4.9.10.8: Learning Objectives, Important Info

In many physical chemistry experiments, the goal is to obtain numerical results by taking measurements and calculating averages or applying theoretical formulas. However, simply producing a number isn't enough – it's essential to evaluate its quality. How accurate is the result? Accuracy indicates the level of uncertainty associated with the measurement. Overstating accuracy without proper justification can be misleading, while understating it can undervalue the result and lead to wasted effort and resources.

This module will deal with two main types of error analyses. You will learn when and how to use each type.

1. Statistical Analyses are done when there are multiple replicates of the same measurement(s).
2. Error Propagation is done when using measured values to calcuate another numerical value.

A major goal of this module is for you to learn when and how to apply these equations to perform statistical analysis or propagate errors through a function. This skill will be required in nearly every future module in this course.

A summary of useful information and equations is below.

1. Statistical Analyses (on replicate measurements):

Average or mean: The average (or mean) of N measurements on the variable \(x\) is: \[\bar{x}=\frac{1}{N} \sum_{i=1}^N x_i\]

Standard deviation (\(S\)) is the spread around the mean of several replicate measurements. It has the same units as the measurement and is commonly used to report uncertainty (see Skoog Ch. 6 p 126) \[S=\sqrt{\frac{\sum_{i=1}^N\left(x_i-\bar{x}\right)^2}{N-1}}\]

Variance (\(S^2\)) is the dispersion between data points. The units are the square of the measured units.

The Confidence Limits (CL) are the values that define the confidence interval (which is the bounds about which an experimental value should be located with a given probability). The confidence limit is useful for small sample sizes. A common confidence limit is the 95% confidence limit, and its value is given by the following (see Skoog Ch7 pg 150 and Shoemaker Ch 2 eq 34). \[CL_{0.95} = t_{0.95} S_m = t_{0.95} \frac{S}{\sqrt{N}}\]

or more generally \[CL = \bar{x} \pm \frac{tS}{\sqrt{N}}\]

where \(t\) is the critical \(t\) value (from Student's \(t\) test) that depends on the degrees of freedom and the given probability. A table of critical \(t\) values can be found online. (See also Shoemaker Ch 2, Table 3)

2. Propagation of error (for values determined from a function):

Using partial derivatives

To propagate error to determine the error (\(\Delta\)) in the value derived from of a function (\(F\)), we must propagate error of each part of the function. This is accomplished by taking the square root of the sum of the squares of the partial derivative of the function with respect to each variable (eg partial derivative with respect to \(x\) is \(\frac{\partial F}{\partial x}\)), multiplied by the error in that variable (eg error in x is \(\Delta(x)\), and the square of the error is \(\Delta ^2(x)\). In mathematical form, this is given below (from Shoemaker Ch 2 p 56, eq 54). If you need to brush up on partial derivatives, check out this video (click here). https://youtu.be/SbfRDBmyAMI

\[\Delta^2(F)=\left(\frac{\partial F}{\partial x}\right)^2 \Delta^2(x)+\left(\frac{\partial F}{\partial y}\right)^2 \Delta^2(y)+\left(\frac{\partial F}{\partial z}\right)^2 \Delta^2(z)+\ldots\]

It is convenient to rearrange this equation as follows to give the error in the value derived from the function:

\[\Delta(F)=\sqrt{\left(\frac{\partial F}{\partial x}\right)^2 \Delta^2(x)+\left(\frac{\partial F}{\partial y}\right)^2 \Delta^2(y)+\left(\frac{\partial F}{\partial z}\right)^2 \Delta^2(z)+\ldots}\]

Simpler shortcuts:

In some cases, there are shortcuts that allow you to approximate errors without taking the partial derivatives. In the examples below, assume that \(a, b, c, n\) are constants and \(x, y, z\) are variables with associated errors \(\Delta x, \Delta y, \Delta z\):

