What do the Experts Say?
The Intergovernmental Panel on Climate Change (IPCC) writes reports which are compiled by synthesizing thousands of peer reviewed scientific papers. The Fifth Assessment Report (AR5) is the most recent report. Based on an assessment of the subset of models that most closely reproduce the climatological mean state and 1979-2012 trend of the Arctic sea ice extent, a nearly ice-free Arctic Ocean in September before mid-century is likely (IPCC AR5 Source). It is important to note that the IPCC AR5 defines "nearly ice-free conditions" as sea ice extent less than 106 km2 or 1,000,000 km2 for at least five consecutive years (IPCC AR5 Source). These estimates are actually old and do not account for the added 6 years of data. A more recent study by Peng G et al. expect an ice free Arctic by 2036 plus or minus 6 years (Source).
My Simple Back of the Envelope Calculation
Most scientific projects analyzing sea ice extent only look at the satellite data which started in 1978 with the first complete year of data in 1979. The data before this period is based on proxies which are not as sturdy as direct observations. Data I used can be found here. It is difficult to see the downward trend when plotting all the data.
The seasonal difference in ice extent is huge creating a sine wave pattern. When taking monthly averages between 1979-2017, March exhibited the greatest Arctic ice extent at 15.3*106 km2 and September showed the smallest Arctic ice extent at 6.14*106 km2. I then chose to plot these two months because September will likely be the month when the ice extent will go to zero. March will likely be the last month to have no sea ice extent. In order to see a trend I fit a second degree polynomial trend line to them. I chose the second degree polynomial equation because it showed the best R-Squared value when compared with a linear regression. I took that same second degree polynomial equation to forecast out when the trend line would hit zero. For September the year will be 2041 and for March is will be 2369. These estimates should be taken with a grain of salt because the equation I used to forecast into the future does not account for positive or negative feed backs that will likely occur
References
Fetterer, F., K. Knowles, W. Meier, M. Savoie, and A. K. Windnagel. 2017, updated daily. Sea Ice Index, Version 3. Daily Sea Ice Extent Data Files. Boulder, Colorado USA. NSIDC: National Snow and Ice Data Center. doi: http://dx.doi.org/10.7265/N5K072F8. 2/10/2018.
Peng, Ge, Jessica L. Matthews, and Jason T. Yu. "Sensitivity Analysis of Arctic Sea Ice Extent Trends and Statistical Projections Using Satellite Data." Remote Sensing 10.2 (2018): 230.
Ether address: 0x1271F7b7Ddfd2dF35e18dA67D2330d56B70c11C4