Calculating improvement and Decline Percentages

Comparing a patient’s decline in a test score from a year ago to today is often done in percentages. Likewise, improvements are described in percentages – although this is less common over time of patients with progressive cognitive disorders. While the arithmetic is simple, the logic can be confusing, so this section attempts to explain how to set up the problem so as to get the right answer.

Why This Can be Confusing
Let’s say a patient’s WLM Delayed Recall score was 5 last year, while this year his score is 3. If you “eyeball” it, think 1 point is 20% of 5. Thus, 2 points is 40%, 3 points is 60%, 4 points is 80% and (of course) 5 points is 100% of 5. So yes, you are correct, that two points is a 40% reduction from the original 5 points he earned before. If you ask what is the percentage loss when one goes from a score of five to three – it is 40%.

Here is the key question: What is the percentage drop of two points compared to five? It is 40% because each of the five points leading up to five is 20% of five. The formula to figure the percentage is what is 2 of 5, or two over five, or 2/5 = .4 and to get a percentage you always multiple that by 100 and you get 40%. Note you are dividing the difference in scores (2) by the higher score of five – to get the loss compared to five.

The confusing part (which is why this math is hard for many people) comes from the alternative question – What is the percentage change if a score went up from three to five? Now the key question becomes: What is the percentage gain of two points compared to three? Intuitively you want to say is is the same, 40% – but it is not. You need to use a different formula and divide by the difference (2) by the lower score of three – to get the gain compared to three.

In brief, a gain between two scores (say 3 and 5) compares the difference to the lower score; 2 up from 3. A loss compares the difference to the higher score; 2 down from 5. It makes no sense to compare a loss of 2 points to the lower score; instead, you want to compare it to the higher score.

Ask the Right Question
The key is asking the right question, to start the math. First recognize you might have two scores (3 and 5) from testing a year ago and from today. We are interested in the difference between those two scores which is 2 points. Lets say a patient improves from 3 to 5. To see what the percentage improvement is, you’ll compare the two point improvement to the lower score of three, meaning you divide the difference by three. If s/he declined from 5 to 3, you’ll compare the two point decline (or difference) to five, meaning you divide the difference by five. It becomes clearer if you first ask the correct question for these two examples. Note the difference is always the same and is used in both calculations.

If we ask: What is a loss of two points compared to five this is like asking what is the loss of two points over five earned this year? Note each point of five is worth 20% of five, so two points means a 40% loss.

Or, you might ask: What is a gain of two points compared to three we asking what is the gain of two points over three earned last year? Note each point of three is a third (33%) of three, so two points means a 66% improvement.

In a nutshell, a gain or loss between two scores will always be the difference between them. A gain will mean you are comparing that difference to the lower score, while a loss will means you are comparing the difference to the higher score. A difference of two points between scores of 3 and 5 means a gain of two points compared to 3 if the score when up -or- it can mean a loss of two points compared to 5 if the score went down. This is how you figure out what to divide into the difference. Replace the word “compare” with “over” you get your math formula.

The trick is to recognize what you are comparing the gain or loss to, as that score becomes the divisor, or what is divided into the difference. Note in our example of two scores of 5 and 3 (whether a gain or loss) the difference is two points and that difference (2) will always be divided by something in the formula. What you divide into the difference changes, based on what you want to know:

1. By what %age did he improve if he gained two points compared to three?

– or –

2. By what %age did he decline if he lost two points compared to five?

Once you have the right question asked, replace the word “compared” with over, and that is your math formula for your calculator. (Don’t forget to multiply by 100 to get a %age.)

For question #1. (2 over 3 is 2/3 = .6) x 100 = 66% improvement
For question #2. (2 over 5 is 2/5 = .4) x 100 = 40% decline

I suggest you print out the above two equations and replace the original and current scores in the correct question (#1 or #2) above. Then the math calculation will become clear, I hope.