For functions with only addition and subtraction of values with assiciated error (eg \(F = ax \pm by \pm cz\)), we can use the following "shortcut": (Shoemaker, Ch 2, p 57, eq 55) \[\Delta(F)=\sqrt{a^2 \Delta^2(x)+b^2 \Delta^2(y)+c^2 \Delta^2(z)}\]

For functions that involve only multiplication and division (eg \(F = axyz\) or \(F = axy/z\) or \(F=a/xyz\)), we can use the following "shortcut": (Shoemaker, Ch 2, p 57, eq 56) \[\Delta(F)=F\sqrt{\frac{\Delta^2(x)}{x^2}+\frac{\Delta^2(y)}{y^2}+\frac{\Delta^2(z)}{z^2}}\]

For functions that involve taking a variable to the power of a constant (eg \(F = ax^n\)), we can use the following "shortcut": (Shoemaker, Ch 2, p 57, eq 57) \[\Delta(F)=F\sqrt{n^2 \frac{\Delta^2(x)}{x^2}} \rightarrow \Delta(F)=F n \frac{\Delta(x)}{x}\]

For functions where a variable is an exponent (eg \(F = ae^x\)), we can use the following "shortcut": (Shoemaker, Ch 2, p 57, eq 58) \[\Delta(F)=\sqrt{a^2 e^{2 x} \Delta^2(x)} \rightarrow \Delta(F)=F\Delta(x)\]

For functions taking the logarithm of a variable (eg \(F = a \ln(x)\)), we can use the following "shortcut": (Shoemaker, Ch 2, p 57, eq 59) \[\Delta(F)=\sqrt{\frac{a^2}{x^2} \Delta^2(x)} \rightarrow \Delta(F)=a \frac{\Delta(x)}{x}\]

3. Reporting Numeric Results

All numeric results in this course should be reported with explicit statement of uncertainty. The following format is expected: 

\[\text{Numeric Value } \pm \text{Uncertainy} \]

In your written assignments for this course use the following guidelines when reporting a "final" numeric value. You will be required to use your best judgement and to provide a written justification of your logic for reporting all final values. More detailed information is found below the guidelines and also in the Shoemaker text.

Guidelines for reporting final numerical values:

• Rules for reporting significant figures should be followed.
• The number of useful decimal places in the uncertainty should match the number of decimal places reported in the value.
• The certainty of a numeric value determined from a function cannot be be more than the certainty of its operands.

A summary of rules for reporting significant Figures (from Shoemaker Ch 2)

The following rules apply only for reporting the final value, not for intermediate rounding steps during calculation. In general, do not round values during the calculation process if it can be avoided. However, if you must round prior to comleting a calcaution, then retain at least one digit more than would be considered "significant", and the more the better. 

Rules for Rounding Final Values:

Follow the following rules depending on the digit immediately to the right of the place you are rounding. The examples below are rounding to the place underlined, and the digit immediately to the right is bolded.

1. Increase the last retained digit by 1 if the nearest dropped digit is more than 5, or if it is 5 followed by any non-zero digits:
   • Examples:
     6.789 rounds to 6.79
     54.9123 rounds to 54.9
2. Leave the last retained digit unchanged if the nearest dropped digit is less than 5:
   • Examples:
     2.346 rounds to 2.35
     0.2123 rounds to 0.21
3. If the leftmost digit to be dropped is 5 followed by no digits except zero, then increase the last retained digit by 1 if it is odd, and leave it unchanged if it is even:
   • Examples:
     15.325 rounds to 15.32
     15.335 rounds to 15.34
4. There are times when you should retain an extra digit so that your reported uncertainty is reasonable.

