# Introduction to Statistical vs. Systematic Uncertainty

The following is based upon, with permission:

Denker, J. Uncertainty as Applied to Measurements and Calculations. Uncertainty as Applied to Measurments and Calculations (2011). Available at: http://www.av8n.com/physics/uncertainty.htm. (Accessed: 26th August 2016).

# A thought experiment and discrete vs. continuous variables

Imagine dropping a ball and timing how long it takes to hit the ground using a watch that’s accurate to about a millisecond. For example, I could do the experiment and get 3.142 seconds. We will call this the time to distinguish it from the time. The true time that the ball takes to fall is a , it can take on any of an infinite number of possible values; there is an infinite number of numbers that round to 3.142 seconds. For example, there is:
• 3.14201 s
• 3.1415926897940001 s
and so on. There is literally an infinite number and each one of them could have an infinite number of digits. For all we know, while this is unlikely, the time that the ball took to fall is exactly equal to π! The consequence of the fact that time is a continuous variable is that the true time it took for the ball to fall is not only unknown, it is impossible to know. Any measurement device is going to have some limited degree of precision and so we can never actually know the true time that the ball takes to fall. Compare this to a dice roll: a die it comes up a five. That is a fixed number. The only fixed options are: 1, 2, 3, 4, 5, and 6. Such variables are called . This distinction will be important for the notion of uncertainty.

# Statistical uncertainty

Now let us take the ball and drop it several times, say four. There are two possibilities:
1. With each drop, I measure a different time. For example, let’s say I get the four observations in the table below. The variation in these observations is the uncertainty. We will learn how to quantify this uncertainty in a later section. But for now, the important point is that the variation in these numbers is the uncertainty.
 Observation Measured Times [s] 1 3.142 2 3.140 3 3.145 4 3.143
2. I get the same value, 3.142 s, each time. This might be because the device i’m using has a limited precision. In our example, our stopwatch only goes to the millisecond. Therefore, even if we got 3.142 each and every time we might not believe that last digit and assign an uncertainty of a millisecond, ±0.001s.

In summary, we can either measure it from the variations or, if we cannot measure it because our device’s limited precision, we can assign an uncertainty. Both of these possibilities are examples of : they are due to the intrinsic randomness of continuous variables and the unknowability of the true value.

# Systematic uncertainty

Now let us repeat the experiment: not only with my watch but also with your watch and with a sophisticated setup using a laser and an atomic clock. The results of the measurements are in the table below.
 Observation My watch [s] Your watch [s] Atomic clock + laser setup [s] 1 3.142 5.312 5.370001 2 3.140 5.002 5.370003 3 3.145 5.687 5.370002 4 3.143 5.479 5.370001
Remember, the true time is still unknowable, but we’re going to assume that the atomic clock and lasers are closest to the true time. My watch has less statistical uncertainty than your watch does: the variation between the numbers is smaller. You can see that the results from my watch are pretty tightly clumped. However, my watch has a : my numbers are always too small, my watch runs slow. This is a systematic effect, always in the same direction as opposed to randomly bouncing around like the statistical uncertainty.
The distinction between statistical and systematic uncertainties is related to the ideas of accuracy and precision that you’ve probably seen in other science courses and exemplified in the figure below. Your watch is very accurate (assuming the atomic clock and the laser are close to closest to the true time), but they have a lot of spread: they lack precision / have a lot of statistical uncertainty. My watch, on the other hand, has a small statistical uncertainty, they don’t fluctuate very much but they have, a systematic effect that is significant: they lack precision.

Summary

• All continuous variables have statistical uncertainty.
• Intrinsic randomness of the Universe.
• Sometimes you can measure it with the variation.
• Sometimes you need to assign it based upon the precision of your device.
• In addition, measurement devices can have systematic uncertainties.
• These are generally harder to get a handle on.
• One way is to try and measure a different way.
• Statistical and systematic uncertainties are related to the ideas of accuracy and precision.
• We will explore quantifying these uncertainties in a later section.