# Notation to report the measurement of a parameter

The estimation of a parameter ($$p$$) is customary reported in Physics and other fields with,

$$p = \hat{p} \pm \Delta p$$,

where $$\hat{p}$$ is an estimator, and $$[\hat{p} - \Delta p, \hat{p} + \Delta p]$$ a confidence interval. For simplicity, we are using a symmetric interval. In addition, sometimes the confidence level used to build the interval is quoted.

The notation in the equation above must have stick for historical reasons, but it is quite odd. First, the sign "=" does not mean a mathematical equality here. The interpretation is rather "the parameter p is estimated by $$\hat{p}$$ and a confidence interval $$[\hat{p} - \Delta p, \, \hat{p} + \Delta p]$$".

Second, a equality in a equation where the left side is a number (parameters have a fixed but unkown value in the frequentist school of statistics) and the right side is an interval, does not make sense from a mathematical point of view.

Indeed this notation is against the clear cut separation Ronald Fisher made more than a century ago between the concepts of parameter and estimator.

So, I would like to ask if there is a more precise notation for reporting results. Or at least, some information on how the current notation originated as it could be a good starting point to discuss such a fundamental topic.

• I have to say, I've not seen that notation before... what field of application are you seeing this in? Commented Jun 24 at 22:26
• This is the usual notation in physics and, I guess, in science and engineering ingeneral. I ammended the question. Commented Jun 24 at 22:28
• I have read an awful lot of physics texts and literature but don't recall seeing this particular solecism. I'm not saying it doesn't exist, but I suspect it might occur only within particular communities of writers.
– whuber
Commented Jun 24 at 22:39
• This notation is normal in high energy physics. Foh example, here is a paper about the measurement of the mass of the Higgs boson with the LHC: arxiv.org/abs/2002.06398. The abstract already contains three instances of such notation. I will check with colleages from other branches of physics. Commented Jun 24 at 23:02
• What's wrong with just saying that $\hat{p} \pm \Delta p$ is a confidence interval for $p$ without bringing the equality sign in? Commented Jun 25 at 10:52