The answer is: yes, the amount of background blur varies. A depth of field calculator gives only the depth of field, namely the depth of the distances where the picture is acceptably in focus (what "acceptably" means of course can vary depending on how large the image is printed, so depth of field isn't an absolute truth).
However, how much exactly the background is out of focus differs from one focal length to another.
There is a video on YouTube (jump to 8 minutes 0 seconds) that demonstrates the effect:
- 24mm focal length, f/10, 2.4 feet away gives a depth of field of 2.07 feet on a full frame camera,
- whereas 100mm focal length, f/11 and 10 feet away gives a depth of field of 2.02 feet, practically the same
Despite the same depth of field, the background is far more blurred with the 100mm focal length.
If you want blurred background, prefer longer focal lengths and fast apertures instead of blindly trusting a depth of field calculator.
You can derive the following equations if the background is at infinity:
b = f^2 / (x_d * N)
= 2 * x_d * C / DoF
DoF = 2 * x_d^2 * N * C / f^2
= 2 * x_d * C / b
b is background blur,
DoF is depth of field,
f is focal length,
x_d is subject distance,
N is aperture F-number, and
C is circle of confusion (usually considered to be 0.019 mm for Canon crop sensors and 0.030 mm for full frame sensors). You can see that long subject distance (which means long focal length given equal framing) allows both deep DoF and high background blur at the same time.
Since a photo is worth more than a thousand words on this site, here are two examples with approximately same background blur, yet very different depths of field.
Here's an image with
x_d = 8400 mm,
f = 250 mm,
N = 5.6:
We can calculate blur disc size at the sensor is 1.33 mm (about 5% of sensor diagonal). We can also calculate DoF is 240 mm. Thus, the whole fence is in focus, and so is the whole seagull.
Here's an image with
x_d = 1000 mm,
f = 50 mm,
N = 1.8:
We can calculate blur disc size at the sensor is 1.39 mm (about 5% of sensor diagonal again). We can also calculate DoF is 27 mm. Thus, parts of the cactus are out of focus. Note that strictly speaking, in this second example the background is not as far away than in the first example, so the real background blur isn't the same as the blur for a background at infinity.