Is it possible to define variables in a wordpress shortcode, and then call the shortcode using a specific variable?

I think you are looking for this-

function star_rating( $atts ) {
$max = 5;
$atts = shortcode_atts(
    array(
        'grade' => $max, // default
    ), $atts );

$star_icons="";

$full_star = floor( $atts['grade'] );
for( $i=1; $i<=$full_star; $i++ ) {
    $star_icons .= '<i class="fa fa-star"></i>';
}

$half_star = 0;
if( (int) $atts['grade'] != $atts['grade'] ){
    $half_star = 1;
    $star_icons .= '<i class="fa fa-star-half-o"></i>';
}

//considering you want to show empty stars as well
$empty_star = $max - $half_star - $full_star;
for( $i=1; $i<=$empty_star; $i++ ) {
    $star_icons .= '<i class="fa fa-star-o"></i>';
}

return $star_icons;

}
add_shortcode( 'star_rating', 'star_rating' );

Note:

1. all fraction numbers are considered as 1 half-star. I mean, 2.5 and 2.6 will produce same output.
2. these type of stuffs should be handled with CSS.
3. Tested and working.