It is widely assumed that the most important factor leading to large family sizes and fast population growth is the lack of availability of contraception.
But Tim Worstall's book Chasing Rainbows says:
...the availability of contraception only explains 10% of the changes in total fertility
The data behind this is sourced from an old world bank report (pdf).
Does modern contraception only account for 10% of the reduction in fertility we observe in actual populations?