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Viewing the differential for variuos rounds allows the scores to be compared based on the difficulty of the course. If you shoot 84 on an easy course, how would that compare with shooting 99 on a difficult course? Using a differential comparison could show that the 99 was a much better score than the 84.
Handicap differential is a term applied to the difference between your score and the course rating, adjusted for slope rating. The number that results is used in the calculations that determine your handicap index.
Handicap differential is not a number you have to worry about unless you are one of those masochists who likes to do the math behind the handicap index.
But just for the record ... the equation to get handicap differential is this: (Score minus Course Rating) x 113 divided by Slope Rating.
Let's say you shot an 82 on a course with a USGA course rating of 72.5 and a slope rating of 128. The equation would look like this: (82 - 72.5) x 113 / 128. The sum would be the handicap differential. Further calculations take the handicap differential and turn it into handicap index.