This page gives a review of uncertainties, how to estimate them, propagate them through calculations and how to use them to determine whether or not the measured value is in agreement with another measurement or prediction.

A **measurement uncertainty** is your best estimate as to how close your “measured” value is to the “true” value. It isn't a random guess or an arbitrary number, but it is judgement you have to make about how much you trust your value. If we say our best estimate for the measured value is $x$ and our estimate for the uncertainty is $\delta x$, then we'd report our final value as $x \pm \delta x$. This does not mean that the true value is *absolutely* within the range from $x - \delta x$ to $x + \delta x$, only that we think it is “likely” in that range. (We can get more specific on what we mean by “likely” and define, for example, the probability of being in a certain range… but let's save that for a later experiment.)

There are typically two ways you might estimate a measurement uncertainty – from looking at the “spread” of values after making repeated measurements, or from estimating the limit of your precision due to the resolution of your measurement device. Let's look at each in turn.

*Repeating measurements*: Suppose that the quantity you are measuring is difficult to pin down. Maybe it fluctuates with time (e.g. a length that is always varying), you have only one chance to measure it (e.g. the time it takes for a ball to drop), or it is inherently fuzzy (e.g. the diameter of a puffy cotton ball). In this case, if you make multiple measurements, each measurement you make might be slightly different than the last. The best estimate for the uncertainty in a case like this is therefore gotten by looking at the “spread” of values you obtain from repeated measurements.

- If you are able only to make a couple measurements, a simple rule is to estimate the uncertainty as (“maximum value” - “minimum value”)/2. Put into words, you look at the range created by the most extreme values, and chose your uncertainty as
*half*of that. - If you are able to dozens or hundreds of measurements, or if you know something about the statistical distribution of these measurements, then we can use more rigorous methods (and we'll teach you those in time.)

*Resolution*: The resolution of a measurement device is usually the smallest unit that the device can measure. On a ruler or analog dial, it is the distance between the two smallest tick marks. On a digital device, it is the smallest displayed digit. If you keep finding the same reading on the device every time you make the measurement (assuming you make the measurement the same way), then the “repeating measurement” technique above doesn't help you much. In this case, you can estimate your uncertainty based on how well you can read your scale on a single measurement. There is no fixed rule here; sometimes the uncertainty is equal to the smallest tick mark, whereas sometimes you can say it is closer to one line than the other, so the uncertainty is half a tick mark (or whatever your case may be.)

Now think about the measurements of the three dimensions of your card. Which of the above two methods is more appropriate here? Will your uncertainty on all three values be the same? (Probably not.) How will you estimate uncertainty on that *smallest* dimension, the thickness?

In addition to measurement uncertainties described above, we also have to be on the lookout for what are called **systematic uncertainties** (or sometimes just **systematic biases**). A systematic uncertainty is an uncertainty in your measurement that is due to some unaccounted for bias in your measurement or your assumptions. It could be that the measurement device you are using is in error (e.g. a thermometer that isn't calibrated correctly or a stopwatch that runs too fast or too slow), that your measurement technique is flawed (e.g. a start or stop signal to a stopwatch always arrives late), or that there is an incorrect assumption about what you are measuring (e.g. measuring the speed of an object that you believe to be at constant velocity, but which is actually accelerating).

Importantly, systematic uncertainties are different from “mistakes” (e.g. writing down a 4 when you meant to write down a 3). Systematic uncertainties are *almost always* present, but they may be difficult to identify or eliminate.

One final comment. We **never** determine an uncertainty by comparing our value to a “known” value. For example, if you measure the acceleration due to gravity to be $g_{\textrm{exp}} = 9.7~\textrm{m/s}^2$, the uncertainty on that value is NOT $\delta g = g-g_{\textrm{exp}} = (9.8 - 9.7)~\textrm{m/s}^2 = 0.1~\textrm{m/s}^2$. Some students may have learned this sort of calculation in high school as the “error” in a measurement, but that is the wrong way to think about uncertainties, and we will develop better ways of comparing two values below.

You now have three length dimensions for your card and three uncertainties on those lengths. How do we report those values?

This is where the question of significant figures appears. If you do some arithmetic and your calculator spits out an answer of 3.45679213, do *all* those digits matter? Probably not. In order to know how many digits _do _matter, you should look at the size of your uncertainty.

The rules for determining the number of digits to report can be summed up as follows:

- Compute your uncertainty. Keep only one digit in the uncertainty, unless the leading digit is a 1 or a 2.
*Example*: If your uncertainty is 0.543 units, then report the uncertainty as 0.5 units.*Example*: If your uncertainty is 0.0237 units, then report the uncertainty as 0.024 units.

