How to set the default for checkboxes element
Simplistically, the issue is that the default value uses the array key, not the array value.  So, there are two ways you can change this to make it work.

One:

$form['average'] = array(
  '#type' =&gt; 'checkboxes',
  '#options' =&gt; array('a' =&gt; 'a','b' =&gt; 'b', 'c' =&gt; 'c'),
  '#default_value' =&gt; array('a'),
  '#title' =&gt; t('The average vote is more than'),
);


or

$form['average'] = array(
  '#type' =&gt; 'checkboxes',
  '#options' =&gt; array('a','b','c'),
  '#default_value' =&gt; array(0),
  '#title' =&gt; t('The average vote is more than'),
);