Standard error of mean when observations are noisy

Question

I have two observations of a normally distributed random variable:

$X_1 = 0.02, \quad X_2 = 0.10$

Obviously the sample mean equals 0.06, and the standard error of the mean $(SEM)$ is equal to $0.04$.

Now things get messy. The observations themselves can only be measured with noise, e.g. due to bad instruments. Let the measurement error of $X_1$ equal $0.04$ and of $X_2$ equal $0.02$.

How can I calculate an adjusted $SEM$ that accounts for the measurement error? Intuitively, it should be greater than $0.04$ in this example. When measurement error becomes smaller, the $SEM$ should converge to its value in absence of measurement error.

Answer 1

Begin by breaking the observation down into Noise + True value

Let $Y$ be the true value and $Z$ be the noise, assume that the noise is normally distributed

$Y \sim N(\mu,\sigma^2)$

$Z \sim N(\nu,\tau^2)$

$X=Y+Z$

The sum of two normal variables is normally distributed and has a mean equal to the sum of their means and a variance equal to the sum of their variances

$X \sim N(\nu + \mu,\tau^2 + \sigma^2)$

Use your data points to get an estimate of the variance of the true value without noise