This page collects useful information about the estimation and propagation of errors, and about the presentation of values to the correct number of significant digits.

*(Portions of this text were contributed by James Geddes.)*

Below are a few useful references about estimating and propagating uncertainties.

- John R. Taylor,
*An Introduction to Error Analysis,*2nd Edition, University Science Books, 1997.

Measurements are never perfect. That is, all measurements have some inherent uncertainty. Uncertainties in this sense must be distinguished from “mistakes” made while measuring. Mistakes may be eliminated, while uncertainties may only be minimized. The analysis of uncertainties is often loosely referred to as “error analysis.”

If you are trying to account for some discrepancy between your results and expected values, “experimenter mistakes” and “equipment failure” are *never* acceptable explanations. If you think you did something wrong, you should repeat the measurement correctly. If you think your apparatus is faulty, you should seek help from your TA or the lab staff. Once these problems are eliminated, uncertainties fall into two basic categories: *systematic* and *statistical uncertainties*.

**Systematic uncertainties** result in a bias in the measurements, usually caused by incorrect assumptions about the measurement.

**Statistical uncertainties** are random fluctuations of data due to

- inherent randomness in the measured quantity, or
- limits on precision which always exist in measurements.

Both kinds of uncertainties can be evaluated in order to determine whether they can account for the discrepancies observed. The uncertainty (or error) analysis formalism given below is used to estimate the magnitude of statistical uncertainties. Systematic uncertainties are usually more subtle, because they often involve assumptions rather than measurements. However, their magnitudes can usually be estimated and you can usually predict how they will affect your results. For example, some bias may make all measured values too large by the same amount. Such bias might affect the intercept but not the slope of a linear plot of data. It is important to think about whether the uncertainties that you cite can really explain the discrepancies you observe. If you are unable to think of anything which seems to explain your discrepancies, you should discuss it with your TA.

Error analysis is the process whereby you determine how confident you are in the results of your experiment, assuming that the experimental design is good. You do so by assigning an uncertainty to each of the data values that you measure and by propagating these uncertainties through any calculations to the final result.

Usually one infers the uncertainty in each measurement from the experimental procedure. For example, the smallest time interval measurable by a stopwatch may be 0.01 seconds; however, due to human reaction time, the uncertainty, $\Delta t$, in a time measurement may be as much as ±0.2 seconds. In this case, the uncertainty introduced by reaction time is much larger than the precision of the stopwatch. The larger source of uncertainty is said to be _dominant. _For the purpose of this lab the smaller uncertainty may be ignored and the reported $\Delta t$ would be 0.2 s. In order to estimate uncertainties such as human response time it is common practice to make repeated measurements and note the scatter.

In some cases it may be difficult to estimate the uncertainty as in the example above, but if we can measure the quantity several times, the uncertainty in each measurement can be *calculated* from the numerical scatter of the data. A commonly-used measure of scatter is the *standard deviation* and is defined as follows:

$\sigma = \sqrt{\dfrac{\Sigma_{i=1}^N(x_i -\bar{x})^2}{N-1}}$. | (1) |

Here each value, $x_i$, is an individual measurement, $\bar{x}$ is the mean value of the measurements and $N$ is the total number of measurements. It can be shown that the *uncertainty in the mean* is

$\Delta \bar{x} = \dfrac{\sigma}{\sqrt{N}}$. | (2) |

One measurement is often not enough to make a reasonable estimate of the uncertainty. A good example of this is a measurement of the time something takes to happen. In this case, if it is at all possible, it is a good practice to measure that quantity a large number of times (ten is usually a large enough number).

The final result of the measurement would then be an average of the different values obtained, and a more informed estimate of the uncertainty of this average is given by the standard error, which is simply the standard deviation of the sample divided by the square root of the number of measurements.

If for any reason it is impossible to measure this quantity a large number of times for every configuration of the system, doing it for one configuration still allows us to obtain a more informed estimate of the uncertainty: the standard deviation of the sample is the uncertainty of each individual measurement.

A good example of this can be found in the Millikan oil drop experiment, where fall and rise times are measured for each oil drop several times. However, while fall times should remain constant, each measurement of the rise time is a separate data point, since the rise times depend on the charge of the drop and that can change between measurements. The uncertainty on the fall times should then be given by the standard error applied to that set of measurements, while a reasonable estimate for the uncertainty on the rise times can be given by the standard deviation on the fall time measurements for a typical drop.

