foreach (@SQL_queries) {

    my $Results = $_;
    
    my $sth_atl = $dbh->prepare($Results) or die "Couldn't prepare que
+ry: ".$dbh->errstr;
    $sth_atl->execute() or die "Couldn't execute query: ".$sth_atl->er
+rstr;
    
    my $cols_for_row;
    
     


    while ($cols_for_row = $sth_atl->fetchrow_arrayref) {

      print OUTFILE "England\t";
      print OUTFILE join("\t",@$cols_for_row),"\n";

      my $last_element = @$cols_for_row[-1];

      my $indicator =  @$cols_for_row[-2];


      my $numerator;
      my $denominator;
      

      if ($indicator eq "SH"){
    $numerator = $last_element;
      }
      elsif ($indicator eq "SL"){
    $denominator = $last_element;
      }
      else {
    print "\nError: There has been an unexpected string within the sec
+ond to last element of the array\n";
      }


    print "\n>>>>>numerator>>>>>>>>>>>> $numerator\n";  #  Works here
    print "\n>>>>>>denominator>>>>>>>>>>> $denominator\n"; # Works her
+e


      if (($numerator > 0) && ($denominator > 0)){
    
    print "\n>>>>>numerator>>>>>>>>>>>> $numerator\n";  #  Does not wo
+rk here
    print "\n>>>>>>denominator>>>>>>>>>>> $denominator\n"; # Does not 
+work here

    my $expected_ratio_figure = ($numerator)/($denominator);
    print OUTFILE "\n\nThe expected ratio figure:\t";
    print OUTFILE "$expected_ratio_figure\n\n";
      }
    }
  }
[download]

      my $numerator;
      my $denominator;
      if ($indicator eq "SH")   { $numerator   = $last_element; }
      elsif ($indicator eq "SL"){ $denominator = $last_element; }
      else { ... }

      if (($numerator > 0) && ($denominator > 0)){ ...
[download]

The first time you execute the loop, $numerator or $denominator may get initialized, but never both. Therefore you get the error message Use of uninitialized value in numeric gt (>) at New_program.pl line 535.

I think you're right, but that would not explain his comments works here/ does not work here, the error would be the same at both points

      my $numerator;
      my $denominator;
[download]

Here both $numerator and $denominator contain undef.

         if ($indicator eq "SH"){
    $numerator = $last_element;
      }
[download]

If that branch is true, $numerator will contain the value of $last_element, which may be undef. $denominator will still contain undef.

      elsif ($indicator eq "SL"){
    $denominator = $last_element;
      }
[download]

If that branch is true (and it will only be true if the first branch is false, in which case $numerator will still contain undef), $denominator will contain the value of $last_element, which may be undef.

      else {
    print "\nError: There has been an unexpected string within the sec
+ond to last element of the array\n";
      }
[download]

If this branch is true, both $numerator and $denominator contain undef.

At this point, at least one of those two variables does contain undef. Both may.

        my $numerator;
        my $denominator;
      

        if ($indicator eq "SH"){
            $numerator = $last_element;
        } elsif ($indicator eq "SL"){
            $denominator = $last_element;
        } else {
            print "\nError: There has been an unexpected string within
+ the second to last element of the array\n";
        }
[download]

You only ever initialise $numerator or $denominator on the first run through. Something like

my $numerator = 0;
my $denominator = 0;
[download]

might work, if 0 is a suitable default value for you.

No, it's initialized to undef in every iteration of the loop.

As requested More code here:

No, that is not what was requested. There is no way the code you posted could create the error message you get, at the point you say you get it. Its impossible.