# 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

- 3.14201 s
- 3.1415926897940001 s

*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

- 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 - 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

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 |

*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.

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.

The indicated measurement is the observational result of a continuous variable as reported by your measuring device, which has a limited precision.

The unknowable actual value of a continuous variable including the infinite number of decimal places.

A quantity, like time or mass, whose values are continuous: they can take on any value with an infinite number of decimal places.

Variables, such as dice rolls or counting atoms, that can only take on specific values. You cannot have half an atom!

Uncertainty arising from any measurement of a continuous variable. The true value is not only unknown, but unknowable due to its infinite number of digits. Moreover, there is intrinsic randomness from measurement to measurement.

A consistent difference between the indicated and true values, usually arising from a miscalibrated instrument or neglected effect. A systematic uncertainty is always in the same direction as opposed to the random bouncing around characteristic of statistical uncertainties.