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my @institutions = qw|hospital1 hospital2|;
my %seen;
...
} else {
    $institution_table{"other"} ++;
}

my $cat_age = int ($age/5) - 9;
my @age_cats = qw|fortyfive_fifty fifty_fiftyfive fiftyfive_sixty|;
...
foreach (keys %institution_table) {
 $institution_table{ $_ }{ $age_cats[$cat_age] }++;
}