# Errors and Uncertainties

## Types of Errors

### Random Errors

These kinds of errors are unpredictable. They mostly cannot be accounted for in advance, since they are caused by factors outside the control of the person conducting the experiment, such as environmental conditions.

The way to mitigate, not solve or remove, these errors is to repeat the same experiment multiple time and calculate averages for all readings.

### Systematic Errors

Systematic errors are the result of faulty instruments or errors in the method of experimentation.

In the case of faulty instruments, the instrument needs to be replaced for the problem to be solved. It cannot be mitigated by averages.

Improper calibration is also a possible error; the solution in that case is to recalibrate the instrument.

In the case of flaws in the method of experimentation, the practitioner needs to revise and correct their methodology. This could include things like avoiding parallax or zero errors.

#### Zero Errors

These errors are the result of reading a measurement a distance in front of or behind the zero. For example, measuring the length of something, but not accounting for the position of the zero on the scale.

There are two kinds of zero errors: positive zero errors and negative zero errors. Positive zero errors occur when the reading starts a distance in front of the zero of the scale. They are solved by subtracting that difference from the final reading. Negative zero errors are the opposite: they occur a distance behind the zero of the scale, and are solved by adding the difference.

## Precision

Precision refers to how close successive readings are to one another (readings of the same instrument).

## Accuracy

Accuracy refers to how close each reading is to the true value.

## Precision and Accuracy

Precision and accuracy do not have to be directly proportional. Assume that a true reading of something lies at 10.00, but the three readings you took fall at 6.20, 6.30 and 6.10. Then your instrument is precise, but not accurate.

Similarly, assume that another instrument takes readings at 8.50, 9.60, 11.40 and 12.00. The true reading is still 10.00. In this case, the readings are accurate, but not precise.

Of course, the specifics will change. The definition of accuracy in a particular measurement is variable. The definition of precision differs too. The examples are something generalized.

## Calculations of Uncertainties

Uncertainties can be represented 3 ways:

1. as a fixed quantity (absolute uncertainty)
2. as a fraction of the measurement (fractional uncertainty)
3. as a percentage of the measurement (percentage uncertainty)

#### Multiplication/Division

The error is read as a fraction of the total measurement. If your error is $$\pm0.5$$, but your original reading is $$2$$, then your percentage error is calculated as $$\frac{0.5}{2}\times100 \rightarrow 25\%$$. In more general terms, $$\frac{\Delta n}{n}\times100$$, where $$n$$ is your measurement, and $$\Delta n$$ represents the margin of error.
If you want to go backwards, and calculate the absolute uncertainty of a measurement from its percentage uncertainty, then simply multiply the percentage uncertainty with the original measurement. Your product will be $$\pm x$$ of the orignial measurement.
Multiply the exponent's value with the fractional or percentage form of the uncertainty, i.e. $$x \times \frac{\Delta n}{n}$$, where $$x$$ is the value of your exponent, $$\Delta n$$ is the margin of error, and $$n$$ is the measurement.