The timing of the average occurrence of the coldest day of the year varies a little bit in the viewing area being in the January 9-20 time frame when you look at 1981-2010 daily records.
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The Top 10 Coldest December Temperatures Recorded at West Lafayette (Snow Depth at the Time).
Purdue University 1879-1953, Purdue Ag Farm 1954-Present
1. -31
December 19, 1884 (12")
2. -23
December 29, 1880 (6")
3. -22
December 22, 1989 (3")
4. -20
December 28, 1924 (3")
5. -18
December 24, 1983 (8")....December 25, 1983 (8")....December 23, 1989 (3")
6. -17
December 3, 1886 (2")....December 30, 1909 (14")
7. -16
December 21, 1963 (5")....December 20, 2016 (Trace)
8. -15
December 10, 1880 (2")....December 30, 1983 (10")....December 31, 2017 (7")
9. -14
December 9, 1880 (2")...December 31, 1880 (6").....December 16, 1951 (3")....December 17, 1951 (3")....December 9, 1958 (2")....December 20, 1963 (5")....December 31, 1983 (8")
10. -13
December 28, 1880 (6")....December 8, 1882 (5")....December 21, 1989 (4")
