way. However, you are expected to use the steps that follow for this exercise in analysis. Uncertainty in Sums and Differences Suppose that x, y, z are measured with absolute uncertainties δx, δy, δz, and the measured values used to compute the function: q = x + y z. The uncertainty in the final value δq is never larger than the sum of the absolute uncertainties. If the uncertainties are random and independent, then we can estimate the uncertainty in the final value δq using the following equation known as the quadrature rule: δq = ((δx) 2 + (δy) 2 + (δz) 2 ) 1/2 (Eq 1) Notice that the uncertainties are squared and then added and then square rooted. This eliminates the problem of negative errors and gives a plus or minus error answer. Uncertainty in Products and Quotients Suppose that x, y, z are measured with absolute uncertainties δx, δy, δz, and the measured values used to compute the function: q = x y z. The relative uncertainty in the final value δq q is never larger than the sum of the relative uncertainties. If the uncertainties are random and independent, then we can estimate the relative uncertainty in the final value δq q using the quadrature rule: δq q = ((δx/x) 2 + (δy/y) 2 + (δz/z) 2 ) 1/2 (Eq 2) Uncertainty in Complicated operations When a calculation involves a combination of several operations, simply break it down into its components and calculate the total error in a series of steps using Eq 1 and Eq 2. Many calculations involve more complicated operations than simply addition, subtraction, multiplication or division. Calculating the errors in this case involves calculus. Soooooo we will not be doing these in general chemistry. Other Notes There are some calculations you will perform that involve an exact number where you know the uncertainty of this number is zero. Can you simply ignore error propagation? No. Just use zero for the uncertainty of that number in your propagation formula.
Combining Systematic and Random Uncertainties In the best possible situation, a scientist eliminates as much systematic error as possible by calibrating all equipment, using equipment with the best precision possible and developing experimental procedures and techniques to reduce the systematic errors to a smaller order of magnitude than the random errors. In this case, only the random errors need to be considered in error analysis. However, this is not possible in a teaching lab, and we must consider how to include both systematic and random errors in our error analysis. There is no single correct method to do this, and no rigorous theory to show which is the better method than some other method. Sooooo. Here is what we are going to do. 1) Any measurements that you take repeatedly should be analyzed statistically to obtain a random error based on the standard deviation of your results. These random errors should be propagated through the calculations to result in a total random error in your final answer. 2) You should estimate an uncertainty associated with each measurement for which you take a single reading. Propagate these errors, in addition to any systematic errors of which you are aware, through your calculation and arrive at a total systematic error in your final answer. 3) Finally, use quadrature to combine your total systematic error with your total random error to arrive at an estimate of the total error in your result. δqtot = ((δq random ) 2 + (δq systematic ) 2 ) 1/2 (Eq 3) Formatting Graphical Analysis All graphs must be computer generated. All graphs must be of a suitable size, scale, and axes choice, so that the relevant features of the data are easily seen. Show units, have a title, label both axes, explain abbreviations, and have a legend identifying any symbols used if the graph shows more than one dataset. Do not have any shading in the graph background or gridlines (the Excel defaults). The title of your graph should be more descriptive than Y vs. X. Most experiments in this lab involve collecting a set of data that you will compare to a mathematical equation used to describe the phenomenon you were investigating. Data analysis of these experiments will involve plotting your data, finding the line of best fit, and deriving quantities from the slope and intercept of that line. If your plot is a series of points that you hope show some relationship, such as a linear calibration curve, then use the computer plot preferences to show the data points, and the line of best-fit. Do not draw any graphs that join the dots this is not good science. Although using your eye to estimate the best line may seem adequate, it is more thorough to use a least squares analysis to find the slope and intercept of the best linear fit to your data. The correlation coefficient R is a measure of how well a set of data fits a linear relationship. The value of R ranges from +1 to 1; values close to ±1 indicate that the points lie close to a straight line; values close to 0 indicate a lack of correlation.
