So how did the Taxpayers' Union come up with such a wildly different result? The problem is that they compared all private sector jobs with all public sector jobs.
The private sector contains many low wage workers, in industries like retail or hospitality, whereas the public sector are mainly salaried professional jobs, often requiring university education. They are comparing apples with oranges.
Williams preempt this by saying "you may be concerned that it's comparing apples and oranges – public servants tend to be more educated, for example... but it doesn't explain why the gap has grown." Drawing from the same data, he claims that the gap between the two has opened up from "18.9 per cent in 1990 to 34.6 per cent in 2017".
Has the gap grown?
This second claim suffers from the same problem, because it doesn't account for the differences in occupation types in the public and private sector. However, we have a great source of data from Statistics New Zealand's Labour Cost Index, which breaks down wage and salary changes by occupation, in both the public and private sectors.
While the data only goes back to 2009, it shows that in every single occupation, public sector salary has grown slower than private sector salary.
Williams' second claim is also untrue.