I am not a professional photographer, architect or geometry professor (one may me the nod to the reviews regarding the distortion, so forgive).
For me it is important what comes out in the end; a good photo!
This can indeed afford the Sigma lens but there are better lenses.
The Sigma lens 1 already said goodbye to 15 photos - total failure of the autofocus.
The replacement of Amazon lies not long in coming and so I could continue my comparison.
The Sigma Objetiv with 512 grams 80 grams heavier than the Nikon.
Personally, I find the heavier lens better already because of the lower risk of camera shake.
The 80 grams does not sound much, feel, in practice, but after much more, the Sigma works definitely heavier and more solid.
The whole feel of the Sigma is better.
But that is the only plus point, in all other disciplines that Nikon took a clear lead.
I've done a lot of comparison photos in identical situations.
The Nikon lens produces pictures with beautiful colors and better contrast, very extreme when shooting outdoors with blue / white sky visible - a very big difference.
The focus is always better when a touch Nikon.
Clearly visible when zooming into the picture.
The improved light sensitivity manifests itself, the Nikon always achieved a faster shutter speed with the smallest focal length than the Sigma.
A clear advantage of the Nikon is his noiseless image stabilizer, which acts perfectly in his role.
When Sigma is a constant "hum" heard (even when the image stabilizer).
That's when photographing unless bad but since my D3100 also has a (good) Full HD video mode, that's disturbing to hear in quiet environments on the recordings.
Also works in video mode, the autofocus of the Sigma fairly sluggish, or unsatisfactory.
Here also a clear victory for the Nikon.
Conclusion:
For me as a "mortal" Fotografierer with high demands on image quality, the Sigma 18-125mm F3.8-5.6 DC OS is not good enough.
For a slightly higher price there are definitely better (visual) quality (Nikon).
On the 20mm longer focal length you can do without it.
Note:
As I now know, there should be a relatively high scatter in the quality of lenses from Sigma.
It may be that I had a negative "outliers".
Of course I do not have the expectation that I only need to order multiple lenses, to find out the best.
The awarded 3 stars for me are so rightly ...