Posterior predictive distribution in eight schools model does not actually compute the distribution

In the Jupyter example Eight_schools.ipynb under the 'Criticism' section we want to compute the posterior predictive distribution. But instead of putting in the entire distribution of theta values into the probability calculation (as the integral would suggest), we just put in the mean theta value. The wikipedia article on this claims that this would underestimate the width of the distribution of the predicted values. I also find that the distribution is more narrow than the distribution of the theta values, which is to be expected.
https://en.wikipedia.org/wiki/Posterior_predictive_distribution
I would like to put in the whole distribution (so e.g. all values of avg_effect_ instead of np.mean(avg_effect_, 0) ), but I couldn't get it to work. I think at the moment what the programme is doing is wrong, especially when compared to what the integral gives for p(y*|y).

Open Eight_schools.ipynb and inspect the 'Criticism' section's posterior predictive calculation, especially the use of np.mean(avg_effect_, 0). Compare that calculation with the full avg_effect_ values and the referenced posterior predictive definition; done means the example computes the distribution rather than only the mean and reflects the expected distribution width.