or, all generalities are wrong, including this one
Don't lose any sleep over any of this; I haven't. Honestly, I'm just intoxicated with the ability to generate data files programatically and have another program generate graphs from them. Any cheap crappy lens will blur backgrounds fine for situations where you want that (like portraits); just shoot it wide open. How about a £100 Nikon 50mm f/1.8?
Does that just fine, thanks. What about a £15 Soviet 58mm f/2ski?
Does that just fine too! If you've got a zoom, stand back and zoom in. Stop thinking about this and get out and shoot instead.
Now, with the normal people gone..
Everybody knows that a wider aperture will give you more background blur:
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Here, we have a selection of hypothetical 50mm lens focused on a subject 1 metre away, with a background four metres from the camera. The numbers at the bottom are absolute aperture sizes, which you get by dividing the focal length by the f/number. The leftmost point on the line corresponds to f/0.7, the rightmost point to an f/5.6 lens.
(If you don't like pretend examples, imagine that I'm referring to the Zeiss 50mm f/0.7, Nikon 50mm f/1.4 AF-S, Zeiss Jena 50mm f/2.8, and Sigma 18-50mm f/3.5-5.6 at the tele end. I'm not, but you can pretend that.)
The y axis is the size of the defocus blur circle for the background. Larger number = more blurred backgrounds.
What we see isn't surprising: the relationship between the degree of background blur and maximum aperture is a simple, linear one. All else being equal, an f/1.4 lens shot wide-open will give you a defocus blur circle precisely twice the size of an f/2.8 lens shot wide-open, the latter being twice as much as an f/5.6 lens.
We could do this with any given focal length, subject distance, and background distance, and we would come to exactly the same conclusion: the relationship between maximum aperture and background blur is always linear. Simples!
Things get more interesting (trust me) when we throw focal length into the mix:
Here we've taken exactly the same subject distance and background distance, and tried it with lenses of the same absolute aperture (50mm f/1.4, 100mm f/2.8, 200mm f/5.6, etc up to 400mm). We're looking at the absolute aperture so that we can examine the relationship between focal length alone and background blur.
So this is interesting, because a direct correlation between focal length and background blur would suggest that a 400mm lens would blur the background 8 times as much as much as a 50mm lens with the same absolute aperture, but it actually does so over twelve times as much.
Thus it is established: an increase in focal length results in background blur disproportionate to the ratios of the focal lengths.
One thing we've forgotten to factor in is the different distances from the subject necessary to get the same framing. There's a direct correlation with focal length and the distance at which you stand; to get the same subject framing that you'd get with a 50mm lens standing 1 metre away, you'll need to stand 4 metres away with a 200mm lens. There's two factors fighting each other here. First, the background blur circle (all else being equal) becomes greater the closer you are to a subject, which would give the 50mm lens the edge. But at the same time, that's also moved the background further away from the camera, which would blur it further.
So let's take a subject that you shot from 1 metre away with a 50mm f/1.4 lens, then try that with different lenses and moving around to keep the same framing. For added fun, we're going to try moving the background as well; we'll try it at 4, 8, 16, 32 and 64 metres. Once again, we're using absolute aperture sizes, not relative ones, just so we can see these effects.
Did you see that? Surprise conclusion! If you maintain the same framing and keep everything else the same (i.e. the same absolute aperture), the amount of background blur decreases as you increase focal length!
Another thing worth noting is that the closer the background is to the subject, the more pronounced this effect becomes. Flip it the other way round: as the background distance approaches infinity, the blur circles become increasingly similar. At infinity, the size of the blur circles of a 50mm f/1.4 and a 400mm f/11.2 (!) are identical if you keep the same subject framing; the amount of background blur increases in direct proportion to the absolute aperture size.
Practical application of this: if you have a long, slow lens (like the f/5.6 long end of a crappy 70-200mm zoom lens), you can get almost the same amount of background blur as you would with a 50mm f/1.4 by just moving some place where the background is far away. Woohoo!
So where does the rule "longer focal lengths blur the background more" come from, if it's so obviously contradicted by the above? Simple: we've dealt with absolute apertures, while lenses are marked, sold, and spoken about in terms of their relative apertures, or f/numbers. There's no such thing as a 40mm f/11.2, that I know of.
