MarkT:
Since there are never two long gaps in a row, its probably not a bit in itself - the code is probably a short pulse for one symbol and a short pulse followed by a gap for the other bit. If we call these 0 and 1 that gives:10101011010100100100001101
10101011010100011011011101Which makes the messages the same length, which is encouraging.
The 101010110101 sequence at the start is probably packet-synchronization preamble.
More data points needed to progress further perhaps?
Whoa, thanks Mark... that makes a ton of sense. I've taken several other samples and they all begin with the same sequence, so I'd agree with you that the 101010110101 sequence seems to be a preamble or perhaps some sort of device ID.
The unit transmits several bursts of the same bit sequence, perhaps 12 or 13 times per sample I've taken - it leaves a longer gap in between each repetition. Because there's so few bits in the sequence maybe that's its means of a CRC check... just say it until it gets right 
I've attached a couple more samples where the display read 86.0F and 16% (top in attachment) and 85.1F and 16% (bottom in attachment). All the samples I've taken begin with 101010110101. With the long gap at the end, do you think the last bit would be a 1?
If I understand them correctly, they come out to be:
10101011010100000001000001
10101011010100110001101101
It's puzzling to me that there is not more bits in the stream... I found this (Hacking the WH2 Wireless Weather Station Outdoor Sensor - Part 1: Reverse Engineering the Protocol - lucsmall.com), specifically the diagram near the bottom of the post, which may or may not prove to be helpful.
Any ideas, anyone?
EDIT: I should mention I did invert the waveform in this attachment vs the one in my original post...
capture4.tiff (57.9 KB)