Should I check the z-score if the p-value of Local Moran's I is significant?

Question

The dataset I'm using contains income data per area. The values are not normally distributed as shown in the following diagram. Global Moran's I indicates significant spatial patterns and Local Moran's I finds significant hot and cold spots (according to the p-value). When I check the z-score, it turns out that the cold spots don't reach significant levels. Could this be due to the distribution of income values? Is there anything I should do differently? Maybe use the log income?

Or can I simply ignore the z-score as long as th p-values are fine (= significant, < 0.05)?

(Using PySAL to compute both Global and Local Moran's I.)

Here's the histogram of log incomes:

Update:

I've recently aquired another income data set from a different country in which income values are normally distributed. Local Moran's I computations for this dataset result in significant hot and cold spots according to both p-value and z-score:

Answer 1

As I understand it .. and I would be happy to be corrected .. the local Morans I looks for spatial autocorrelation in local values (i.e. relative to the adjacent areas), abit like a GeoWeighted version of Global Morans I. As compared with say Gettis Ord which identifies spatial clusters of globally extreme values.

If so the result would seem to accord with your map, significant Z for the small region of very high incomes, while the blue area is only a broad basin within a more gradual local surface.

So the importance of the Z value depends if you are just looking for clusters of high and low incomes, or looking for clusters with steep boundaries e.g if comparing actual vs perceived income inequality?