# Confidence interval vs. expanded uncertainty

#### sagevahi

##### New Member
Hi,

I'm struggling with the difference between confidence interval and expanded uncertainty.

I understand a confidence interval around the mean. For example, a 95% confidence interval around the mean says that if N samples were taken 100 times and a confidence interval was recalculated each time, at least 95 of my confidence intervals would actually contain the true mean.

The expanded uncertainty is X +/- k*s, where:
• X is my best estimate determination of the parameter of interest
• k is my coverage factor, e.g., 2 for 95%
• s is my combined standard uncertainty of X
Is the expanded uncertainty the same concept as confidence interval, but for a specific measurement (whereas the confidence interval was for a collection of measurements, i.e., the mean)? For example, does the expanded uncertainty of 95% mean that if I were to take 100 more measurements, at least 95 of them should fall within the previously determined expanded uncertainty interval?

I have scoured the internet with things like "expanded uncertainty vs. confidence interval" and can't find much.

#### GretaGarbo

##### Human
I have heard of confidence intervals many times. But I have never heard of "expanded uncertainty". So I don't know what that is.

#### Dason

Agreed. Never heard the term until you posted it.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I also have not heard of them. Can you provide a source/citation? From the way you present them, they seem like an industrial engineering or information theory concept.

#### sagevahi

##### New Member
Hi all, thanks for responding.

Sure thing, here are some sites that refer to the "expanded uncertainty".
The more I read about it, the more I'm starting to think there's no "meaning" associated with it. It's just a common practice when reporting uncertainty. For example, note that they use 2 for a 95% confidence instead of 1.96.

Here's an example of my question written another way. Let's say I want to measure an area, and this value has 2 sources of uncertainty: length of X and length of Y. My combined standard uncertainty is the square root of sum of squares of the error contributions of X and Y, i.e.:

uncert = sqrt( var(X) + var(Y) )​

Does "Area +/- 1.96*uncert" have any meaning? It corresponds to 95% of the area under the normal curve, but does it also mean anything else?

#### GretaGarbo

##### Human
Here is a link to the National Institute of Standards and Technology: https://www.itl.nist.gov/div898/handbook/mpc/section5/mpc57.htm
Frankly, I don't understand that. What about "Definition of standard uncertainty" What if the variabels ar correlated. Should the covariance not be included?

"Expanded uncertainty assures a high level of confidence" But if the variances and/or the coefficients (the a_i) are large, then it will not assure a high level of confidence. And what do they mean by confidence? A confidence interval?

#### sagevahi

##### New Member
And what do they mean by confidence? A confidence interval?
You have hit my question exactly! What do they mean by confidence? I can't tell at all. I can't figure out what this value represents.

Regarding your first question about the covariance, I think they are assuming the covariance is zero on a previous page. For example, see the section for "Treatment of covariance terms" on https://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm