How much gain does a 5/8th wave mobile antenna really have?

As with many things in life the answer is "it depends".

A 5/8th wave is claimed to have 3dB gain over a 1/4 wl antenna. Let's see if that is true.

 Above....Perfect infinite ground 1/4 WL. 5.15 dBi gain

Above pattern 5/8th wave over perfect infinite flat ground. Peak gain 8dBi.  Gain over 1/4 wave is 2.85db assuming ZERO matching coil loss.

The same infinite-groundplane antenna, with a matching coil Q of 100, has 2.6dB gain over the reference 1/4 wave.

With Height above Ground

A 1/4 wave centered on a 1 meter radius plate five feet high (that's a big, flat, and high vehicle roof) has 4.4dBi gain at zero degrees. Notice the pattern change.

A 5/8th wave centered on a 1 meter radius plate five feet high (that's a big, flat, and high vehicle roof). Five feet over perfect lossless earth. This antenna has

6.81 dBi gain at zero degrees. Notice a similar  pattern change. A vertical antenna' pattern is largely created by the reflection from earth around the antenna.

The 5/8th wave gain is 2.4dB dB better than the 1/4 wave in this case.

A 5/8th wave centered on a large flat roof over good flat soil has -4.56dBi gain at 2 degrees.

Maximum gain is now only 3.07dBi.

At 2 degrees takeoff the 1/4 wl over good soil with perfectly flat terrain has -6.75 dBi gain. Maximum gain is now only 3.08 dBi. 

The gain of a 5/8th wave over a 1/4 wl antenna at 2 degrees is now 2.19 dB, but the maximum gain is only .01dB different than the maximum gain of a 1/4 wave antenna!

With a normal mid-size roof contour 5/8th wave gain is now  -4.64dBi at 2 degrees elevation.  Maximum gain is now up slightly to 3.2dBi.

With a normal sloped roof shape and mid-car size ground, the 1/4 wave has -6.59dBi at 2 degrees TO.  Maximum gain is 3.05dBi.

5/8th wave gain is now 1.95 dB over a 1/4 wave antenna at 2 degrees, but the peak gain is only .15dB better!

The smaller the roof and/or the less centered the antenna, the less gain advantage for a 5/8th wave.

Lower height roofs also have less gain difference, and shape of the roof greatly affects pattern and gain.

The 2.85dB theoretical gain is only for a perfect lossless loading coil antenna over perfect infinite flat groundplane.

In the real world the actual gain difference between a 1/4 wave antenna and a 5/8th wave can be anywhere from the same up to a maximum of 2 dB.