Math Geek Mom : A Mother Lion Does Statistics

When I entered graduate school, I once proudly proclaimed to someone I had just met that “I don’t want to be normal”. I have no idea what they must have thought of that statement, or of me, but it was basically true. I saw myself as changing the world, as saving the world from its economic messes with my little equations. Who wanted to settle for “normal” when they could go down in the history books? Of course, I have since become more realistic in my thoughts on the matter, but I remembered my statement when I recently spent some time researching “Attention Deficit/Hyperactivity Disorder” in books, magazines and on the internet.

When my daughter’s kindergarten teachers suggested she might have AD/HD, I found myself transformed into the “mother lion” seeking to defend her cub, and did so in the only way I knew how; by gathering information. Although her pediatrician eventually told us that she probably does not have the illness (at this time), and gave us a “diagnosis of a normal child for her age”, it was not until after I had devoured every bit of information I could find on the subject, from books, magazines, other mothers and from the internet.

Most of the information on the web dealing with AD/HD can be divided into three categories. First there are people thinking that AD/HD does not exist, that it is a “made-up” illness. Second, there are those sharing horror stories of either their children or of the medical mazes they found themselves thrust into. And finally, there are those who feel sorry for all the poor children who need to be diagnosed with AD/HD and have not yet been so diagnosed and properly medicated. Most of this information I found useless. But eventually I did stumble upon a posting that bears repeating. When I read it, after almost 20 years of teaching statistics, I asked myself why I had not thought of it first!

The posting was from someone named Dan, and appeared on March 18th of 2009 in an on-line chat section of an article from the Atlanta- Journal Constitution. He proposed that ADHD is definitely over diagnosed, and used statistics to explain his thoughts.

His analysis is based on the assumption that we can think of the temperaments of all children as forming a normal distribution. Imagine a number line describing their temperaments, with the average child in terms of hyperactivity and ability to focus in the center. Above this line is drawn a graph that describes the proportion of children falling in each section of the number line. This graph will look like the classic “normal distribution”, often called a “bell curve”. Close to the center, where the average, or the mean is found, will be found most of the children and the largest fraction of the area under the curve, with the proportion of children and area under the curve found in each region falling off as we move away from the average value. This leads to a graph that extends out to infinity in each direction as the area under it approaches zero.

In statistics, we usually think of about 2/3 of observations as falling near the center of this graph, and think of the central 95% of observations as being “normal”, with any “abnormality” occurring in the outer 5%, with 2.5% on either tail. This means that anyone who is not in the upper 2.5% of people in terms of “hyperactivity” is actually “normal”, while anyone who is not in the lower 2.5% in terms of “ability to concentrate” is also “normal”. To be an “outlier” in a statistical sense, an observation must lie in one of these tails. Different disciplines used different values for the area in the tail, often using values even smaller than 2.5% to define outliers.

But between 5 and 8 or more percent of children in the U.S. are diagnosed with ADHD, a disease that is diagnosed by using questionnaires that decide if a child’s behavior is out of the ordinary, and enough out of the ordinary to make them an “outlier”. Remember, statistically, we usually define only 2.5% of observations as being outliers. So, since there are more than 2.5% of children being diagnosed with AD/HD, statistically there ARE too many children being diagnosed, which is what Dan’s point was.

While this analysis is based on a normal distribution, a chi-square test, that does not require the assumption of a normal distribution, leads to the same conclusion. Comparing the expected number of children diagnosed with AD/HD (again, assuming we only want to label the upper 2.5%) and the actual number diagnosed yields results that would not be expected to occur randomly. Further, the percentage of children with the illness is not uniform across the states. Indeed, in one chat room I found a large percentage of parents chatting who were from Ohio. When someone pointed this out, someone else suggested it was due to the required standardized tests that are the norm in Ohio. I was intrigued by this discussion, sine we, too, live in Ohio.

Some doctors actually think that AD/HD is under diagnosed, in that many of the people who actually have it are not being found and treated. And, certainly, I am not proposing that people who are truly ill be denied medical care, or that medicine not be given when it is needed. After all, I have lived seizure free for more than seventeen years thanks to the miracle of modern medications. However, as someone who likes to play with statistics, I have to question how we can come to a result where more than 2.5 percent of observations are classified as “outliers”. And, as a “mother lion”, I feel I must do everything in my power to protect my child, as she is the way that I will truly change this world.