# Combining Uncertainties - Adding and Subtracting

In our experiments as physicists, we often need to perform calculations with values we have recorded. But what happens to the uncertainty associated with those values?

Let's say we wanted to measure the length of a football field and all we had to hand was a meter stick.

Each time we use the stick we would be able to measure another meter until finally crossed the field from end to end. But, every time we would have to carefully align the stick at the zero point, and read off the value at the other end, creating uncertainty each time.

Every time we use our stick we would add more and more uncertainty to our result. How do these uncertainties combine? Well, fortunately, it's very simple! We have to factor in more and more uncertainty.

If the field is 100 m long then we must use the stick 100 times, adding an uncertainty of ± one resolution each time. By the end of it, our result would be 100 m ± 10 cm.

So when adding measurements we add their associated uncertainty. How about subtraction? Imagine we have to work out the difference between two people's heights; Chara and Levi. Chara's height is 180 ± 5 cm. Their friend, Levi, has a height of 200 ± 10 cm. So to calculate the difference we would need to subtract 200 - 180 = 20 cm.

But what happens to the uncertainties? Even though this calculation is subtraction, the uncertainties are added. This makes sense, since we aren't sure of Chara's height, or Levi's height, so we must be even more unsure of the difference between their heights!

The uncertainty on the difference in their heights would then be 5 + 10 = 15 cm giving us a final result of 20 ± 15 cm.