In this International Year of Statistics, the subject of statistics, probability, and math will receive increased press attention. It’s a wonderful opportunity to celebrate the field of statistics and raise awareness of the role statistics plays in shaping the world around us. Unfortunately some news reports will miss the point, becoming lost in translation between spoken language and the beauty inherent in the precise language of mathematics.

English, or for that matter any language, is inherently ambiguous. The language of mathematics is more precise. For example, in plain English, a straight line isn't a curve, and a curve isn't a straight line. But in the language of mathematics, a straight line is the simplest example of a curve – one without curvature.

The language of mathematics has its own vocabulary and requires study, the same as learning any language. To quote Albert Einstein, “Pure mathematics is, in its way, the poetry of logical ideas.” In this post, I'd like to share a few of my favorite linguistic mathematical conundrums.

The nothingness of zero

Some have questioned if zero is a number.  Stupid notion?  It’s a matter of perspective.  In the language of mathematics zero is now classified as a number.  Zero is a natural number (counting numbers) and the smallest of the Cardinal numbers (set of numbers), but it hasn’t always been that way.  The ancient Greeks and Romans wondered how nothing could be something.  To accommodate this nothingness, ancient cultures used a variety of symbols and blank spaces to represent what we now call zero.

Chance, probability, and odds

In conversation the terms chance, probability, and odds are often used interchangeably.  But there are subtle differences.  Chance is a word used in everyday language to express an event taking place.  For example, there is a good chance when I turn the key in my Maserati (I can dream can’t I) the engine will start.  By contrast, probability which is a special branch of mathematics provides a precise measurement of the likelihood something will happen.

During the International Year of Statistics, there is the opportunity to shed some light on the relationship between mathematics and statistics.  While probability is considered a branch of mathematics, statistics is not.  Statistics of course uses math, but it is a discipline in its own right.  An excellent post on the JMP blog by Anne Milley, Statistics: The Language of Science, covers this in more detail.

Probability is a measure or estimate that something will happen and is expressed as a number between 0 and 1.  The values represent 0% chance that an event will happen, to the extreme of 100% certainty that an event will occur.  Consider a case where there are a given number (n) of equally likely outcomes, the probability of a particular outcome is 1/n.  For example, if you roll a die with six sides, the probability of rolling a “two” is 1/6.  In the case of rolling dice, the probabilities are known.  In other situations they must be estimated based on available data.  Predicting the results of an election outcome is a case in point.

Odds are yet a little different.  Sometimes odds are stated in favor of an event happening.  But sometimes they are expressed as odds against.  Going back to the example of rolling dice, if the probability of an event is p, then the odds on that event happening are p/(1-p).  For example, if the probability of rolling a “two” is 1/6, then the odds on that event are (1/6)/(5/6) = 1/5 or in words, “1 to 5”.  If the probability of an event is 1/2, then the odds are said to be “even.”

Making this all the more confusing is when considering true odds versus odds often cited for any given situation.  A simple example is a horse race and the odds posted by the track.  The true odds represent the actual chances of where a given horse will finish in a race.  However the odds posted by the track represent something different – how much the track will pay out for a given bet.  The track needs to make money and the horse’s owners must be paid.  So in many cases when the word odds is used, it’s really a subjective estimate rather than a precise mathematical computation.

Yogiisms and percentages

Percentage is another term often misused.  Yogi Berra famously said, “Baseball is ninety percent mental and the other half is physical.”  From Yogi it’s funny.  But the press too often uses percentages incorrectly when trying to make a point.  Chuck Pirrello wrote an excellent post for the JMP blog on How Percentages Can Exaggerate.  In his post, Chuck examines the difference between market growth and market share.  News editors and copywriters always push the limits with attention-grabbing headlines.  But when the headlines are misleading they do a disservice to their readers.

Having mentioned Yogi Berra and baseball, this post wouldn’t be complete without mention of another mangled baseball term, that of average.  Most elementary school students can work through a math problem and calculate average.  But as Chuck Pirrello points out, a deeper understanding of the data is required to gain true insight.  Of all sports, baseball is the most mathematically scrutinized.  And of all baseball stats, that of batting average is the most well-known.  (Hits divided by at-bats.)  For a hundred years batting average was the most important of baseball statistics.  The movie Moneball drove home the weak correlation between batting average, an individual player or that of a team, and how it relates to runs scored and wins.  Here’s a short video about how the Oakland A’s turned the baseball world upside down by delving deeper into the data.

As we celebrate 2013 The International Year of Statistics let us also celebrate the beauty and precision inherent in the special linguistic expression of mathematics.