I recently attended a conference where one of the speakers apologized for sharing a finding with percent. I will admit that apology surprised me.
Should we apologize for the product generated by multiplication? Do we say “sorry” when we receive the quotient from a division problem?
It seems odd to express regret for math. But here we had a professional doing just that.
Percent simply refers to a notation for hundredths. A percentage applies to the number obtained by finding the percent of another number.
Example 1: I did a random sample and counted 2 people out of 100 have red hair. Therefore, 2% of the people in my sample have red hair.
Example 2: First grader Sarah spelled 8 words correctly out of 10. Sarah spelled 80% correct words on her spelling test.
Example 3. For the “Biggest loser contest” at work, Mike weighed 220 pounds and lost 20 pounds (final weight 200). Amy weighed 115 pounds and lost 15 pounds (final weight 100 pounds). Mike had a 9% reduction in total weight. However, Amy had a 13% weight reduction and won the contest.
The percentages in the examples above give useful information. No reason at all to apologize for using them. Then why all the fuss? Wait, did I just use “fuss?” (Rick’s mental note, update references and never use “fuss “again, it sounds like I live in the 50s).
Saying "2% of a sample has red hair" quickly communicates a relationship. Namely, 2% tells us we have a small number of people with red hair.
Likewise, Sarah spelling 80% of her words correctly also offers useful information. The percent value quickly and simply conveys size or scale.
A further advantage occurs when examining the Biggest Loser Contest (example 3). If the contest just looked at the absolute amount of weight lost, Mike would have won. But Mike weighed more to start with. Amy losing 13% percent of her weight compared to Mike’s 9% means when comparing relative weight loss, Amy did better.
As we have seen, no reason to become upset at the percentages. They do what they do. But at times, percentages can pose problems in certain situations.
The Problem with Percentages for Time-Series Data
In Education and Psychology, many practitioners measure behavior over time (produces time-series data). Time-series analyses involve carefully measuring behavior over a period of time.
Example 1: First grader Sarah spelled 8 questions words correctly out of 10 (80%) on Monday. Tuesday through Friday Sarah had the following percentages: 80%, 70%, 90%, 90%.
Example 2: Fred and Jill want to stop smoking. Fred recorded the following percentage decreases of cigs smoked using a nicotine patch: Monday 10% , Tuesday 12%, Wednesday 13%.
Jill used the cold turkey method (which involved lots of encouragement from her friends). Her reduction for Monday, Tuesday, and Wednesday respectively: 6%, 6%, 8%.
With time-series data, professionals (e.g., teachers, school psychologists, behavior analysts, psychologists) need the most precise information they can bring to bear to understand the effects of intervention on behavior.
Adding information to Sarah’s spelling performance demonstrates a problem. Look at her data in time:
Monday: 8 correct, 2 incorrect in 40 seconds
Tuesday: 8 correct, 2 incorrect in 42 seconds
Wednesday: 7 correct, 3 incorrect in 39 seconds
Thursday: 9 correct, 1 incorrect in 55 seconds
Friday: 9 correct, 1 incorrect in 56 seconds
While Sarah improved her words spelled correctly, she did so at the expense of time. It took Sarah much longer to spell more words correctly. Percentage completely ignores the time element.
If time matters, and it should to everyone serious about behavior change, ignoring how long it takes to perform a behavior will lead to less effective interpretation and subsequent decisions.
What about Fred and Jill? Again we have a problem. Percentage only tells us the size or scale of each measure. We don’t know the difference between how many cigarettes Fred and Jill smoked. If Fred had a pack a week problem whereas Jill smoked a pack a day, Jill’s percentage decrease could dwarf Fred’s in terms of the magnitude of cigarettes not smoked (absolute amount of change). Therefore, how can we know if the nicotine patch or the cold turkey method work better without more precise numbers?
Percentage has its place in the world of math and can help people understand some phenomena with a number. Yet in other situations, such as an intensive analysis of behavior with time-series data, percentage hides important features of behavior change. Understanding when to use percentage and when not to will facilitate a more productive analysis of data. As the famous Psychologist Alfred Adler said, “Mathematics is pure language - the language of science.” Let’s make sure we always speak as clearly as possible!
For much more detail — and a continuing education credit, to boot — check out Dr. Kubina's webinar, The Truth About Percent Correct. He explains the limitations of percent corrrect and the benefits of dimensional features of behavior. Enrollment is free.