Understanding 1,537cM match to person and 1,950cM match to their uncle

Greg , DNA doesn’t lie - you have presented the information and the return replies reflect your situation . Perhaps for real piece of mind you make contact with the female ( half sibling ) and both do DNA samples within your GPS or pathology departments .. best of luck.

If you are sharing the 1537cM with this female It would be half sibling based on both your ages. Share the same father , different mother . You both share the same Paternal Uncle . Based on the information you have provided.

The Shared cM Project 3.0 tool v4 suggests for 1,537 cM that the relationship has these probabilities (based on stats from The DNA Geek):

• 100.00% Grandparent Aunt / Uncle Half Sibling Niece / Nephew Grandchild
• ~0% Great-Aunt / Uncle Great-Niece / Nephew

and that the:

[second] set of relationships is just within the threshold for 1537cM, but has a zero probability in thednageek's table of probabilities

Based on the age of your match in comparison to your own age (which is three years difference) I think your first theory should be that you may be half siblings.

The Shared cM Project 3.0 tool v4 suggests the same relationships for 1,950 cM and you already know this match is an uncle (and you think that uncle is paternal) to the first match (who may be a half sibling to you), and this shared centimorgan value would suggest that her uncle may also be your uncle.