Answered by Shaykh Faraz Rabbani
If I’m living on the floor above a mosque and an Alim from Dar al Uloom India who lives on the next block (and who considers both East and NE qiblas to be equally valid, though he prays towards the East) tells me that the qibla in that mosque faces East and I am in New York City, then should I pray in my room based on the NE qibla found using my compass or should I join the jamaat downstairs?
If what my compass tells me is correct, all of the mosques that I have prayed in NY are facing at least 15 degrees to the east of the exact direction, and probably more than that.
Since I have only recently begun to use a compass with some intelligence (I hope), I don’t know what is better for me to do, join the jamaat facing a dubious Qibla (that an Alim has informed me faces East), or trust in my use of the compass and pray in my room facing NE.
It is obligatory to face the direction of the Qibla, by scholarly consensus. [Radd al-Muhtar] The Qibla in North America is towards the North East, varying in each place.
It is valid to be within 45 degrees of the qibla [ibid.], though it is not proper to face away from the exact direction when one knows it. [Sh. Mahmoud Usmani, and others]
Note that when using a compass, you have to account for the angle of declension. As mentioned in Encarta:
“In order to obtain readings of true north on the magnetic compass, it must be corrected also for variation, the angle between the magnetic and true meridians. This angle, also called the magnetic declination, varies in amount, in direction from east to west of the true meridian principally with geographical position, and to some extent with time.”
This would account for the variance you found.
[See answer below, too.]
And Allah knows best.