IMHO, nothing works better than using their own data and showing them an insight that they have not discovered themselves. Having been in the trenches building analytics in organisations, it is likely your first inroad will actually be data visualisation.
In fact, data visualisation is built just for that. It’s strength (if proper data viz principles are adhered to) are that it hits your reader at the perception level, not the cognitive level.
To illustrate with a simple example, say I want to do sell analytics to a customer manager who has used his gut and and his spreadsheet to resource his team and set his budget.
For the past decade.
You are tempted to do some statistical modelling of call center volumes, with every single complexity real life you can imagine – multiple layers of seasonality, missing values, dirty and noisy data, anomalies and outliers. You could craft a business case outlining your process, your product, your methodology, and present it along with your qualifications and a hefty chunk of research paper backing.
The problem is, if you are speaking to Tough Tom, who has thrived pretty well without your ‘fancy stats’ longer than you have been in the workplace, and your product looks like this:
You don’t have a chance. Chances are engaging Tom at the cognition level takes a whole lot longer than 30 minutes of brilliance.
On the other hand, you could take the same algorithms, run their own datathrough it, keep all the advanced analytics under the hood for now (it’s painful – I know!) and go: ‘hey Tom, check this out – using this you can see that you need to call in 3 temps on Friday’.
A murmur ripples through the room – Tom blurts out ‘how did you know we are having a special this Friday afternoon?’. You smile, go through the process to update the forecaster – in less than 3 minutes – and introduce the analyst who will be happy to help Tom run through the software the first couple of times through.
From the analytics meetups, forums and even right here on Quora, the more advanced analytics capabilities seem to be in vogue this couple of years (*cough*deep learning*cough), but in truth it is seldom the first thing that corporates adopt. The inroad is almost invariably data visualisation / BI style work to start allowing the organisation to evolve towards being more data driven.
I hope that helps.