- Look at your value, and truncate your value to the same digit place as the final digit in your uncertainty.
*Example*: If your value is 123.72 units and your uncertainty is 0.5 units, then you should truncate your value to 123.7 units.- Example: If your value is 0.53325 units and your uncertainty is 0.024 units, then you should truncate your value to 0.533 units.

- Put your value and uncertainty together
*Example*: 123.7 ± 0.5 units*Example*: 0.533 ± 0.024 units

Each of your lengths was *directly* measured and the uncertainties were *directly* estimated. But what happens when you compute a new quantity using these measured values? That new quantity must have some uncertainty, and it must depend on the uncertainty of the individual measured components. This process of determining the uncertainty in a *calculated* quantity is called **propagation of uncertainties.**

We'll introduce the general formula for uncertainties in a minute, but first let's look at the two most common cases.

**Products and quotients**

If our quantity is computed by multiplying or dividing variables, then the final uncertainty is given by the sum of the fractional uncertainties (i.e. the uncertainty divided by the value, $\delta A/A$) of each variable added up in **quadrature** (which is just a fancy way of saying *square each term, add them, and take the square root.*

So, if we have a quantity $f(x,y,z) = \dfrac{xy}{z}$, then the uncertainty in $f$ is

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

Notice that this equation is the same whether a given variable is multiplied or divided.

**Sums and differences**

On the other hand, if our quantity is computed by adding or subtracting variables, then the uncertainty is given by the sum of absolute uncertainties, again added in quadrature.

So, if $f(x,y,z) = x - y + \dots +z$, then the uncertainty in $f$ is

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

Notice that the same formula applies whether you add or subtract; the uncertainty in the final sum or difference is **always** bigger than the uncertainty in any individual term.

**General formula**

Those two formulas above handle *many* (or maybe even *most*) situations, but if you are calculating a more complicated equation, we must turn to the general formula for calculating uncertainties.

For a generic function $f(x_1,x_2,...x_n)$ with measured values and uncertainties $x_1 \pm \delta x_1$, $x_2 \pm \delta x_2$ ,… $x_n \pm \delta x_n$, the final uncertainty is

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

We see that each term is the *slope* of the function with respect to a given variable multiplied by the *uncertainty* in that variable, and that these terms are squared, summed, and square-rooted (i.e., added in quadrature). You can check for yourself that this formula reproduces the formulas above when we have products/quotients or sums/differences.

We will return to the propagation of uncertainties again in later labs and go into more depth on what these equations mean and where they come from. For now, though, we can just use these equations as tools. If you are interested in where this formula comes from, you can read some more details here.) |

Now that you have finished your measurements and uncertainty calculations, it is time to draw some conclusions. This is the point in the scientific process where you present your final results, and discuss what they mean in the broader context. It can be a chance to claim agreement with prediction (if that's the case) or to discuss why there may be disagreement or why the results remain inconclusive.

Today's activity was a very simple measurement and not a full lab where you were testing a model or investigating a phenomenon. Therefore, there isn't really much to “conclude”. However, we can at least compare your measurement to those of some hypothetical classmates, and try to discuss why you agree or disagree with those students (if either of those are the case).

Often in experimental physics, we want to compare our number to someone else's.

- Maybe we have a model which makes a prediction for a value. How does our value compare to the prediction?
- Maybe two groups are trying to measure the same thing. Do the two groups agree?
- Maybe we actually expect a result to differ from another known result. How different is different enough?

In order to be quantitative about these sorts of questions, we need to establish some criteria.

For this course, we will use a measure called $t^{\prime}$ . (This is related to – but not equal to – something called the *Student's t-test*. If that doesn't mean anything to you, don't worry about it.)
Suppose we have two quantities with uncertainties which we want to compare: $A \pm \delta A$ and $B \pm \delta B$ . I this case, $t'$ is defined as

$t' = \frac{A - B}{\sqrt{(\delta A)^2 + (\delta B)^2}}.$ |

In the case where only one value has an uncertainty (for example, when you want to compare a measured value $A \pm \delta A$ to a predicted or literature value $B$ ), this simplifies to

$t' = \frac{A - B}{\delta A}.$ |

If the values are within one uncertainty of each other, it is possible that the difference is due only random chance. We will consider this to constitute agreement: $ |t'| \leq 1$. |

Note that agreement might turn into disagreement if more data is taken and the size of the uncertainties shrink. Remember that we can never *prove* something to be true… we can only say that the current data supports agreement.

If the values are more than three uncertainties away from each other, it is statistically unlikely that the difference is due only random chance. We will consider this to constitute disagreement: $ |t'| \geq 3$. |

If the values are between one and three uncertainties of each other, we cannot say with certainty if the difference is random chance or a real disagreement. We will consider this to be inconclusive: $1 < |t'| < 3$. |