Peter T. Leeson makes multiple claims about economic and public health improvements since the collapse of its government in his paper Better Off Stateless.
This paper investigates the impact of anarchy on Somali development. The data suggest that while the state
of this development remains low, on nearly all of 18 key indicators that allow pre- and post-stateless welfare
comparisons, Somalis are better off under anarchy than they were under government.
The specific claims of improved welfare are included in Table 1 and show a variety of metrics. Do other metrics tell a different story? Are these metrics insufficient to establish an improvement in Somalia's development or quality of life according to standards of social science?
Even if the numbers in this table were right, it wouldn't mean that "Somalis are better off under anarchy than they were under government."
After doing some searching on the Internet, the guess of total population of Somalia in 2016 is about 10 mio. It's difficult to get good numbers, but most sources agree on that one.
According to Amnesty International, in 2011 there were 1.5 mio refugees within the country and about 900.000 refugees who left the country. Most of them are in Kenya. Again, numbers differ from source to source, but I think we can assume that more than 2.5 mio Somalis are removed from their homes and everything they owned. Are the 600.000 Somalis that live in Kenyan refugee camps taken into account in that table? Probably not.
If you remove about 1/5 of your population, almost all those numbers in the table look better afterwards. Especially when you consider that those better off had less desire to flee the country, for example Physicians. That naturally results in more Physicians per 100.000 citizens.
You can change nothing else in the country but make the poor flee, and woosh! according to the statistics used, your country appears to be better off.
You can even have a less total of something, but more per capita, if you have people fleeing your country.
As a side note, the meaning of numbers can be distorted in the other direction too. The table shows that 8000 people died due to measles between 1985 and 1990. Somalia's population at that time was about 6 mio Population Somalia. Between 2000 and 2005 the population rose from 7.4 mio to 8.3 mio, and 5600 died from measles. With even more citizens, less deaths than 15 years before. This is not visible in the table. Or maybe that is not the full picture either. What about similar countries in that area? Was there a global vaccination campaign? Turns out there was!
Launched in 2001, the Measles Initiative is a partnership — led by the
American Red Cross, United Nations Foundation, U.S. Centers for
Disease Control and Prevention, UNICEF and WHO — committed to reducing
measles deaths worldwide.
In April of 2012, the partners of the Measles Initiative introduced a
new global plan to jointly tackle measles and rubella using the same
strategy and a combined measles-rubella vaccine. This new strategy is
represented in its new name, the Measles & Rubella Initiative. The
Initiative’s goal is now to reduce measles deaths worldwide by 95%
between by 2015 and to eliminate measles and rubella in at least five
of six WHO regions by 2020.
Somalia is right in one of those WHO regions, and the campaign started just at the beginning of the period taken for the table. Btw, further down in the WHO article it's stated that Somalia had a major measles outbreak in 2011, with 5 times as many outbreaks than neighboring Ethiopia. Ethiopia's population is 10 times as big as Somalia's
I strongly doubt that from the data presented, one could conclude that "Somalis are better off under anarchy than they were under government." It looks cherry-picked to make this point.
After reading William Grobman's comment, I searched if someone within the scientific community actually responded to Peter Leeson's paper. I only found a post on it by Chris Blattman, Professor of Global Conflict Studies at The University of Chicago
Blattman argues that there were decentralized governments in Somalia, so there was no real "stateless" period. He also questions the quality of the data. He agrees though that some governments are so bad, the people would be better off without.