Asbaty, Phillips, and Zhong | 2

STAT 512 Final Project: Quarterback Performance Based on AgeBy: Jeffrey PhillipsBrian AsbatyArnold Zhong

Quarterback is the most important position in all of sports. For NFL teams, the one player under center can change an entire franchise for over a decade. Years of great successes (and failures) are defined by quarterbacks for every team. This is why having an elite quarterback is vitally important for an entire teams success. Teams often select quarterbacks very high in each years draft, in the hopes that they can instantly turn their failures into successes. One would expect instinctively, however, that the veteran quarterbacks in the NFL would have more success. Our project examines the difference between the success of quarterbacks over and under the age of 30, an age commonly perceived as over the hill by analysts and fans. We are analyzing the ability to perform as a quarterback by examining several statistics from quarterbacks who played at least 3 years (minimum 672 pass attempts) both over and under the age of 30 (30 was included in over). This analysis will help us find out if teams are better off finding a quarterback through the draft at a young age and breeding him into an elite quarterback, or if they should search for one in free agency in the hopes of an instant turn around. This can also help teams with young quarterbacks decide if they want to build their team in a win now mode with a young quarterback, or try to get mostly young players at once and coach them to grow together in the hopes of building a great team in the future.To gain all the necessary statistics for this analysis we used several common quarterback statistics. These were completion percentage, win percentage, touchdown/ interception ratio, passer rating, yards per attempt, and ANY (Adjusted Net Yards). We indexed completion percentage, touchdown percentage, interception percentage, yards per attempt, passer rating, and ANY. These indices are all standardized variables comparing a quarterbacks performance on a normal approximation of all quarterbacks performances in the same year where a score of 100 is league-average and each standard deviation is worth 15 points. All of our variables are discrete.We assumed every statistics distribution was normal except for completion percentage and win percentage, which we estimate to be binomial distributions. These two we estimate to be binomial because there are only two possible outcomes for each scenario (complete/ incomplete; win/ loss). For win percentage, we assume an unbiased theta of .5 because every game ends with one winner or one loser. The indices should all be normal with a standard deviation of 15. Our study was done using statistics from Pro Football Reference. For our study, we used the 85 quarterbacks that have thrown 672 pass attempts while younger and older than the age of 30 since the NFL-AFL merger of 1970 to control for individual performance outliers. We eliminated all players in the age groups under 672 pass attempts (3 years if averaging 224 pass attempts per season, which is 14 pass attempts per game, which is the passing level required to qualify for statistical accolades by the NFL) to eliminate any players who did not play very long once they entered the league, which would have skewed our data with presumably lower statistics. For each quarterback, we calculated the difference in the statistics for the seasons they were 29 or younger then when they were 29 or older (a positive difference represents a decline in the statistic over time). By the properties of normality and the central limit theorem, all of the calculated distributions should have a mean equal to the difference of the general means for under 29 and over 30 quarterbacks and a variance equal to the sum of the two sets overall variances.For all of our statistics, we focused only on the sample means, which is are significant estimators for both normal and binomial distributions. We start with a definition of a sufficient statistic. We say T is a sufficient statistic if the statistician who knows the value of T can do just as good a job of estimating the unknown parameter as the statistician who knows the entire random sample. The mathematical definition is as follows. A statistic is a sufficient statistic if for each t, the conditional distribution of given T = t and does not depend on . For example,which is the sample mean, may in some cases contain all the relevant information about, and in that case T(X) is called asufficient statistic. That is, knowing theactual n observationsdoesn't contribute any more to the inference about, than just knowing theaverageof thenobservations. We can then base our inference abouton T(X), which can be considerably simpler thanX. To conclude, a statistic T=T(X) is said to be sufficient for a family of distributions if and only if the conditional distribution of X given the value ofTis the same for all members of the family (that is, doesn't depend on).GivenX1, X2,. , Xnis a random sample from a binomial distribution with parametersm and ,is a sufficient statistic for.We consider a normal population for which the mean is unknown, but the variance 2 is known. As shown with the proof done in class, using the factorization theorem on the joint density of a normal distribution: is a sufficient statistic and the same sample mean n is sufficient for this distribution as well.Complete tables and graphical representations of our data can be found attached in appendices 1-5. We found the data to be:

After standardizing each statistic by dividing by their respective overall means, we found the data to be (square is for when the quarterbacks were 29 and younger, triangle is for over 30):

You can see for the first four rate statistics, quarterbacks produced decently better when they were 30 or older than when they 29 or younger, albeit with very high sample variances. We believe this is a result of an interesting trend in the NFL rather than the quarterbacks themselves; numerous studies have shown that passing offenses have steadily increased over time, giving the older quarterbacks on average an extra 6 years of time-based offense padded into their stats. After adjusting for time with the indexes all of the under 29 statistics except for completion percentage show to be slightly better than the over 30 rates, as well as time-independent winning percentage. While completion percentage is an important indicator of success, the pairing with a decrease of all other rate indices (including an increase in interception percentage) hints to a more conservative approach bore out of a lesser ability to make the harder, further passes. Overall, our data shows that quarterbacks perform in the aggregate better when they are 29 or younger than when they are over the age of 30, with better touchdown, interception, rating and yards gained metrics per pass attempt at a young age. With the NFL draft coming up at the end of April, our data shows that teams with quarterbacks over the age of 30 should all consider investing in a young quarterback to take over soon, granted that they are able to find a skilled passer.