The calendar year has the first day of Winter falling on December 21/22. The last day of Winter is March 19, so that makes the middle of winter on or about February 5, according to the calendar.
But, if you want to know what is the middle according to temperature, it is a little more complicated. My research has revealed that the most accurate measure is to use degree days. A person could calculate the total number of degree days from mid summer to mid summer, divide that by two, and find out what day half of the degree days falls on. For example, during the winter of 2007/2008, there were a total of about 7900 degree days, according the the NOAA's National Weather Service Weather Forecast Office. Half of those degree days (about 3950), shows up on the NOAA chart on January 25, 2008. In other words, as much fuel would be used to heat a home after January 25, 2008, as that was used from Summer 2007 to January 25, 2008.
This example is for that year only, but my investigation shows that to be an average year. Of course, an extremely cold or warm fall and warm or cold spring could change this date of January 25 back or forward a day or two, but January 25 is the day that would be used in an average year.
So, when you get to January 25, you can say that we are half way through the cold, and be optimistic that we are on our way out of winter.
The data used for this analysis was weather for the Alpena, Michigan area.