The problem you've got is that your data is uncalibrated, even for directivity (let alone absolute gain). Since your data represents only a small 1-D slice of the entire sphere, you can't derive anything.

Let me ask this - is your data real-world measured data, or something arrived from a model?

This is real-world data...

I had the impression, though, that Directivity was simply a measure of the radiation in a certain direction compared to an average radiation intensity.  Shouldn't that be completely described by the radiation pattern, assuming I choose my 'average radiation' to be, say, an ideal dipole?

Hmm, maybe I should ask what I CAN get out of a plot such as this.  I can easily calculate the side lobe levels, as well as the half-power beamwidth...  What other useful metrics could be gleaned without further knowledge about the antenna itself?  At present, I don't have access to what the antenna manufacturers' claim is the gain, etc.  It's coming, but not for a while.

You can assume omni directionality in the other plane, which would give you the lowest directivity.
A quick gain formula for an omni in one plane is;
Gain (dB) = 10*log(112/Beamwidth in degrees at -3 dB down).
Hence if you have a measured antenna which rolls off 3 dB in 5.6 degrees from antenna boresight and has therefore 11.2 degree beam pattern, then the gain is 10 dB for your antenna.
Realistically, your antenna directivity is at least a few dB higher because most 1D arrays have 90 degree beamwidths in the other plane. If this is the case, your directivity is probably closer to 16 dB than 10 dB since the other plane has beamwidth not 360 degrees but 90 degrees, or 4x less hence 10 log (4) = 6 dB higher.

Maybe repeat your dipole directivity calculation by assuming a -3 dB beamwidth in the other plane of ?90 degrees or whatever you estimate.

It helps to memorize some gain formulas too, they come in handy. Directivity = 41,000/(Az*El) with Az and El being beamwidths in degrees for orthogonal planes.

What type antenna is it? Notch, patch, etc. You can make pretty accurate assumptions with the antenna type and size for the beamwidths in the other plane.

kch
PS: Knowing directivity, adding power divider losses gives you gain, which can be many dB less then directivity.

Since your data only covers the 'front' half of the pattern (+/- 90 Az), do you know if the 'rear' of the pattern is similar to the 'front' (like a dipole), or closer to zero (more like a dish)? Obviously if you assume wrong, then you could be about 3dB off.

Has someone left off the 'rear' data because there are no significant rear lobes, or because they're terrible and they would prefer not to mention them?

Many of the short-cut formulas are based on assumptions. You need to know those assumptions and compare them to your best guess of the total picture.

The main point is the total energy in the areas other than the main beam. Lower amplitude over a wide area can take away as much as a narrower and higher peak.

The difference between gain and directionality is loss.

If it is manufacturer's data that you're checking, you should be highly skeptical (depending somewhat on the application and particular industry's habits).