Rules for tracking significant figures for calculated values

Addition: When adding numeric values, the result's precision is limited by the least precise value involved. The number of decimal places in the result should match the fewest decimal places in any component. For example:
\[\begin{aligned}
& 15.4\\
+ & 2.87 \\
+ & 7.621 \\
= & \hline 25.916 \rightarrow 25.9
\end{aligned}\]

Subtraction: When subtracting, the result's precision is limited by the least precise value involved, and the precision of the result will always be less than that of the operands. For example:
\[\begin{aligned}
& 289.461 \\
- & 288.58 \\
& \hline 0.881 \rightarrow 0.88
\end{aligned}\]

Let's do an analysis to illustrate what this means. In the example above, let's assume an uncertainty of \(\pm 3\) in the rightmost digit of each number. Based on this assumption, the least precise operand has uncertainty of ~ 0.01%, while the final reported value has an uncertainty of 3.4%!

When multiplying and dividing: In multiplication and division, the precision of the result is limited by the least precise number involved. The number of significant figures in the result should align with the component having the fewest. Note that relative precision isn't solely dependent on the number of significant figures; for instance, 99 with two significant figures is approximately as precise as 101 with three. When unsure, do an analysis of the relative uncertainty, and if the uncertinties are approximately equal, use the larger number of significant figures.

Conventions for Reporting and Interpreting Reported Values

1. Numerical values are typically considered uncertain in the last digit by ±3 or more, with perhaps slight uncertainty in the next-to-last digit by up to ±2.
2. If the last digit to be retained is a one or two, an additional digit is retained so that precision is not inappropriately reported due to rounding. 

 

Guidelines

1. Before you leave the lab, remember to do the following:
   (a) Make note of the uncertainties in each of the physical quantities you measure. Follow the suggestions in step (2) below. Consult your TA concerning systematic errors that may arise from instrumental calibration and measurements.
   (b) Be aware of the consequences of systematic errors, such as starting the clock too late in one of the kinetics experiments.
2. Some rules of thumb on estimation of uncertainties in mass, volume, pressure and temperature measurements
   (a) Find and record the uncertainties associated with your equipment and instrumentation. This information is often listed right on the device, or is easily obtained from the manufactures statement of equipment specifications. In the rare event that such information is not available, you may assume that single experimentally measured values are uncertain in the last digit by at least ± 0.5 of the smallest unit or measurable digit. For example, if a digital voltmeter reads 5.432 V, and no other information is available, the uncertainty may be estimated to be ± 0.005 V.
   (b) The uncertainty of our analytical balances is either ± 0.0001 g or ± 0.0002 g, You can often find it stated on the individual balance and associated with the letter d (for deviation). For example, the label might say d=0.1 mg.
   (c) The position of the liquid level in a burette or a graduated 1 mL Mohr pipet can be estimated to ± 0.05 mL (half of the smallest unit measurable). Initial and final readings must be made, however, so that the probable error in the volume measured will be, \[ \sqrt{\left ( 0.05 \right )^2 + \left ( 0.05 \right )^2} = 0.07 \]A similar analysis can be applied to pressure determination by measurement of mercury levels in a U-tube manometer.
   (d) For non-graduated volumetric pipets, the error limits can be found right on the pipet itself, or in the " Comparison of Class A and B Volumetric Glassware” link, found in the External Links section of your Sakai site or from other resources readily available online. You might also look in the Baxter Scientific catalog found in the lab. This catalog is a useful source for tolerances and error limits of thermometers, balances, pipets, burets, volumetric flasks and graduated cylinders.
3. In the case of Replicate or Single Measurements
   (a) If you make a series of replicate measurements of the same quantity, you should report the mean of that quantity together with its precision (standard deviation, variance or 95% confidence limits) calculated by statistical analysis. Unless otherwise requested in the lab manual, use the standard deviation as a measure of precision. If the accepted value of the quantity has been reported in the literature, you might also report the percent error or absolute error as a measure of the accuracy of your measurement(s). If you feel that one of the replicate measurements should be rejected, use the Q-test to justify your decision.
   (b) If you make a single measurement of a quantity, give your best estimate of the uncertainty in that measurement, whether random or systematic. If appropriate (see 2.2.3(a) above), you should also report the accuracy of that measurement. The suggestions given in Step (2.2.2) may help. Consult your TA if in doubt.
4. Graphical Analysis
   (a) If you use graphical analysis to determine the value of any quantity, you must make a decision whether or not to use a weighted least squares regression analysis. In most cases, a non-weighted analysis (using the command lsq in Matlab) will suffice. If you perform a weighted least squares analysis, you will have to decide on what weighting factor, e, to use. If the data deviate systematically from linearity, a linear regression analysis is meaningless, and should not be performed.
   (b) Outliers may be rejected only on the basis of a Q-test performed on the residuals.
   (c) The standard deviation may be used as a measure of the uncertainty in the slope and intercept calculated by linear least squares regression.
   (d) In most cases, you will use the values of the slope and/or y-intercept to calculate a physical quantity. In either case, you must perform a propagation of error treatment to determine the uncertainty in that physical quantity.
5. Propagation of Error
   Whether or not you perform a graphic analysis, you will use experimentally determined variables to calculate one or more physical quantities. In each case you must estimate the uncertainty in each measured variable and then propagate that uncertainty through to the final result. In a few simple cases you may take advantage of Equations (55) - (59) (Shoemaker et. al., 1996). Usually however, you will use Equation (54) (Shoemaker et. al., 1996), and take partial derivatives to calculate the probable propagated error. Physical constants, such as the gas constant, R, may be assumed to be error free.
6. Reporting the Final Numerical Result(s) in your Notebooks and Reports
   (a) Each numerical result should be reported together with the associated uncertainty, which is usually the probable propagated error. In addition, and where possible, you should report the accuracy of your result by comparison with literature values and calculation of the absolute error.
   (b) In reporting your results, ensure that the use of significant figures is appropriate to the uncertainty in those results. Use the conventions for significant figures and for rounding off data described earlier in this text.
7. Discuss Qualitative Uncertainties in your Notebooks and Reports
   In some cases, it may be difficult to quantitatively estimate the uncertainty in an experimental variable. In this instance, you may be asked to provide a qualitative discussion of the sources of uncertainty in the experiment. This discussion should be as thorough as possible. You should include "reasonable" estimates of the uncertainties in measured parameters together with a discussion of how these uncertainties will affect the final result. Any estimates of uncertainty should be substantiated by hard facts!!

