(adsbygoogle = window.adsbygoogle || []).push({}); Question on "estimated uncertainty" and significant digits

1. The problem statement, all variables and given/known data

The value is 136.52480, and the "estimated uncertainty" is 2. How many digits should be included as significant?

2. Relevant equations

Not sure whether estimated uncertainty refers to a 1σ (68.26%) confidence interval or what.

Rounding implies a 100% confidence interval, because you're taking a known number between [.5, 1.5) and rounding it to 1.

Edit: I should also mention that stating one significant digit after rounding, for instance 137 ϵ [136.5,137.5) implies a uniform distribution, so the likelihood of the actual value being 136.6 is the same as the chance of havinga value of 137, whereas 136.52480 ± 2 tells you that your point estimate for the value is 136.52480, and the likelyhood of the actual value being 136.6 is greater than the likelyhood of it being 137.

3. The attempt at a solution

The confidence interval based on the estimated uncertainty is (134.52480,138.52480) so there is, I'm guessing a 68.26% chance the data is within that region. If so, the 95% CI, is (132.52480,140.52480)

1 sig fig: x≈100 -> x ϵ [50,150); The interval contains the 95% CI but is roughly 10 times too wide.

2 sig figs: x≈140 -> x ϵ [135,145). The interval contains the 1σ CI, but misses part of the 95% CI. The width of this interval is close to the width of the 95% Confidence interval.

3 sig fig: x≈137 -> x ϵ [136.5,137.5);

4 sig fig: x≈136.5 -> x ϵ [136.45,137.55); This interval contains only a tiny fraction of the Confidence Interval.

I think the best answer is that4 digits are significant, because very little information is lost if we say our measurement is 136.5 ± 2. Any other answer would be rounding to a point where your result is unnecessarily inaccurate.

I wonder if there is a general rule? Though this question appears in the homework, I cannot find any reference to "estimated uncertainty" in the examples or text.

Additional information

I am a math and physics professor, so I can probably get hold of the answer key next week. I'll be curious what the book's answer is, but I wonder if it is wise to mix the concepts of rounding with the idea of "estimated uncertainty," or whether it should be treated so nonchalantly, like this question actually has a multiple choice answer.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Question on estimated uncertainty and significant digits

**Physics Forums | Science Articles, Homework Help, Discussion**