What is an uncertainty?

An uncertainty is the term used to describe the range of margin of error in a measurement from its true value.

Example

Sample space -

Range = 0.6 (maximum - minimum || 0.95 - 0.35)

Mean = 0.56 (Average of all the values)

The uncertainty of a set of elements is given by the range divided by two

So from the example, we could say that the value is

NOTE: The uncertainty of a value cannot have a greater precision that that of the value

Random and Systematic Uncertainties

Random Uncertainties

Random uncertainties are a type of uncertainty that has a zero mean.
What this means is that the error of each measurement do not have a common factor

Systematic Uncertainties

Systematic uncertainties have a non-zero mean, and produce a consistently incorrect measure of a value that is either too big or too small. Often these types of uncertainties are caused by poor technique in measuring, incorrectly calibrated equipment, or zero errors

Dependent and Independent Uncertainties

Dependent uncertainties occur when the same equipment is used in all of the measurements.

Independent uncertainties occur when different measuring equipment are used.

In first-year physics (aka this course), uncertainties are classified either as dependent or independent.
However, in real-life, the type of an uncertainty is more of a range.

Calculations with Uncertainties

Forms to represent an uncertainty

Absolute -

Fractional percentage -

Where is the error margin (), and is the mean value

Rule of Thumb

When manipulating uncertainties, the end result is some sort of action involving the addition or subtraction of the uncertainty
The addition or subtraction of uncertainties utilises the absolute uncertainty, whilst,
The multiplication or division of uncertainties utilise the fractional percentage form

Dependent Uncertainties

Adding/Subtracting

When combining uncertainties, we simply just need to sum the uncertainties from their absolute form

Multiplying/Dividing

We must use the fractional percentage form of the uncertainty

Independent Uncertainties

To account for the differing error margins of each measurement, we use Pythagoras' Theorem to acknowledge both the positive and negative case

Adding/Subtracting

Take the square root of the sum of the square of the absolute uncertainties

Multiplying/Dividing

We must use the fractional percentage form of the uncertainty

Can we use Pythagoras' Theorem when dealing with Dependent Uncertainties?