The answer will depend on the country or region whose crime data you are interested in. And since you have not bothered to specify which one, I cannot provide a more useful answer.
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some sources of data are information
Simple, Primary and Secondary Data
In general, the answer is no, both negative and positive z score values should be expected. A z-score (or standardize score) is the raw data value minus the mean and then this result divided by the standard deviation. If the data can be considered normally distributed and a random sample is taken from a population, then as the sample size becomes large, approximately half the z-scores should be negative and half of the z-scores should be positive. There are some exceptions. Small data sets may have only positive values. A non-normal (skewed) distribution if skewed to the right, may have, after normalizing, may have a higher portion of z scores as positives.
The FBI collects data used by researches to compare crime in different countries.
Certainly, if some of the data is negative