So what happens if we take our above experiment, but use a relative aperture of f/1.4? This does:
We notice that as the background distance increases, there's an increasingly direct correlation between focal length and blur circle size. This is the same as earlier: as the background distance approaches infinity, the size of the blur circle grows in an increasingly linear way with absolute aperture size. In other words, as the background gets further away, "longer lens means more background blur" becomes more true. (And once again, with a background at infinity, that's exactly what happens.)
But wait, what's happening with the background at 4 metres? It's starting to level out. Let's extend our focal length range to 3200mm (at f/1.4!) and see what happens:
Notice how the red line goes nearly flat, and the blue and green lines are well on their way there? Beyond a certain point, with background distances that you may well encounter in the real world, increasing focal length has very little effect on background blur.
(That's even though we're pretending that there's such a thing as a 3200mm f/1.4: there isn't. But you can feed in any other relative aperture value for the same effect.)
There might be something interesting worth looking at below 50mm. Let's see:
So why is it so hard to get background blur on point and shoot cameras?
Just knowing the focal length and maximum aperture will tell you relatively little about how nice the background blur will look. One thing frequently overlooked is that we perceive all things in a photograph relative to the other things in a photograph (this is one reason HDR sucks so much). In this case, humans will, to some degree, perceive the amount of background defocus relative to how sharp the subject is. Nearly all lenses are less sharp shot wide-open, thus we'll perceive the defocused parts of a photograph as being sharper as the subject area becomes softer.
Then there's the issue of "bokeh", which has been done to the death by so many people that I won't go into it very far, other than to mention that it's about the quality of the background blur. Ultra-fast lenses (in the 50mm domain, around f/1.4 and faster) often have pretty ugly background blur when shot wide-open; therefore under certain circumstances and with certain lenses, the background becomes less distracting as you stop down, despite the blur circle being smaller.
It happens that two random lenses that I could be bothered to find illustrated this quite well. Here's a boring test shot of a very small teddy bear, taken with a Nikon 50mm f/1.8D wide-open on a Nikon D2H:
And here's almost the same framing (I accidentally pulled some sky into the above shot, which is bad) with a 55-200mm VR, at 200mm and f/5.6:
Calculations show that with a background 6 metres behind the subject and keeping the same framing as a 50mm lens with a subject distance of 1 metre, the difference between the size of the defocus blur circles on the two lenses should be negligible (1.25mm for the 50mm f/1.8 vs 1.13mm for 200mm and f/5.6). Yet the backgrounds look completely different. It's up to you which you prefer. For me, the 55-200mm VR looks slightly more pleasant, especially on the repeating patterns, though the 50mm defocuses the twigs (actually tomato plants) in the background somewhat better. If they were the only two lenses I owned, I'd still grab the 50mm for a portrait, since on DX digital the crop factor gets me just far away enough that the perspective is pleasing while not being so far away that communication with the subject becomes difficult without shouting. Real-world factors win every time.
(Talking of repeating patterns: you can drive yourself crazy worrying about spurious resolution. I actually ran into that problem with this shot; there was a chainlink fence in the background.)
If any of this bothers you, make your own experiments if you already own a lens, or find real-world photographs (not boring test shots) taken by a skilled photographer if you do not. As for me, I know that I would go nuts worrying about all the factors involved, so I don't hesitate to shoot any lens, even the crappy Ukrainian lenses that I got with my Kiev 88, wide-open if I really need to.
Generalities are nice. Once the volume of information increases beyond a certain point, we need generalities to stop our heads from exploding. Things like "increasing your focal length gives you more background blur" simplifies a much more complicated situation such that we can make intelligent decisions within a limited scope, just like "all politicians are liars" helps us deal with the news in a way that makes sense.
So what did we learn today, children?
I didn't enter a single value into a spreadsheet to generate these graphs, thank goodness. I wrote a Python script to generate data files, which was then fed into a gnuplot script to generate these tables. Data at random points generated by the Python script were fed into VWDOF to verify that they were correct.
In the spirit of openness, you can have the source code, including all generated data files in a zip file here; this is so you don't have to take my word for it. The source code and files generated therefrom are in the public domain to be used without any restrictions. (A simple "sh build.sh" will generate the files on Unix systems; you'll need Python, gnuplot and rsvg installed.)