Does it make sense to average data taken continuously as the system evolved? I have a simulation where a bunch of particles are moving. Every time step I fit a gaussian (through non-linear least squares) to the distribution of speeds. Here I end up with a temperature $T_i$ and a variance $\sigma_i ^2$ on this parameter.

I do this for every time step during a period of time, i.e. I end up with an array of temperatures and variances.
Does it make sense to average these in order to quote a final value of the temperature:
$$T = \left< T_i \right> \pm \sqrt{\left< \sigma _i ^2 \right>}  $$
My concern is that these are not some random values sampled from the system, each subsequent $T_{i+1}$ is somewhat correlated to the previous temperature, as the system dynamically evolves from each time step to the next one.
For example imagine I wanted to quote the average kinetic energy of the particles. Would it make sense to compute the average kinetic energy at each time step; hence I end up with an array of kinetic energies $K_i$ (one $K$ for each time step). Finally referring to the average kinetic energy as:
$$K = \left< K_i \right> \pm \sigma,$$
where $\sigma$ is the standard deviation of the $K_i$ in the array. For some reason, this feels even worse than the previous example of the temperatures.
Are both cases equally wrong? Is there any difference between them?
 A: In terms of estimating averages, it doesn't matter how correlated your samples are, you still have a correct unbiased estimator for the ensemble average by just summing them up and dividing by their number. This is true, because the expectation value of a sum of variables is the same as the sum of the expectation values:
$$\Big<\frac{1}{N}\sum_{i=1}^{N}X_i\Big> = \frac{1}{N}\sum_{i=1}^{N}\Big<X_i\Big>=\Big<X\Big>$$
for any observable $X$ whose ensemble average is $\Big<X\Big>$. The situation is different for the variance, and naively estimating the variance from correlated samples will typically give you an estimate that's too low. In order to fix this, you need to calculate the covariances for each sample:
$$Var\Big[\frac{1}{N}\sum_{i=1}^{N}X_i\Big] = \frac{1}{N^2}\sum_{i=1}^NVar\big[X_i\big] + \frac{1}{N^2}2\sum_{i<j}Cov\big[X_i,X_j\big]$$
The first term is just the variance of the mean ($\frac{\sigma_X^2}{N}$), and this can be estimated from all samples. The second one is more tricky, because you don't know the covariances a priori, so you have to estimate each one them by, e.g correlation functions. Describing how this is done is a bit more involved (but not necessarily difficult) so I am not going to go into further detail. Suffice it to say that you can instead simply estimate an effective decorrelation time and use this as an interval for your variance estimation. This has been done before and you can check this paper out for more information:
A simple method for automated equilibration detection in molecular simulations (which also addresses the question of estimating the sample mean by taking equilibration into account). Of most interest to you will be the sections "Autocorrelation analysis" and "Practical computation of statistical inefficiencies". They also describe a Python library that will do these automatically for you if you pass it an array of your data.
A: When your time increment remains constant, the average that you determine is a "time-averaged" value.
To obtain a sense of what you are getting, you can graph different analytical functions of temperature $T$ versus time $t$, calculate the analytical average of the functions, and plot that analytical average value on the graph. A straight line $T(t) = $ constant $C$ will return $\langle T \rangle = C$. A sin wave $T(t) = A\sin(kt)$ averaged over one cycle will return $\langle T \rangle = 0$.
When your time increment is not constant from step to step, you may want to consider using a weighting factor.
As for what this means or should mean physically for your system, that is for you to decide. You may have merit to use average temperature and average kinetic energy as metrics of two different characteristics of your system when observed over a given time period.
