3.3: Estimating and Reporting Error (Statistics, Propogation, and Rounding)

In many physical chemistry experiments, the main goal is to obtain numerical results through processes like calculating averages or applying formulas derived from theory. However, obtaining a numerical result is not enough; we must also assess its quality. "How accurate is the result?" Accuracy expresses the degree of uncertainty in the result. Claiming excessive accuracy without justification is misleading, while understating accuracy diminishes the result's value and wastes 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 propagate error. 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 is found online here (click). (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:

1. Increase the last retained digit by 1 if the leftmost digit to be dropped is more than 5 or 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 leftmost digit to be dropped 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 15.33
     27.450 rounds to 27.45
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 34%!

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.