## If you want an average answer…

One of my early mentors in the area of analytics and management information systems always referred to an old adage:

If you want to get average answers to your questions, you should always use the AVERAGE.

The truth underlying this quaint aphorism is that whenever we use any aggregate measurement to represent data, we (necessarily and by definition) give up and obscure some of the details. The hope, of course, is to exchange detail for simplicity so that it’s easier to understand or extract meaning from piles of data. On centerlinescores.com, we show the details of dressage tests, but we also try to summarize the score data to allow an “at-a-glance” view of performance at different levels, for different tests and even for different horses ridden, or riders competing.

Shortly after centerlinescores.com launched (and still today several years later) we received emails challenging and questioning the usage of Median scores vs. Average (or “mean”) scores. After the most recent query, I decided it was time to put together a quick blog-post to not only clarify the choice but to explain the differences between a Median and an Average.

## Measures of Central Tendency

Both the Median and the Average (which is what most people call the “Arithmetic Mean”) are measures of “central tendency” for a set of numbers. For data that is perfectly “Normally” distributed (i.e. the popular “bell curve”), there is no difference between the Median and the Average. However, most data does not fall into a perfect “bell curve” or “normal” shape, and so, the need for different measures.

### Average (or Arithmetic Mean)

The “average” (also known to Math geeks and Statisticians as the “arithmetic mean”) is the most commonly used measure of central tendency and is calculated by taking the SUM of all of the numbers in a set and dividing by the number of items in the set.

So, for the set of scores shown here (pulled anonymously from a recent email inquiry) the “Average” score is equal to:

Average Score = Sum of all Scores divided by the Number of Scores Average Score = 546.9 / 8 Average Score = 68.35%

### Median

The “median” is another measure of central tendency. However it is calculated differently than the average. The Median for a set of numbers is simply the “middle value” when you arrange the numbers in ascending (or descending) order. If there is an odd-number of items, you simply take the middle value, if there is an even number of items, you take the average of the middle two values. So, again, for the scoreset listed;

Median Score = Middle Score for odd-numbered (or average of the two middle scores) for even-numbered Median Score = [67.1 + 67.2] / 2 Median Score = 67.15%

Here you can see the relationship between the Scores, the Average and the Median. In this case, there is a 1.2% point difference between the Median and the Average. The median is often referred to as a “resistant” measure of central tendency because it is less influenced by outliers. Here I will quote another blogger to explain why this is important for a Dressage rider:

…it is because an outlier score (an overly high or overly low number) would tip the mean up or down which could create a number that doesn’t really reflect a typical average.

Karen Sweaney ~ Not So Speedy Dressage

The median score is less likely to be subject to the dramatic swings that an average score is when abnormally high or abnormally low scores are added to the set. For example, given the same data set, if this rider were to have their worst day ever (they overslept, forgot to feed that morning, only warmed-up for 10 minutes before going into the ring, the judge wore a hat that blew off her head in the middle of the extension, the ring steward lost control of the blue tent and it went flying and finally, they just plain got confused during the test) and scored a 55%.

We see in the graph that the new score of 55% – when added into the set of scores – barely affects the median at all (moving it by less than 0.1% points). The average, on the other hand – because it is not resistant to the influence of outliers – drops down by almost 1.5 points from 68.4% to 66.9%.

Because we are more interested in an accurate overall summary picture of the trends in the scores, we use the median in almost all of our summary statistics. It provides a more robust measurement of a rider or horse’s “typical” performance and is less influenced by abnormally high or low scores.

CenterlineScores.com is the most complete and most accurate source for dressage scoring and show results data for United States Equestrian Federation recognized Dressage Shows. The site has over 1.8 million dressage scores for shows from 1993 on and serves as the premiere destination for American Dressage enthusiasts interested in researching scores and show results for Horses, Riders and Trainers.