Is It for Real (S3 ""Hardball"")

NUMB3RS - Season 3 - "Hardball" - Is It for Real

3 - "Hardball"

Topic: Data Analysis & Sabermetrics

Difficulty: Algebra 2 / Pre-Calculus / Statistics

In the NUMB3RS episode "Hardball," an amateur mathematician uncovers an equation that uses baseball statistics to identify when an athlete began using performance-enhancing substances. This moment of sudden, dramatic divergence in player data is called a 'change point'. While the math in the episode is dramatized, we can use a very real sabermetric statistic—the Cumulative Home Run Ratio ( $CHR$ )—to track historical players, visualize long-term trends, and identify when significant shifts occur.

To analyze a batter's home run productivity over time, we calculate their Cumulative Home Run Ratio (CHR). Unlike a simple single-season ratio, the $CHR$ at any given age is computed by dividing the total cumulative home runs ( $HR$ ) hit up to that age by the total cumulative at-bats ( $AB$ ) up to that age.

$\text{CHR}_{\text{age}} = \frac{\sum \text{HR}}{\sum \text{AB}}$

Let's look at Babe Ruth's early career stats:

Age 19: $10$ At-Bats ( $AB$ ), $0$ Home Runs ( $HR$ )
Age 20: $92$ At-Bats ( $AB$ ), $4$ Home Runs ( $HR$ )

At age 19, Babe Ruth's cumulative stats are simply his age 19 stats. Thus, his $CHR$ up to age 19 is:

$\text{CHR}_{19} = \frac{0}{10} = 0.00000$

NUMB3RS - Season 3 - "Hardball" - Is It for Real

Step 1: Understanding the Cumulative Home Run Ratio (CHR)

Cumulative Home Run Ratio (CHR) Trends