Calculating with numbers: Decimal numbers
Extra: a bird's eye view of error analysis
Random measurement errors are made in scientific experiments; this is already apparent when measurements are repeated and the exact same measurement result is not always obtained. This phenomenon is called spread or dispersion. In this theory page we transcend precalculus with a brief treatment of statistics-based error theory. The error analysis below is the real deal, but the previously described method of significant digits has a few simple rules with which an underpinned result can be obtained quickly.
A measurement of the same mass of a metal block on 3 different balances, measured in quick succession, can for example lead to three different results: \(34.7251\), \(34.7256\) and \(34.7193\) (grams). Possible reasons are that the measurement was carried out by different people, that the balances are of a different type or are adjusted differently (e.g., the zero point), intermediate contamination of the object, and so on. It seems that the numbers \(34.7\) can be considered "certain", but that the following decimals are uncertain. The second decimal is a 2, but perhaps also a 1 or 3. The measurement result therefore fluctuates around \(34.72\) with a dispersion of \(0.01\). This is often denoted as \(34.72\pm 0.01\) and the result can therefore only be meaningfully written in 4 significant digits. If one only writes down the number \(34.72\) with 4 decimal places as the measurement result, it is usually assumed that the last digit is uncertain and a 1 can just as well be added or subtracted.
To link a numerical value to the spread, different measures of spread can be used. In the example above, we looked at the measurement data and made a choice based on this. But we could also have looked at the range, i.e. the difference between the largest and smallest data value. In this example that is \(34.7256-34.7193=0.0063\) and from that you would also get a smaller spread of \(0.004\) around the average value of \(34.723\) (rounded up to 3 decimal places). In this way we use a more accurate measurement result of \(34.723\pm 0.004\).
A statistical measure for the spread of measurement results Assuming random measurement errors, a statistics-based approach with concepts such as variance and standard deviation is preferred. For a quantity \(X\) one first calculates the mean \(\overline{X}\) of \(n\) results via the formula \[\overline{X}=\frac{1}{n}\sum_{i=1}^n X_i\] Then the so-called variance \(S^2\) is calculated via the formula \[S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i-\overline(X))^2\] Hereafter the square root is taken and the standard deviation \(S\), also called absolute error, is calculated as a statistical measure for the spread of the measurement results.
Without going into details, we mention the calculation rules for calculating the standard deviation in addition, subtraction, multiplication and division of two or more measured quantities.
Rules for statistical spread in a linear combination If a quantity \(U\) is an arbitrary linear combination of directly measured independent quantities, say \[U=aX+bY+cZ+\ldots\] with coefficients \(a\), \(b\), \(c\) and quantities \(X\), \(Y\), \(Z\), then the variance of \(U\), denoted as \(S_{U}^2\), is given by the following formula in terms of variance of \(X\), \(Y\), \(Z\), etc. (denoted as \(S_{X}^2\), \(S_{Y}^2\), \(S_{Z}^2\) ): \[S_{U}^2=a^2S_{X}^2+b^2S_{Y}^2+c^2S_{Z}^2+\cdots\] One says: "quadratic addition of absolute errors." This formula is especially useful for a sum or difference when the coefficients are \(\pm1\).
Rules for statistical spread in a product and quotient with exponents For a quantity \(U=kX^aY^bZ^c\cdots\) with an arbitrary coefficient \(k\), exponents \(a\), \(b\), \(c\), and quantities \(X\), \(Y\), \(Z\), the variance of \(U\), denoted as \(S_{U}^2\), is given by the following formula in terms of mean of \(X\), \(Y\), \(Z\), and so on (denoted as \(\overline{X}\), \(\overline{Y}\), \(\overline{Z}\) ) and variance of \(X\), \(Y\), \(Z\), and so on (denoted as \(S_{X}^2\), \(S_{X}^2\), \(S_{X}^2\) ): \[\frac{S_{U}^2}{\overline{U}^2}=a^2\frac{S_{X}^2}{\overline{X}^2}+b^2\frac{S_{Y}^2}{\overline{Y}^2}+c^2\frac{S_{Z}^2}{\overline{Z}^2}+\ldots\] This formula can also be written as follows: \[\left(\frac{S_{U}}{\overline{U}}\right)^2 =a^2\left(\frac{S_{X}}{\overline{X}}\right)^2+b^2\left(\frac{S_{Y}}{\overline{Y}}\right)^2+c^2\left(\frac{S_{Z}}{\overline{Z}}\right)^2+\cdots\] One speaks of: "quadratic addition of relative errors." This formula is especially useful for a product or difference when the exponents are \(\pm1\).
Suppose you read an initial value \(12.34\pm0.05\) and a final value \(5.67\pm0.05\) (mL) during a tritration with a class A burette. The consumption is the difference between these measured values. You may tend to think here that the error would be \(0.05 +0.05=0.1\) because it is the biggest possible error; you have then applied the so-called variation method. But because random errors can cancel each other out, it is better to use the calculation rule for statistical spread for a difference: you have to add the absolute errors quadratically and get the standard deviation \[S=\sqrt{(0.05)^2+(0.05)^2}\approx 0.07\qquad\text{rounded to 2 decimal places.}\] The statistical spread is smaller than the largest possible error. So the result becomes \(6.67\pm 0.07\,\).