If we are talking about exact numbers, then you know that 1.4 = 1.40 = 1.400 (and so on) for any number of zeros after the four. However, in an experimental physics setting, most numbers are measurements made about the physical world, and therefore the number of digits you present has a meaning; you write down *only* those digits that are significant. When the value is an experimental measurement, it must have an uncertainty associated with it, and then determining which digits are significant is an easy task: we should only present one or two digits in the uncertainty, and the value of the measurement is significant only up to that precision. A few examples are shown in Table 1.

$7.40 \pm 0.3$ | wrong |

$7.4 \pm 0.32$ | wrong |

$740.345 \pm 32.189$ | wrong |

$7.4 \pm 0.3$ | right |

$6.0 \pm 0.05$ | wrong |

$6.00 \pm 0.05$ | right |

**Table 1:** How many digits to keep after uncertainty starts.

Note that these rules should be obeyed *whenever* you present a value – be it in the text, in a table, or on an annotation on a plot. This doesn’t mean, however, that all other digits should be thrown away. For intermediate values entering calculations, we should always keep as many digits as possible, even if we don’t show them. If all your analysis is made in the same platform (for example, a python script) this should happen automatically.

Two rules for using significant figures are given in the Taylor book, *An Introduction to Error Analysis,* on page 15:

**Rule for Stating Uncertainties:**“Experimental uncertainties should almost always be rounded to one significant figure.”

*Exception*: If the leading digit in the uncertainty is a 1 or a 2, then it may be better to give two digits in the uncertainty, since, for example, rounding, from 1.4 to 1 would be a large proportional reduction of the uncertainty.

**Rule for Stating Answers:**“The last significant figure in any answer should usually be of the same order of magnitude (in the same decimal position) as the uncertainty.”

For example, the expression $1.234 \pm 0.06$ is WRONG! If the uncertainty is 0.06, then the 4 in the value is meaningless. Contrast this with the expressions $1.234 \pm 0.006$ and $1.23 \pm 0.06$ which both have the correct number of significant digits.

Often, the goal of an experiment is not obtaining the data alone, but arriving at some quantity *calculated* from the data. Uncertainties in the data give rise to uncertainties in the calculated quantity also. Determining the uncertainty in the calculated quantity from the uncertainties in the measured quantities is is called uncertainty *propagation.* Correct propagation of uncertainties is needed for proper evaluation of the final result.

Let us represent the final calculated result as the function, $f(x_1,x_2,\dots,x_n)$, which is a function of the (uncorrelated) measured quantities $x_1,x_2,\dots,x_n$. By the chain rule of calculus, the change in a function $f(x_1,x_2,\dots,x_n)$ given in terms of the change $dx_i$ is

$df = \left(\dfrac{\partial f}{\partial x_1}\right) dx_1 + \left(\dfrac{\partial f}{\partial x_2}\right) dx_2 + \dots + \left(\dfrac{\partial f}{\partial x_n}\right) dx_n$. | (3) |

If we average over a large ensemble of such measurements, and assume

- the x’s are Gaussian distributed, and
- the x’s are uncorrelated,

then the root mean square (rms) deviation of the changes in *f* is given in terms of the rms deviation of each of the $x_i$ by

$\Delta f = \sqrt{\left(\dfrac{\partial f}{\partial x_1}\Delta x_1 \right)^2 + \left(\dfrac{\partial f}{\partial x_2}\Delta x_2 \right)^2 + \dots + \left(\dfrac{\partial f}{\partial x_n}\Delta x_n \right)^2 }$. | (4) |

Examples of Eq. (4) applied to common forms of $f$ are given below. You should use Eq. (4) to derive them yourself.

If $f(x,y,z) = x - y + \dots +z$, then Eq. (4) becomes

$\Delta f = \sqrt{(\Delta x)^2 + (\Delta y)^2 + \dots + (\Delta z)^2 }$. | (5) |

If $f(x,y,z) = \dfrac{xy}{z}$, then Eq. (4) becomes

$\dfrac{\delta f}{f} = \sqrt{\left( \dfrac{\delta x}{x}\right)^2 + \left( \dfrac{\delta y}{y}\right)^2 + \left( \dfrac{\delta z}{z}\right)^2}$. | (6) |

If f(x) = x^n$, then Eq. (4) becomes

$\dfrac{\delta f}{f} = n \dfrac{\delta x}{x}$. | (7) |

If $f(x) = 1/x$, then Eq. (4) becomes

$\dfrac{\delta f}{f} = \dfrac{\delta x}{x}$. | (8) |

(Note that reciprocals are just a special case of products or quotients.)

Given below is a practical procedure for approaching the error analysis of all of the experiments in this course.

We use as an example the measurement of the acceleration due to gravity using a simple pendulum. The relevant relationship is

$T = 2\pi\sqrt{\dfrac{L}{g}}$. | (9) |

where $T$ is the period of the pendulum, $L$ is the length of the pendulum and $g$ is the acceleration due to gravity. Measuring $L$ and $T$ enables us to calculate $g$.
**STEP 1**: Write the theoretical expression for the desired (calculated) quantity in terms of all the measured quantities of the experiment.

**EXAMPLE:** Re-write Eq. (9) to isolate $g$ as follows:

$g = 4\pi^2LT^{-2}$. | (10) |

Here, the calculated $g$ is expressed as a function of the measured quantities, $L$ and $T$.
**STEP 2:** If the function describing the experiment is…

- …a sum or difference, use Eq. (5) to calculate the
*absolute*uncertainty. - …a product or quotient, use Eq.(6) to calculate the
*fractional*uncertainty. - …a power, use Eq. (7) to calculate the
*fractional*uncertainty. - …another function, take partial derivatives and use Eq. (4).

**EXAMPLE:** Eq.(10) is a product of two factors, $L$ and $T^{-2}$, therefore we can calculate fractional uncertainties using Eq. (6) as

$\dfrac{\Delta g}{g} = \sqrt{\left(\dfrac{\Delta L}{L}\right)^2 + \left(\dfrac{\Delta(T^{-2})}{(T^{-2})}\right)^2}$. | (11) |

At this point, we need to figure out what our second term involving $\Delta T^{-2}$ is. We can treat this as a new equation with the form of Eq. (7) with $n=-2$. With that, we find that

$\dfrac{\Delta T^{-2}}{T^{-2}} = -2 \dfrac{\Delta T}{T}$. | (12) |

Combining Eq. (11) and Eq. (12), we finally get our uncertainty in $g$ to be

$\dfrac{\Delta g}{g} = \sqrt{\left(\dfrac{\Delta L}{L}\right)^2 + \left(-2\dfrac{\Delta T}{T}\right)^2}$. | (13) |

In order to evaluate Eq. (13) we need some numerical data. Suppose you have measured $L$ to be $99.0 \pm 0.1$ cm, and $T$ to be $2.000 \pm 0.004$s. We may obtain our “best value” of $g$ using Eq. (10) as

$g_{best} = \dfrac{4\pi^2(99.0 cm)}{(2.00s)^2} = 977\,\mathrm{ cm/s^2}$. | (14) |

To obtain the uncertainty in $g$ we apply Eq.(13) to find

$\dfrac{\Delta g}{g} = \sqrt{\left(\dfrac{0.1}{99.0}\right)^2 + \left( 2 \times \dfrac{0.004}{2.0}\right)^2} = \sqrt{(0.001)^2 + (0.004)^2}$. | (15) |

**STEP 3:** Ignore any contributions to the uncertainty which are very small compared to the others.

** EXAMPLE: ** Note that the first term in Eq. (15) is smaller than the second by a factor of 16 and may be neglected. Thus, Eq. (15) becomes

$\dfrac{\Delta g}{g} \approx 0.004$ . | (16) |

Therefore

$\Delta g = 0.004 g = (0.004)(977\,\mathrm{cm/s^2}) \approx 4\,\mathrm{cm/s^4}$ . | (17) |

**STEP 4:** Report the final result by combining the value and the uncertainty.

**EXAMPLE:** Combining Eq. (14) and (17), we have

$g = 977 \pm 4\,\mathrm{cm/s^2}$. | (18) |

**STEP 5:** Make your conclusions and state your case.

Are your data consistent, within the range of your uncertainties, with literature values or with a theoretical model which is supposed to describe your experiment? At this point the scientist has to take a stand. If your data are not consistent with theory or with another investigator’s results, then who is wrong? Only with proper error analysis can we determine the answer to this crucial question!