... or other mobile devices such as the iPhone 5s, Nexus 6 or iPad Air 2?
Everyone says that the dynamic range of mobile devices is lower, but how much lower?
For comparison, an APS-C camera like the Sony NEX-5R does 13.1EV, while a camera with a 1-inch sensor like the RX100 does 12.4EV.
I think the simple answer here is "you can't". "Dynamic range" isn't a concept with a simple definition, so unless you can get numbers from DxOMark as you've quoted for the NEX-5R and RX100, then you can't compare them. As an example of this, compare DPReview's test of the Sony a6000 where their tests give a dynamic range of 8 2/3 EV for the A6000 (the green line runs from -4 2/3 to +4 on the first graph on the page), but DxOMark quotes 13.1 EV for the same camera. Are either of them "wrong"? No - they're just each measuring dynamic range in their own way. DxOMark don't publish the same kind of technical measurements for mobile phone cameras as they do for standalone camera, so we can't do a sensible comparison.
Getting slightly off-topic here, but there's an obvious question as to "why don't DxOMark publish technical measurements for mobile phone cameras?" I can't find a direct answer to that from DxOMark, but given that it's known that they work purely from RAW data and a significant number of mobile phones don't (as of October 2014) offer a RAW option, they obviously had to find a different methodology for mobile devices. There's also the fact that mobile photographers are (generalizing wildly) less concerned about the technical details of a photo and more about the content when compared with photographers using standalone cameras, so DxOMark are pitching their measurements towards the things that the majority of users care about, which is just business sense.
Take a newspaper, and shoot it from a normal reading distance, starting from an exposure that results in only traces of the letters over paper rendered bright white and all the way down to the paper rendered black. Bring the shots to computer, open them so that the scale is 1:1 and see the range of exposures where you can read the articles, including small type in captions. That range is the practical photographic range. It is usually 3-4 stops less than what DxO are indicating, and depends on the lens, colour/spectrum of light, and so on.
You can do what DxO and others do - make test photos and analyze the result. Dynamic Range can be measured using the test chart displaying a range of fields of known brightness. imatest is one of applications that automate the process (see: http://www.imatest.com/docs/q13/) but this is all maths and image processing and could be performed "by hand" by a determined amateur researcher.
On iPhone you can use an app that saves uncompressed photo instead of JPEG, this would be still 8 bits per pixel, but without JPEG compression artifacts. The fact that your input picture is not RAW does not mean you can't measure dynamic range, it only means that the dynamic range will be inferior compared to what image could be extracted from the RAW data.
