It seems that there are no actual measurements of unemployment during the Great Depression; all the numbers are later estimates by economists. Eric Rauchway notes that some folks are relying on earlier estimates rather than more contemporary ones.
Rauchway has a graph on his blog, but not the actual numbers. The Historical Statistics of the United States has their data online, but they charge a fee. Fortunately, my city taxes fund a useful educational institution known as a public library, which I pass every day on my bicycle ride home from work, and which holds a usefully broad collection of standard reference works. In few minutes, and for little more than two bits, I obtained photocopies of the relevant tables.
There are four editions of HSUS. The first three were published by the US Department of Commerce; the most recent of these was the Bicentennial Edition in 1975. The most recent edition was the Millenial Edition, published by Cambridge University with the permission of the Department of Commerce in 2005.
The earlier edition was based on a 1964 reconstruction by Stanley Lebergott, while the revision relies on a 1992 study by David Weir. The key difference is the handling of WPA and CCC workers, HSUS reports, but Weir also drew on a broader range of sources to assess non-manufacturing unemployment.
Below are the unemployment estimates from the two editions, from 1928 to 1943.
Historical Statistics of the United States, Millenium Edition
Unemployed, as a Percentage of Civilian Labor Force:
1928 - 4.74
1929 - 2.89
1930 - 8.94
1931 - 15.65
1932 - 22.89
1933 - 20.90
1934 - 16.20
1935 - 14.39
1936 - 9.97
1937 - 9.18
1938 - 12.47
1939 - 11.27
1940 - 9.51
1941 - 5.99
1942 - 3.10
1943 - 1.77
Historical Statistics of the United States, Bicentennial Edition
Unemployed, Percent of Labor Force
1928 - 4.2
1929 - 3.2
1930 - 8.9
1931 - 16.3
1932 - 24.1
1933 - 25.2
1934 - 22.0
1935 - 20.3
1936 - 17.0
1937 - 14.3
1938 - 19.1
1939 - 17.2
1940 - 14.6
1941 - 9.9
1942 - 4.7
1943 - 1.9
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