# Averaging out noise and SNR improvement

Suppose I have a sequence of identical symbols carrying the same information at a sample rate equal to the symbol rate, why do we always talk about averaging them to average out noise, why not just adding them all together without averaging them? I suppose the improvement in SNR is going to be the same assuming that noise is uncorrelated from one sample or symbol to another, is this related to the limited dynamic range in DSP?

Adding $N$ independent random variables (finite mean $\mu$, finite variance $\sigma^2$ produces a result $\Sigma X$ with mean $N\mu$ and variance $N\sigma^2$. Averaging these same $N$ random variables produces a result $\bar{X}$ with mean $\ mu$ and variance $\frac{\sigma^2}{{N}}$. The average is thus a much more compact result -- very nearly equal to the mean $\mu$ when $N$ is large, whereas the sum will vary quite considerably about the large mean $N\mu$.