Examples of the DOF calculator on previous page, just to show you. You should go out and see how this works too. f/1.8 may be a bit difficult in bright sun, may need up near 1/8000 second at ISO 100. The first picture below is f/8, and all are uncropped, full frame. It is probably about the right field size for a head and shoulders portrait. The vertical metal ruler is one foot for scale, about same height as a human head. You'll have to use your imagination. :)
The situation for the example photos shown below are the the Nikon D300 sensor size 23.6 x 15.8 mm (DX or APS size). That computes CoC 0.0197 (if Divisor is 1442), and Crop 1.523 (to compute numbers below the pictures).
The far end of the house and Crepe Myrtle bush is measured to be 40 feet behind the "subject" (you might make out the blurred yellow tape on ground). Focus is on the near subject. Camera is measured to be at 6 feet for the 50 mm lens, and 24 feet for the 200 mm lens (6 x 200/50 = 24) - which both are the Same size Field of View at the subject, but 200 mm is vastly less view of the background. The bottom yard stick shows field of view. The top yard stick is angled about 25 degrees so that the ends are about 7 inches closer or farther distance than the bottom straight yard stick (hoping to show DOF). DX Nikon D300, 50 mm f/1.8 and 70-200 mm f/2.8 lenses.
At the background, the first f/8 image below computes 7.76x times the 0.0197 CoC limit, and we can see it is not quite sharp back there (f/8 is only 1.36 feet DOF at 6 feet, but the background is at 46 feet).
In the f/1.8 image, we see 34.83x CoC is much more background blur, and DOF is much less, only 0.3 feet at the subject.
The 200 mm f/4 background is at 64 feet now, for 46.4x background CoC (lager CoC was one of our goals, to blur the background). It has larger Circle of Confusion at the background, an even greater blur. We can sort of see the circles. Yet (at the same FOV), the f/4 subject DOF is over 2x greater than if at f/1.8, or 0.67 feet at f/4, better. It's still a tough situation, but we made progress toward our goals (more blur on background, and more DOF around our subject). Just a little experience noticing should learn to know what to expect.
I'm sure you must see the point. We stood back with a longer lens, and still have the same view of the "subject", but we improved the background drastically... Simply removed most of it, and blurred the rest as well or better. There really wasn't much we could do with the short lens, but the longer lens allows only a step or two sideways to select the best small part of the background to include.
These images are not cropped either. The greater subject distance gives the 200 mm essentially the same field of view at the subject. But the 200 mm certainly does crop the background, a plus. And sort of like moving a spot light around, we can easily move the camera slightly to choose the best part of it. And the lens can enhance the DOF, another plus at the subject, and still be more blurred at the background.
IMO, even the f/5.6 background looks better than f/1.8. And f/5.6 certainly has strong DOF advantages at the subject. We have choices. We could of course pick better backgrounds, and subjects too, and there are many numerical combinations, but the longer lens is a very strong technique. It helps portrait perspective too. This is just a quickie test, with a hopelessly poor background to make a point.
Which above do you think could be the better portrait background? Did you notice how much of the wide background is simply missing? The telephoto background is so narrow, making a step or two sideways is easy to select the most desirable part of it to show (or omit). Do you imagine stopping down a couple of stops from f/1.8 could help the subject DOF?
Here are some blowups to better see the subject detail: (same pictures, full frame width, 33% size, 950 pixels wide)
50mm 6 feet f/8, DOF 1.368 ft, BG CoC 7.76x
50mm 6 feet f/1.8, DOF 0.30 ft, BG CoC 34.83x. I don't see much appeal of f/1.8
200 mm 24 feet f/2.8, DOF 0.48 ft, BG CoC 63.07x
200 mm 24 feet f/4, DOF 0.67 ft, BG CoC 44.6x
There really is nothing to debate here. Trying to eliminate the background with 50 mm at f/1.8 seems like throwing out the baby with the bath water. Standing back with the longer lens is a good first step. The real point is, suffering with f/1.8 is self inflicted, not normally necessary, or desirable. There are better goals.
For those who get up tight about slight rounding errors:
That's me too, but it's not all that easy. This D300 takes 4288x2848 pixel images. That part always comes out right. That of course computes 12.212 megapixels. The manual calls it 12.3. The 4288x2848 is 1.5056 aspect ratio. The sensor dimensions of 23.6 x 15.8 mm computes 1.4937 aspect ratio. I've been calling it 1.5 ratio. The math is not hard, but the hard part is determining the right numbers to input. So there may be some tiny rounding errors. :)