# What is it used for?

The Noll-Scully metric is used for measuring competitive balance in sports. As James Quirk and Rodney Fort put it in Pay Dirt — this metric (developed by Roger Noll and Gerald Scully) “compares the actual performance of a league to the performance that would have occurred if the league had the maximum degree of competitive balance (in the sense that all teams were equal in playing strength)”.

# How does it work?

The Noll-Scully metric works by looking at the following aspects of a sports league:

• The average number of wins per team.
• The number of teams in the league
• The number of games each team plays

Based on these three things we can determine what an ideal league would look like. Important Side Note: An ideal league is not one where every team wins the same number of games! We’ll jump more to that in a second.

# How to measure it:

The Noll-Scully compares what a league actually looks like to what it would ideally look like. In a perfect world — i.e. one where actual balance matches the ideal — the metric would return a 1.0, meaning the league is balanced. If the number is above 1.0 it means the teams are further away in wins than we would expect given the ideal. If the number is below 1.0 — and we should stress this is hardly ever the case — it means the teams are closer in wins than we would expect.

# That annoying in depth math part:

If you’d like the full steps for Excel you can find them here!  We begin with the standard deviation of wins.  In statistics the standard deviation essentially tells us how spread out our data is. We’ll first figure how spread out wins should be based on the league size and number of games and then compare that to reality. It’s not so bad I promise!

First calculate the Idealized Standard Deviation.

Ok here’s where we figure out what things should look like. First we’ll take the average number of wins each team has. For most leagues this is pretty simple, just take the number of games each team plays and divide by 2. For hockey or soccer you have to take the average number of points, which is a little more work but still pretty easy.

Alright so we have our average number of wins/points, all we do now is divide it by the square root of  the number of games each team plays and we’re rocking. More “formally”:

`Idealized Std. Deviation = [Average wins]/[Schedule Length]^0.5`

Let’s give some concrete numbers

2010-2011 Idealized Standard Deviation for major U.S. sports
League Measurement Idealized Standard Deviation
NFL Wins 2.00
NBA Wins 4.53
MLB Wins 6.36
NHL Points 10.15

In case you’ve forgotten Standard Deviations here’s a quick way to use them. If you take your standard deviation and subtract it from your average, and then also take your standard deviation and add it to your average, you get the range most teams should fall into. So here’s how our league should look

• NFL: Most teams should be between 6 and 10 wins.
• NBA: Most teams should be between 36.5 and 45.4 wins
• MLB: Most teams should be between 74.6 and 87.4 wins
• NHL: Most teams should be between 81.8 and 102.0 points

Now on to reality! Now you need to calculate the actual standard deviation for the league you’re looking at. I’ll give a quick cheat. If you have Excel or use the spreadsheet in Google Docs you can put the wins or points in a column and then just do

`=stdev(A1:A30)`

if your numbers were in first 30 rows of column A and you’ll get the standard deviation. If not I’ll let you run over this quick tutorial for figuring out standard deviations. Take your time, I’ll be here when you get back.

Alright so let’s use the real world again here’s how our leagues actually looked in 2010-2011

Actual Standard Deviations for U.S. major sports leagues in 2010-2011
League Measurement Standard Deviation Idealized Standard Deviation
NFL Wins 3.00 2.00
MLB Wins 11.42 6.36
NBA Wins 13.18 4.53
NHL Points 13.27 10.15

The final step is the easiest. Just take your real standard deviation and divide it by the Idealized Standard Deviation. The closer it is to 1.0 the more competitively balanced your league is.

As we can see, hockey and football are fairly balanced.

Noll-Scully for U.S. major sports leagues in 2010-2011
League Noll-Scully
NHL  1.31
NFL 1.50
MLB 1.80
NBA 2.91

Baseball starts to have some lopsidedness but is still not terribly unbalanced (and it is much more balanced than it was in the early part of the 20th century). Basketball however is wildly unbalanced, which is of course a major subject in the book Wages of Wins and a popular subject on this blog.

Hope the tutorial helps and you understand why the Noll-Scully is used in explaining competitive balance and how to interpret it.