NY tax is calculated in the following manner (see IT-203, the NY nonresident return).
- Take all income for the year, earned anywhere. Lets say that figure is $70,000
- Apply the standard deduction
- NY income is now $62,100
- Tax on $62,100 is $2500 (hypothetical figure)
- Apply rate of NY taxable income actually earned in NY (10,000 vs 70,000) =14.29%
- Multiply % (14.29%) by calculated tax on all income ($2500)
- That figure is your ultimate NY tax liability ($357, in this example)
New York calculation doesn't simply take your NY source income and ignore the rest. It performs an allocation.
You can see this on the form, going down to line 45:
https://www.tax.ny.gov/pdf/current_forms/it/it203_fill_in.pdf