Values such as R = 0.999943 are not uncommon and should be reported with the first one or two digits that are not 9s. In some experiments, you are given a table of literature values, say density of a liquid at different temperatures, and you need a value for the density at a different temperature that falls between the values in the table. In this situation, you use interpolation. To interpolate, you plot the literature values, find a line of best fit and use the equation of your fit to calculate the value you need. You do not have to show sample calculations for least squares analysis in your lab reports; you may just quote the answers from excel. If it seems obvious to you that there is a linear and a non-linear part to your data, then you may choose to only fit the linear portion of your data. That is your call, but you must discuss this in your conclusion section. Using the line-of-best fit in your data analysis 1) Determine the scientific equation that applies to the experimental data you are going to plot. 2) Decide which of the terms of that equation is the one that you varied during the experiment; e.g., time, temperature, pressure, frequency. That is your X variable. 3) Decide which of the terms of that equation is the one that you measured during the experiment; e.g., period, viscosity, pressure, capillary rise. That is your Y variable. 4) Rearrange your scientific equation so it is in the form Y = MX + B. If it is not possible to rearrange in this form, then your equation is not linear; e.g., if your rearrangement resembles Y = AX2 + BX + C, then you have a quadratic equation. 5) Use a computer program (Excel, etc.) to plot Y vs. X, noting that Y is always the vertical axis and X is always the horizontal axis. 6) If the data you have collected can be explained by a linear equation, and then if you collect enough data under appropriate conditions, you will see a linear trend in your data. Unless you have the "perfect" experiment, you will never measure experimental data that fits perfectly on a single line. Some experiments produce "better" data than others do. Use a plotting program to calculate the line of best fit equation using the least squares technique. Plot the line of best fit on your graph with your data, and show the equation of the line of best fit and the associated correlation coefficient on your graph. 7) By comparing your rearranged scientific equation with the equation of your line of best fit, you may derive the scientifically interesting information from the slope and intercept of the graph. The slope and intercept of the graph do have units. The y-intercept of a graph is a value of y when x is zero; therefore, the units of the intercept should be the same as the units of y. The units of the slope of a graph should be the same as the units of y divided by the units of x.
Error Analysis for Lab Reports The following items are required for full marks on your report error analysis: 1) Estimate the appropriate uncertainties (random and/or systematic) in each type of measurement you made. Make sure to label them as to what they are so that anyone grading your report can tell which type of error you suspect it to be. If you made repeated measurements of the same phenomenon, the random error in those measurements is the standard deviation. Use the uncertainties given in Tables of fundamental constants or unit conversions for any calculations that involve these values (label these as random error). Use the significant figures supplied to assign an uncertainty to all numbers used that you didn't measure, e.g., numbers supplied in the experiment Literature Values sections. 2) Use the Q-test to reject data if appropriate or use your secondary feeling technique making sure you qualify your analysis. (If you want to publish your work later, you would have to redo the experiment for better analysis options.) 3) Propagate the uncertainties in your data through all calculations. Write out one complete sample calculation, clearly showing an example of each step of your analysis and summarize the results of these calculations for all your data points in a table or spreadsheet. All calculations must include units. 4) Determine the uncertainty and hence the appropriate number of significant figures in each answer you report. 5) When using a best-fit line for data, calculate the error in the slope and intercept, (In excel regression, they are in the Standard Error part just to the right of the slope and y intercepts: Standard Coefficients Error Intercept 0 #N/A X Variable 1 0.993639968 0.001849388 and use them where necessary in your error propagation. In this example, the slope would be 0.993 g/ml +- 0.002 g/ml. You do not have to show sample calculations for the errors in least squares analysis in your lab reports. You may simply quote the answers from your computer spreadsheet. Then use these errors in further calculations if required for a final answer. 6) Create a separate section of your report for each lab, labeled Error Discussion. Categorize these possible sources of error as random, systematic, or gross. By considering the error propagation you have performed, identify which of these sources of error would have made the most significant contributions to the uncertainty in your final results. Justify your choices, and describe specifically how each of these significant errors would have affected your results (i.e., increased or decreased your answers). Give an explanation of reasonable methods you could use to eliminate or reduce each of these significant sources of error. 7) When you compare your results to literature values (or unknown actual values supplied by me) as part of your conclusions, comment on whether or not your literature values fall within the uncertainty range you calculated for your experimental results.