Thanks to all the responses, I got this working having the image src call a separte cgi program.
<img src="test4.pl?sta=$station&const=$constituent" alt="$station $constituent" border="1">

#!/usr/bin/perl -w
# test4.pl for testing real-time graph capabilities
# November 9, 2004
use CGI qw( :standard );
use CGI::Carp 'fatalsToBrowser';
my $q = new CGI;
my $station = $q->param('sta');
my $constituent = $q->param('const');
my $combined = "$station" . "$constituent";
my $img = `/usr/local/bin/gnuplot $station $constituent $combined`;
print $q->header("image/png"), $img;
[download]

However, because I need to change the gnuplot set commands based on user input, I want to do this differently. Something like

#!/usr/bin/perl -w

# test5.pl for testing real-time graph capabilities
# November 17, 2004
use CGI qw( :standard );
use CGI::Carp 'fatalsToBrowser';    #will print error message to user
$CGI::POST_MAX = 512;   # limit maximum posting size to 512 bytes
$CGI::DISABLE_UPLOADS = 1; # disables uploading files entirely
my $q = new CGI;
my $data = "05056670.txt";
print $q->header("image/png\n\n");
my $img = plotdata();
sub plotdata {
  open my $graph => "| /usr/local/bin/gnuplot" or die "Failed to open 
+pipe: $!\n";
  print $graph <<"gnu";
  set terminal png color
  set output
  set xdata time
  set timefmt "%Y%m%d"
  set key left top title "Legend" box
  set grid xtics ytics
  set yrange [700:]
  set format x "%Y"
  set xlabel "Year"
  set ylabel "Sodium, water, filtered, milligrams per liter"
  set title "05056670 Western Stump Lake Major Ions"
  plot "$data" using 2:3 title "P00930 Sodium dissolved"
gnu
  close $graph or die "Failed to close pipe: $!\n";
}
[download]

When I try this, I get:
�PNG

IHDR�
�����sBIT�Lˢ0PLTE�����U�U��UUU��U�UUU�UUU������������I�N�.tEXtSoftwaregnuplot Unix version 3.7 patchlevel 1C����IDATx��Kң���`�3��&5��e�'5�e�HJ@ �!����tt�� �?���i � � � ���C���ۗn�4��o�a���m��]�����$붛l�ϐ>�a~�o�f&����w0߇�N�����ܺ ����aٍۇ):M��}ݍ:��MQ�u���aJn�>u�F}z�YO/�$����$Gg�\5l��=o����Oe����-L�f�e�Ƨ�’�.��&�<�)��9�դ�1���mt6�v0Qf
N���.T��#F�}�!���YuS�ԏzc���n�@7�(�M�r;�

in the browser.

What am I doing wrong?

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