# What does it mean that “point estimation resulted in $\hat{\mu} = 12.5 \pm 1.0$”?

I'm trying to figure out what the following statement means:

As a result of point estimation of the mean value of a certain quantity, the value $$\hat{\mu} = 12.5 \pm 1.0 \ [u]$$ was obtained. Assume $n=36$ measurements, each with the same precision.

I know that for a iid normal sample, the sample mean $\hat{\mu} \equiv \bar{X}$ (which is a random variable) has a $N(\mu, \sigma^2/n)$ distribution. Likewise, the sample variance $\hat{\sigma}^2 \equiv s^2$ is such that $(n-1) s^2/\sigma^2$ has a $\chi^2 _{n-1}$ distribution.

I'm confused because I thought "point estimation" referred to specifying one number, in this case $12.5$. So I'm not sure what the $1.0$ in this case - is it $s/ \sqrt{n}$, the standard error of the mean (which is the estimate for the standard deviation of the random variable $\bar{X}$ )? What is the convention used here?

As a result of point estimation of the variance of a certain quantity, the value $$\hat{\sigma} = 12.3 \pm 4.1 \ [u^2]$$ was obtained. Assume $n=36$ measurements, each with the same precision.

In this case, does the $\pm$ part refer to the estimate of the standard deviation of the random variable $s^2$, obtained from the $\chi^2$ distribution, which is $s^2 \sqrt{2/n-1}$ ?

• Can you provide the reference for the statements? – Thomas Jan 26 '18 at 13:00
• @Thomas Unfortunately there isn't any, it's just a problem from a problem set – Spine Feast Jan 26 '18 at 13:23
• You ought to consult the underlying authority, then: either the teacher or the textbook. – whuber Jan 26 '18 at 14:10
• @whuber So I take it this isn't a standard way of writing things? – Spine Feast Jan 26 '18 at 15:16
• It isn't nonstandard, either. It's a little idiosyncratic. It's also ambiguous, because the meaning of the numbers after the "$\pm$" isn't clear: would those be standard errors or confidence limits (and if they are the latter, with what confidence level)? The ambiguity might be resolved within the context; for instance, maybe your textbook has previously explained what these numbers mean. That's why you need to go back to it and why we can only guess. – whuber Jan 26 '18 at 16:37

The notation $\mu$ is highly suggestive of the population mean whose estimate $\hat{\mu}$ is usually the sample (arithmetic) mean and has a standard error estimated by the sample standard deviation divided by the root of the sample size. Confusingly, I have seen written $\hat{\mu} \pm \text{SE}$ as well as $\hat{\mu} \pm 1.96 \times \text{SE}$ in lieu of reporting an actual confidence interval, the former only really being appropriate if there is no set level of confidence or if it is in fact 66% and the latter only if the confidence is 95%. So alas, all I can definitively say is that the notation is ambiguous.