Database schema:

CREATE TABLE institution_details (
	unitid INTEGER -- example: ['100663', '100654'],
	chronname TEXT -- example: ['University of Alabama at Birmingham', 'Alabama A&M University'],
	city TEXT -- example: ['Birmingham', 'Normal'],
	state TEXT -- example: ['Alaska', 'Alabama'],
	level TEXT -- example: ['2-year', '4-year'],
	control TEXT -- example: ['Public', 'Private not-for-profit'],
	basic TEXT -- example: ['Masters Colleges and Universities--larger programs', 'Research Universities--very high research activity'],
	hbcu TEXT -- example: ['NULL', 'X'],
	flagship TEXT -- example: ['NULL', 'X'],
	long_x REAL -- example: ['-86.80917', '-86.568502'],
	lat_y REAL -- example: ['33.50223', '34.783368'],
	site TEXT -- example: ['www.uab.edu', 'www.aamu.edu/'],
	student_count INTEGER -- example: ['4051', '11502'],
	awards_per_value REAL -- example: ['14.2', '20.9'],
	awards_per_state_value REAL -- example: ['18.8', '17.8'],
	awards_per_natl_value REAL -- example: ['21.5', '22.5'],
	exp_award_value INTEGER -- example: ['105331', '136546'],
	exp_award_state_value INTEGER -- example: ['75743', '92268'],
	exp_award_natl_value INTEGER -- example: ['66436', '101725'],
	exp_award_percentile INTEGER -- example: ['97', '90'],
	ft_pct REAL -- example: ['93.8', '72.7'],
	fte_value INTEGER -- example: ['3906', '10032'],
	fte_percentile INTEGER -- example: ['33', '67'],
	med_sat_value TEXT -- example: ['1146', '823'],
	med_sat_percentile TEXT -- example: ['0', '84'],
	aid_value INTEGER -- example: ['6088', '7142'],
	aid_percentile INTEGER -- example: ['50', '72'],
	endow_value TEXT -- example: ['24136', 'NULL'],
	endow_percentile TEXT -- example: ['93', 'NULL'],
	grad_100_value REAL -- example: ['10.0', '29.4'],
	grad_100_percentile INTEGER -- example: ['15', '67'],
	grad_150_value REAL -- example: ['53.5', '29.1'],
	grad_150_percentile INTEGER -- example: ['14', '66'],
	pell_value REAL -- example: ['71.2', '35.1'],
	pell_percentile INTEGER -- example: ['39', '98'],
	retain_value REAL -- example: ['80.2', '63.1'],
	retain_percentile INTEGER -- example: ['17', '70'],
	ft_fac_value REAL -- example: ['82.8', '92.4'],
	ft_fac_percentile INTEGER -- example: ['98', '89'],
	vsa_year TEXT -- example: ['2010', '2011'],
	vsa_grad_after4_first TEXT -- example: ['22.3', '14.7'],
	vsa_grad_elsewhere_after4_first TEXT -- example: ['2', '2.9'],
	vsa_enroll_after4_first TEXT -- example: ['34.2', '36.5'],
	vsa_enroll_elsewhere_after4_first TEXT -- example: ['16.1', '19.2'],
	vsa_grad_after6_first TEXT -- example: ['33', '42.6'],
	vsa_grad_elsewhere_after6_first TEXT -- example: ['10.5', '5.3'],
	vsa_enroll_after6_first TEXT -- example: ['12.5', '7.9'],
	vsa_enroll_elsewhere_after6_first TEXT -- example: ['13.1', '14.6'],
	vsa_grad_after4_transfer TEXT -- example: ['NULL', '15.7'],
	vsa_grad_elsewhere_after4_transfer TEXT -- example: ['NULL', '1.5'],
	vsa_enroll_after4_transfer TEXT -- example: ['NULL', '40.9'],
	vsa_enroll_elsewhere_after4_transfer TEXT -- example: ['NULL', '17.2'],
	vsa_grad_after6_transfer TEXT -- example: ['NULL', '36.4'],
	vsa_grad_elsewhere_after6_transfer TEXT -- example: ['NULL', '5.6'],
	vsa_enroll_after6_transfer TEXT -- example: ['NULL', '17.2'],
	vsa_enroll_elsewhere_after6_transfer TEXT -- example: ['11.1', 'NULL'],
	similar TEXT -- example: ['196060|180461|201885|145600|209542|236939|126818|2', '232937|100724|405997|113607|139533|144005|228501|1'],
	state_sector_ct INTEGER -- example: ['16', '13'],
	carnegie_ct INTEGER -- example: ['386', '106'],
	counted_pct TEXT -- example: ['56.0|07', '99.7|07'],
	nicknames TEXT -- example: ['NULL', 'UAB'],
	cohort_size INTEGER -- example: ['882', '1376'],
	PRIMARY KEY (unitid)
)