And finally, if you read nothing else .....

1. Estimate the uncertainty in each experimentally determined variable. Usually this involves one or more of the following:

(a) Calculate the standard deviation of a quantity determined from the mean of a series of replicate measurements.

(b) Find and record the uncertainties associated with your equipment and instrumentation. This information is often listed right on the device, or is easily obtained from the manufactures statement of equipment specifications. In the rare event that such information is not available, you may assume that single experimentally measured values are uncertain in the last digit by at least ± 0.5 of the smallest unit or measurable digit. For example, if a digital voltmeter reads 5.432 V, and no other information is available, the uncertainty may be estimated to be ± 0.005 V.

(c) Perform a simple propagation of error analysis (usually equation (54) (Shoemaker et. al., 1996)) if a quantity (e.g., volume, pressure) is determined from a difference of two or more readings.

(d) Extraction of a Physical Quantity from Graphical Analysis: Calculation of the standard deviation of the slope (or intercept) is followed by a propagation of error treatment of the functional form of that slope (or intercept) to give the uncertainty in the physical quantity to be extracted. If the data deviate systematically from linearity, a linear regression analysis is meaningless and should not be performed.

(e) If the source or estimation of uncertainty seems more obscure, -- CONSULT YOUR TA BEFORE YOU LEAVE THE LAB.

2. Once you have determined the uncertainties associated with each individual measurement, perform a propagation of error treatment to estimate their combined effect on the uncertainty of the physical quantity of interest. Use equation (54) or some combination of equations (55) - (59) (Shoemaker et. al., 1996). Physical constants such as the Gas Constant, R, may be assumed to be error free.

3. Report your final result(s), the probable propagated error associated with that result, and the percent error or absolute error if appropriate literature values are available.