How to find the appropriate fit?

Imagine we are given this set of data:

data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}, {6, 4}, {7, 5}}; 

We fit the data with:

nlm = NonlinearModelFit[data, Log[a + b x^2], {a, b}, x], 

and we can obtain $ a$ and $ b$ .

Now, if we are given that the $ y$ coordinates in the data list have the variance of, for example, $ 0.5$ , $ 0.3$ , $ 1.3$ , $ 0.2$ , $ 0.9$ , and $ 0.7$ , can we use this additional information to improve fitting?