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Peak Performance: The Math of Sports

Sports have manifested in all cultures throughout recorded history.  Sports pit man vs. man, man vs. machine and man vs. himself. As sports games and activities unfold, a variety of opportunities emerge to apply math principles.  Very simple math can help us answer such questions as “how fast?”, “how often?” and “how many?”

Basic Concepts

  • Units of Measure
    • We measure distances in units like feet and meters and time in units like seconds and minutes. This information helps us understand scale and scope of a numerical value.
    • For example: I walked 50 meters or I waited for 30 minutes.
  • Average
    • Also called the arithmetic mean, an average is a simple statistic that gives a general idea of where a set of values are centered.
    • For example, the height of the average man in the US is 5’10”. Some people will be taller and some shorter but many adult males will be about 5′ 10″ in the US.

The World’s Fastest Man

Usain Bolt of Jamaica has demonstrated amazing athletic ability in the sprint events of track and field. One memorable event featured in the video is a world record 9.58 second performance in the 100 meter dash. You can watch the video  and get a sense of how his performance compares to other world class athletes, but it is hard to get a sense of how fast this actually is. Perhaps we could transform the units of meters/second to something more familiar like miles/hour.

To do that we will need to do a process called unit conversions. A few facts will help in our conversion calculation:

  • 60 seconds = 1 minute
  • 60 minutes = 1 hour
  • 3.2 feet = 1 meter
  • 1 mile = 5280 feet

Note that we can move the right side to the left or the left side to the right and the value of the fraction will be the same (e.g. a form of one). The calculation to convert the unit looks like so:


We can cancel out units of the same type. We show cancellation by putting a line through the unit. For clarity we made the color for each unit unique. Cancellation looks like so:


After cancellation the computation looks like:


Completing this calculation we can see that Usain bolt covered as much ground in 9.58 seconds as a car traveling at 23.3 miles/hour would cover. This means that in some jurisdictions, Mr. Bolt runs so fast he could get a speeding ticket in a school zone!


Going the Distance

A race for Usain Bolt is over before you can finish a commercial. How about distance runners? What type of speed can long distance athletes achieve? For this question we will consider the athletic feats of Bennisa Bekele of Ethiopia. Mr. Bekele is the world record holder in the 10,000 meter race with a time of 26:17.53.

We can do the same type of calculation we did for Mr. Bolt’s race to determine a speed in miles per hour. We’ll go a little faster since we have already covered the calculation in detail.

First, let’s express 26:17.53 in seconds. There are 1560 seconds in 26 minutes. That gives us 1577.53 seconds in 26:17.53


With this number we can see that a distance runner has about a 40% drop in speed but a 100x increase in distance. Even with this information, one may still not be able to appreciate just how fast this world record holder is going. A more common way to think of pacing in the time to complete one mile. For example a student in a exercise class might record a “mile time” that they would improve upon during a year. In terms of units this means we want time/distance, or specifically seconds per mile. This means we want the inverse of the value we calculated.

The calculation changes to:


This means the world record holder has nearly a 4 minute mile pace for the duration of the race. Bennisa Bekele can run a full mile in just about the time it takes to boil an egg.

With some simple math we can transform sports outcomes into familiar units of measurement.

The Sultan of Swat

The pace and flow of baseball allow ample opportunities to capture a wide variety of statistics. One of baseball’s most famous players was Babe Ruth. Dubbed the “Great Bambino” and “The Sultan of Swat”, Ruth had a storied career of ups and downs that included many championships and records that are still in place.


Of the many statistics gathered in baseball, a batting average is a common measure of how skilled a batter is at connecting with the ball. A batting average is calculated by dividing the number of hits by the the number of at bats or attempts by a player. The batting average represents the probability or chance that a hitter will connect when they face a pitcher.

Babe Ruth had 8,399 at bats over his 22 year career. Of those at bats, he had 2,873 hits. With these numbers we can calculate his batting average over his entire career as (2,873 hits)/(8,399 at bats). This means Ruth had a batting average of .342. Now if a student came home from school and presented a grade of 34.2%, they would surely be scolded by their parents. But in the game of baseball, where it is pitcher vs. batter and there are a myriad of choices playing out each moment, connecting 34.2% of the time is pretty remarkable. This can be approximated to say that Ruth got a hit every three times he made an attempt.

The Cyclone

Another great player in baseball history was Cy Young. Cy was short for “Cyclone, which was a nick name given to Young during his first try out as a professional baseball player. The speed of his fastball had damaged the grandstands to the point that it looked like a cyclone had occurred. The Cyclone name stuck and Cy Young went on to have a long career as a pitcher. Cy Young established many records, some of which still exist today. Cy Young’s name was given to an award that recognizes pitching excellence in Major League Baseball.


While hitters are measured by how many times they can connect with a pitch, pitchers are measured by how few runs they allow. This statistic is called an earned run average or ERA. Like a score in golf, a player aims to have a low ERA. A lower ERA means fewer runs are scored while a pitcher plays. The ERA as a statistic is calculated by taking the proportion of runs allowed to innings pitched multiplied by nine. This scaling to a full game of nine innings makes it possible to compare pitchers who may pitch a variable amount of innings over each game in a season.

Over the 22 years that Cy Young played, he pitched 7,356 innings. In those innings a total of 2,147 runs were earned. The value of 2,147/7,356 can be scaled by 9 to give a career ERA of 2.63. While there have been pitchers with lower career ERAs, no one can match the number of inning pitched by Young.

Using simple math we can get an idea of the long term performance of these legends and appreciate their accomplishments.

Counting Games

Counting is learned at such an early age that we may not even think of it when we think of “math”. But counting plays a key role in score keeping for a variety of sports. Games with variable scoring events introduce a element of strategy to optimize efforts for the best reward. While this can be an important or even interesting part of a game like basketball or football, score optimization can also have startling outcomes when it is applied to tournament play.

Many sports have tournament style play where advancement depends on the outcome of a number or games played within a small group. For example, the world cup has pool play that determines how the rest of the tournament unfolds. Tournament organizers must however be careful with how they structure tournament scoring. It is possible to create a scenario where the best long term outcome is to LOSE in the short term. This may sound confusing, isn’t the point of playing a game to WIN? How would losing offer any kind of advantage?

This very scenario played out in great prominence during the London 2012 Olympics in the sport of Badminton. Several team had already secured positions in the next round and winners would face higher ranked players sooner in the knockout rounds. Since teams wanted to be more likely to advance to a medal round, they attempted to lose on purpose to secure an easier match up. The full details are included in the references. The ethics of losing on purpose is sure to be a topic that will be debated for years to come.

Even a simple math principle like counting plays a key role in selecting a strategy for a given game.


Ultimately, success in sports is determined by time spent in the weight room and on the field playing. We can however use math to appreciate and understand sports in greater depth.



Categories: Math
  1. February 20, 2016 at 6:40 pm

    nice post!

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