When one looks at a.radiating array pattern one can see that the yagi
is very inefficient. Does anybody know of the relative volume
contained in the main radiation lobe versus the total volume of the
entire pattern? I know there are a lot of different type antenna gains
and
arrangement but I am trying to determine in an informal way the
efficiency ratio and compare it to what would appear to be a very
efficient
antenna such as a dish.

I think the parameter that you are searching for is GAIN !!!!

An antenna only has gain by compressing power more into one direction more
than another. It is a case of robbing Peter to pay Paul, the more power you
have in the main lobe the less you have in other directions.

73
Jeff