Consider the length measurements you made in today's experiment. When you make such a measurement, it is unlikely that the value you find is exactly on a tick mark of your ruler. And even it looks to your eye like it's on that tick mark, it may not be exactly that length. (If you zoomed in with a microscope would it still be perfectly on the line? Is the edge of the card perfectly flat? Is your ruler so perfect that the tick marks are exactly in the right place? The answer to all these questions is “no”.) Therefore, its not sufficient to simply report a value for a length, you have to also provide your estimate of the uncertainty in that measurement.

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 (like the standard deviation and standard error defined above.)

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.)

QUESTION: Now think about the measurements you made today. Which is the appropriate type of uncertainty for a single measurement of paper displacement? Which is the appropriate type of uncertainty for the average paper displacement?

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.

QUESTION: Think about your measurements today. What (if anything) could be biasing your values and how might you come up with a way to test for that? (You do not need to actually perform any additional tests. We're just looking for ideas.)

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.

Once you have a value and its uncertainty, how do you report this final result?

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

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

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

Agreement

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.

Disagreement

Inconclusive

Let's define some useful statistical measures. For the following, we will assume that you measured a quantity $N$ times and got a list of values $x=[x_1, x_2, \ldots, x_N]$.

• The mean (or average) of a list of numbers is given by $\mu = x_{\textrm{avg}} = \dfrac{1}{N}\sum_i^N x_i$.
• The standard deviation of a list of number is given by $\sigma = \sqrt{\dfrac{\sum_i^N (x_i-\mu)^2}{N-1}}$.
• The standard deviation of the mean (or standard error) is given by $\mu_{\sigma}=\sigma/\sqrt{N}$.

When we make observations of something which varies with random fluctuations, that data typically obeys a Gaussian (also called a Normal) distribution. Such a distribution can be characterized by parameters like the average, $\mu$, (or mean) and standard deviation, $\sigma$. The standard deviation tells us something about the typical “spread” in values. It tells us how wide the distribution is.

Suppose we a student who can collect data for a long time. The figures below show an example of how this student's Gaussian distribution evolves as they add more and more data.

10 drops	50 drops
100 drops	500 drops

As the number of drops increases, the distribution looks more and more like a smooth Gaussian and we get better estimates for both the average and the standard deviation. But notice that the standard deviation (the width of the distribution) doesn't change much; taking more data doesn't make the distribution more narrow.

For that reason, we need to define a different parameter… one that will scale as the number of data points increases to reflect the fact that our estimate of the uncertainty also improves with the number of counts. This parameter is called the standard deviation of the mean, $\sigma_{\mu}$, (or, sometimes called the standard error):

If we look now at those data distributions again, we can see that the standard deviation of the mean does decrease as we add more data. Hence, we can now more carefully state our average and its uncertainty as $\mu \pm d\sigma_{\mu}$.

10 drops	50 drops
100 drops	500 drops

Using the plot with 500 drops, we can now say that the average fall time is $T = 0.997 \pm 0.006$ seconds.