Math CentralQuandaries & Queries


What is the percentage difference between 64.24% and 74.24%?


The short answer is 74.24% is 10 percentage points larger than 64.24%.

The more insightful answer is that it depends on which perspective you take.

Let us say that Paulo has 64.24% on his first math test and 74.24% on his second test.

From the perspective of the first test, his second score is a (74.24 - 64.24) / 64.24 = 15.6% improvement.

From the perspective of the second test, his first score was (74.24 - 64.24) / 74.24 = 13.5% worse.

From the perspective of most teachers, students and parents, his mark was 10 percentage points higher. This is solely because the two numbers expressed are percentages (74.24% - 64.24%).

It is perhaps even easier to see what is going on with another example. Let's say two mp3 players out of every 100 you manufacture are defective. That's a 2% defect rate. Now if you improve the reliability so that only one out of every 100 is defective, you have a 1% defect rate. What is the percentage difference? The rate of failure has been reduced by 50%. The old rate of failure was 100% higher than the new rate of failure. But the difference in failure rate is 1 percentage point.

As you can see, one must be wary with "percentage differences" because they are so ambiguous. If I raise the price on a pair of socks by 25% and then drop it by 20% (of the new price) later on, my final price is the same as the original price. Crafty marketing people use this ambiguity, with subtle wording, to give the impression that sale prices are better than they really are.

Stephen La Rocque.>

I want to add my 2 cents worth to Sue's response. It can be extremely misleading to compare percentages. Here is a hypothetical example.

I a recent salary settlement the clerical staff got a 15% increase and the senior management got a 10% increase. You can calculate the "percentage difference" one way and get (15 - 10)/10 = 50% or the other way and get (15 - 10)/15 = 33.3%. You can even say that the increase for the clerical staff is 5 percentage points larger than the increase for the senior management. Each of these comparisons however ignores the fact that the bases are different. The clerical staff have an average salary of $40,00 per year so their average increase is 15% of $40,000 which is $6,000 per year while the senior executive's average salary is $150,000 per year and hence their average increase is 10% of $150,000 which is $15,000 per year.

I hope this helps,

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