A Scale That Can't Remember

A precision scale on a packing line weighs one bag at a time and reports in micrograms, so a 1 kg bag reads about $1\,000\,000\,000$. Its tiny controller cannot store past readings: between bags it may keep only a few running numbers.

After every bag, the display must show two statistics for all bags weighed so far:

• the mean, and
• the population variance: the average squared deviation from the mean, dividing by the number of readings $N$, as in the chapter.

Task: write running_stats(readings) that walks through readings in order and returns a list with one (mean, variance) tuple per reading (the statistics of every reading up to and including that one), with both numbers rounded to 4 decimal places.

Your method has to stay accurate when the readings are huge but close together, like bags that all weigh about a kilogram and differ by only a few micrograms.