You have said Chicago’s daily normal temperatures were smoothed so that they change by no more than one degree per day. How much variation would there be if they were unsmoothed?
You have stated the “smoothed” situation correctly. Chicago’s daily normal temperatures at O’Hare International Airport (calculated from the 30 years, 1981-2010) range from 23 degrees (the average of a day’s high and low temperatures) in mid January to 74 in July. The unsmoothed averages range from 20 degrees to 77, and they fluctuate (up or down) by zero through four degrees per day. The greatest fluctuation, four degrees, occurs on three occasions (from 23 to 27 degrees on Jan. 21-22, from 50 to 54 on April 23-24 and from 23 to 27 on Dec. 26-27). Fluctuations of three degrees occur an additional 13 times; the remaining days fluctuate from zero through two degrees.