How to Calculate with Uncertainty

“It is better to be roughly right than precisely wrong.”
— Alan Greenspan

While calculating with exact numbers is easy 4th grade math, scientific calculations with approximate measurement values require some special considerations. Here is how to do it roughly right.

The correct notation

Every physical quantity, such as the gravity constant1 $$\mathrm{G} \approx 6.674\,08(31) \cdot 10^{-11} \;\mathrm{m^{3}\,kg^{-1}\,s^{-2}}$$

should be correctly rounded, state the uncertainty, and specify the SI units using exponents. Use small spaces to separate digits and units. Use a decimal point . and not a comma. If we want to discuss and express uncertainty, we need to know two properties of a given value:

  1. Significant Figures: All digits without leading zeros,
    e.g. $\underline{1.2}$/$0.0\underline{20}$ have both 2 significant figures.
  2. Decimal Places: All digits after the decimal point,
    e.g. $1.\underline{2}$/$0.\underline{020}$ have 1 and 3 decimal places respectively.

Measurements

Since no measurement of a physical quantity is infinite accurate, it is important to determine how much a measured value will most likely deviate from the unknown, true value of the quantity. There are three important metrics for the quality of a repeated measurement:

  • Accuracy: Closeness of the measured average to the true value.
  • Precision: Closeness of repetitive measurements to each other.
  • Resolution: Amount of steps in which a measurement can be expressed.

Examples of measurements with low and high accuracy, precision, and resolution. The center represents the true value and each yellow dot is a measurement sample.

Determine Uncertainty by Standard Deviation

If you have three measurements $X = { 1.27316, 1.43869, 1.32584 }$ of a quantity $x$, your resolution is $6$ digits but you need to determine accuracy and uncertainty of this measurement. The average is $\mu_x = 1.34589\overline{6}$ and the standard deviation is $\sigma_x = 0.084568...$. The uncertainty is normally set to the standard deviation and so you can determine the final result according to these rules:

  1. Round the uncertainty $\sigma_x$ to one significant figure. If the first significant figure of the uncertainty is a 1 or you are very confident in the accuracy of your uncertainty estimation, you may round the uncertainty to two significant figures
    $\sigma_x = 0.0\underline{8}4568... \approx 0.08$
  2. Round the estimated value to the same decimal places as the uncertainty.
    $\mu_x = 1.3\underline{4}589\overline{6} \approx 1.35$
  3. Report the final value as: $x = \mu_x \pm \sigma_x = 1.35 \pm 0.08$

If the measured value has a unit, e.g. resistance in $\Omega$, then the following notations are valid:

$x = 1.35\,\Omega \pm 0.08\,\Omega = (1.35 \pm 0.08)\,\Omega = 1.35(8)\,\Omega = 1.35(0.08)\,\Omega$

However, the notation $1.35(8)\,\Omega$ should be preferred because the $\pm$ symbol has been traditionally used to indicate a higher accuracy than the standard uncertainty.

Calculating with Uncertainty

When calculating with estimated values you need to follow the error propagation rules and thus adding $a = 3.44 \pm 0.23$ and $b = 0.72 \pm 0.13$ results in

$$a + b = (\mu_a + \mu_b) \pm \sqrt{\sigma_a^2 + \sigma_b^2} = (3.44 + 0.72) + 0.26419... \approx 4.16 \pm 0.26$$

Since this is very tedious and verbose, many calculations involve just rounded values without giving the uncertainty explicitly. To do so you need to round the estimated value such that its significant figures closely match the uncertainty. Since, a rounded value $3.4$ implies $3.40 \pm 0.05$, we use the following rule:

Round the estimated value to one digit before the first significant figure of the uncertainty.
Examples: $3.\underline{0}0 \pm 0.0\underline{2} \approx 3.0 \qquad 3.\underline{1}4 \pm 0.0\underline{9} \approx 3.1$

Calculating with Rounded Values

Calculating with rounded values is the lazy way to demonstrate that you somehow know and care about accuracy and uncertainty without specifying it accurately. It allows you to roughly determine the uncertainty of a value in magnitudes of $10$. So $3.1$ corresponds to $3.10 \pm 0.05$ and $3.14$ corresponds to $3.140 \pm 0.005$.

Multiplication / Division: Reduce the sig. fig. of the result to the least sig. fig. of the operands.
Example: $\underline{12.3} \cdot 0.5323 = \underline{6.54}729 \approx 6.55$

Addition / Subtraction: Reduce the dec. pl. of the result to the least dec. pl. of the operands.
Example: $18.585 - 16.\underline{13} = 2.\underline{45}5 \approx 2.46$

Exponent: Reduce the sig. fig. of the result to the dec. pl. of the operand.
Example: $3.\underline{05}^{2.010} = \underline{9.4}068... \approx 9.4$

Logarithm: Reduce the dec. pl. of the result to the sig. fig. of the operand.
Examples: $\log_2(\underline{10}) = 3.\underline{32}1928... \approx 3.32 \qquad \log_2(\underline{109}) = 6.\underline{643}856... \approx 6.644$

Series of Calculations

Only round the result, not the intermediate values! If you continue calculation, calculate with the value before the rounding but use the rounded value when estimating the significant figures of your new result. More specifically if you have a long equation with several operations:

  1. Calculate without any rounding, this is your correct result
  2. Calculate with rounding after each operation, this is your wrong but correctly rounded result. You can also just calculate the number of significant digits for each operation by appling the rounding rules directly after each operation without calculating any value.
  3. Round the result from 1. to the sig. fig. / dec. pl. obtained from 2. This is your correct result, correctly rounded.

References


  1. The NIST Reference on Constants, Units, and Uncertainty: Newtonian constant of gravitation ^