The variable controls the amount of sampling analyze table performs.

If ANALYZE table with histogram collection is too slow, one can reduce the
time taken by setting analyze_sample_percentage to a lower value of the
total number of rows.
Setting it to 0 will use a formula to compute how many rows to sample:

The number of rows collected is capped to a minimum of 50000 and
increases logarithmically with a coffecient of 4096. The coffecient is
chosen so that we expect an error of less than 3% in our estimations
according to the paper:
"Random Sampling for Histogram Construction: How much is enough?”
– Surajit Chaudhuri, Rajeev Motwani, Vivek Narasayya, ACM SIGMOD, 1998.

The drawback of sampling is that avg_frequency number is computed
imprecisely and will yeild a smaller number than the real one.