How do I find the geographic mean of a multipoint feature in QGIS?

There are multiple ways of accomplishing this.

For SQL, see https://postgis.net/docs/ST_Centroid.html.

Select st_centroid(geom) as geom from table

For a point collection (rather than a multipoint), use:

select st_centroid(st_collect(geom))
from table

Also, there's a Geometric Median function, which is useful on messy data with extreme outliers https://postgis.net/docs/ST_GeometricMedian.html.

And as per Erik's comment, there's also this method in the Field Calculator:

centroid($geometry)

And this one, in the Vector - Geometry menu: