# Uncertainties accumulate when measuring longer lengths?

• I
• fog37
fog37
TL;DR Summary
Correctly reporting uncertainties that accumulate measuring longer lengths
Hello,
It is clear that as distance being measured increases, more graduations are needed to represent a unit of measure for such distance. Each graduation in a measuring device is subject to uncertainty and error, and these errors and uncertainties accumulate with the increase in the number of graduations. For example, the meter stick has an instrumental accuracy of 0.5 mm (half the graduation of 1 mm).

How would we report the uncertainty taking into account the concept described above, i.e. the larger the measurement the greater the error since we include mode graduation units causing the error to accumulate?

For example: ##2.30 \pm 0.05 cm##. Would the uncertainty still be ##0.05 cm## if the measured distance was ##55.00 cm##? Based on what I shared, it should be higher...How much higher? How do we determine that?

Thank YOU

I recommend you get a book like Taylor's Introduction to Error Analysis. Since you have many questions on uncertainties, it will be more efficient for you to read the relevant chapters than to wait for us to type in much the same thing,

gmax137 and fog37
Fair! I got that book but I could not find the answer. I will look more carefully.

fog37 said:
How would we report the uncertainty taking into account the concept described above, i.e. the larger the measurement the greater the error since we include mode graduation units causing the error to accumulate?
Uh ... ya think maybe HOW your measuring device was created might make a difference? Repeated use of a meter stick is NOT the same as use of a 25 meter steel tape.

That's just ONE aspect of the issue.

fog37 said:
Each graduation in a measuring device is subject to uncertainty and error, and these errors and uncertainties accumulate with the increase in the number of graduations
I don’t think this is true.

hutchphd and Frabjous
Dale said:
I don’t think this is true.
Probably not.

There was the famous case when the ruler printed by the PDG was 3% too small. Normally, the relative error will go down as you get a longer length measured with a meter stick. But this is why I suggested working through Taylor. What you are measuring and how you are measuring it matters. "Just give me the formula" works badly here.

Dale and phinds
Sometimes you may use the longer range to cancel errors, such as 'triangulation' in surveying, when you can build a virtual 'truss frame' to span vast field, with 'closure' errors being fed back to mitigate them...

fog37, phinds, berkeman and 1 other person

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