CREATE TABLE institution_grads (
	unitid INTEGER -- example: ['101028', '100760'],
	year INTEGER -- example: ['2012', '2011'],
	gender TEXT -- example: ['B', 'M'],
	race TEXT -- example: ['X', 'W'],
	cohort TEXT -- example: ['4y other', '2y all'],
	grad_cohort TEXT -- example: ['446', '185'],
	grad_100 TEXT -- example: ['NULL', '73'],
	grad_150 TEXT -- example: ['105', '40'],
	grad_100_rate TEXT -- example: ['16.4', 'NULL'],
	grad_150_rate TEXT -- example: ['21.6', '23.5'],
	CONSTRAINT fk_institution_grads_institution_details FOREIGN KEY(unitid) REFERENCES institution_details (unitid)
)


CREATE TABLE state_sector_details (
	stateid INTEGER -- example: ['0', '1'],
	state TEXT -- example: ['Alabama', 'United States'],
	state_post TEXT -- example: ['U.S.', 'Ala.'],
	level TEXT -- example: ['2-year', '4-year'],
	control TEXT -- example: ['Private for-profit', 'Private not-for-profit'],
	schools_count INTEGER -- example: ['1180', '632'],
	counted_pct TEXT -- example: ['NULL', '61'],
	awards_per_state_value TEXT -- example: ['NULL', '18.8'],
	awards_per_natl_value REAL -- example: ['21.5', '22.5'],
	exp_award_state_value TEXT -- example: ['NULL', '75743'],
	exp_award_natl_value INTEGER -- example: ['66436', '101725'],
	state_appr_value TEXT -- example: ['NULL', '290.68'],
	state_appr_rank TEXT -- example: ['NULL', '11'],
	grad_rate_rank TEXT -- example: ['23', '18'],
	awards_per_rank TEXT -- example: ['42', 'NULL'],
	PRIMARY KEY (stateid, level, control), 
	CONSTRAINT fk_state_sector_details_institution_details FOREIGN KEY(state) REFERENCES institution_details (state)
)


CREATE TABLE state_sector_grads (
	stateid INTEGER -- example: ['2', '1'],
	state TEXT -- example: ['Alaska', 'Alabama'],
	state_abbr TEXT -- example: ['AL', 'AK'],
	control TEXT -- example: ['Private for-profit', 'Private not-for-profit'],
	level TEXT -- example: ['2-year', '4-year'],
	year INTEGER -- example: ['2012', '2011'],
	gender TEXT -- example: ['B', 'F'],
	race TEXT -- example: ['A', 'Ai'],
	cohort TEXT -- example: ['4y bach', '4y other'],
	grad_cohort TEXT -- example: ['0', '1'],
	grad_100 TEXT -- example: ['0', '2'],
	grad_150 TEXT -- example: ['0', '3'],
	grad_100_rate TEXT -- example: ['0', 'NULL'],
	grad_150_rate TEXT -- example: ['0', 'NULL'],
	grad_cohort_ct INTEGER -- example: ['9', '16'],
	CONSTRAINT fk_state_sector_grads_institution_details FOREIGN KEY(state) REFERENCES institution_details (state),
	CONSTRAINT fk_state_sector_grads_state_sector_details FOREIGN KEY(stateid) REFERENCES state_sector_details (stateid)
)

Question: State the average median SAT value for institutes in the state with the most male graduate cohort in 2013.
Evidence: median SAT value refers to med_sat_value; average = AVG(med_sat_value); male refers to gender = 'M'; graduate cohort refers to grad_cohort; most male graduate cohort refers to MAX(COUNT(grad_cohort WHERE gender = 'M')); in 2013 refers to year = 2013;

SQL:
SELECT AVG(T1.med_sat_value) FROM institution_details AS T1 INNER JOIN state_sector_grads AS T2 ON T1.state = T2.state WHERE T2.year = 2013 AND T2.gender = 'M' GROUP BY T2.grad_cohort ORDER BY COUNT(T2.grad_cohort) DESC LIMIT 1