I have a robot that takes a measurement of its current pose in the form
$$ z = \begin{bmatrix} x\\ y\\ \theta \end{bmatrix} $$ $x$ and $y$ are the coordinates in $XY$ plane et $\theta$ the heading angle.
I also have ground truth poses $$ z_{gt} = \begin{bmatrix} x_{gt}\\ y_{gt}\\ \theta_{gt} \end{bmatrix} $$ I want to estimate the standard deviation of the measurement. As the standard deviation formula is:
$\sigma=\sqrt{\frac{1}{N}\sum_{i=0}^{i=N}(x_{i}-\mu)^2}$
Is it correct to calculate in this case where I have only one measurement, i.e
$N=1$:
$\sigma_{xx}=\sqrt{(x-x_{gt})^2}$
and the same for $\sigma_{yy}$ and $\sigma_{\theta\theta}$ ?
Edit The measurement is taken from a place recognition algorithm where for a query image, a best match from a database of images is returned. To each image in the database is associated a pose where this image has been taken, i.e $z_{gt}$. That's why I only have one measurement. In order to integrate it into the correction step of Kalman filter, I want a model of measurement standard deviation to estimate the covariance having the following form:
$$ \Sigma = \begin{bmatrix} \sigma_{x}^2 & 0 & 0 \\ 0 & \sigma_{y}^2 & 0 \\ 0 & 0 & \sigma_{\theta }^2 \\ \end{bmatrix} $$