1. The statistics behind the game
Baseball Findings
The statistics behind the game
Harlan Thompson
Sungjin Cho
Ryan Fagan

2. An Introduction
Throughout its long history, baseball has been the subject of many statistical studies. It lends itself well to statistics because very careful records are kept of everything that happens in every game.
The topics that have been studied range from the affect of interleague play on team standings to the role of chance in streaks and slumps
Other topics of study include records and predicting the outcomes of games.
We thought that looking at home runs and salary would be interesting because the great number of home runs hit and the inflation of salaries are both controversial topics.

3. Home Runs Per Year
-How has the total number of home runs in major league baseball changed from year to year?

4. Test #1
We ran a regression with the year as the independent variable and the number of home runs as the dependent variable to find out the rate at which the number of home runs in the league is increasing.

5. Scatterplot

6. Results Source | SS df MS Number of obs = 25
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
hr | Coef Std. Err t P>|t| [95% Conf. Interval]
year |
_cons |

7. Interpretation
The 95% confidence interval for the coefficient of year is totally positive - this shows that the number of home runs is definitely increasing each year.
An R2 value of clearly shows a positive relationship, although not a very strong one. This could be because many other factors can affect the number of home runs hit -- weather, injuries to certain players, etc.
The coefficient of year is , so each year about more home runs are hit in each game. This is over 4 more home runs per year.

8. Test #2
We split up the home run data into 2 separate groups and
Then we ran a hypothesis test on the two groups to find out if their variances are equal to determine whether or not we could use a paired t test on the data.
We used the following hypotheses:
H0 : var(HR (‘76 - ‘87)) = var(HR(‘88-’99))
HA : var(HR(‘76 - ‘87)) not= var(HR(‘88-’99))

9. Results
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
hr1 |
hr2 |
Comb. |
Ho: sd(hr1) = sd(hr2)
F(11,11) observed = F_obs =
F(11,11) lower tail = F_L = F_obs =
F(11,11) upper tail = F_U = 1/F_obs =
Critical values at .05 significance level: (.288, 3.47)
Because the F statistic does not lie outside of this region, we cannot reject the null hypothesis!!

10. Interpretation
The variance in home run hitting has not changed significantly over the past 25 years.
Therefore we can use these two sets of data in a paired t test to determine whether or not the number of home runs hit has increased.

11. Test #3
Because we found that the two groups did not have an appreciable difference in variance, we can use a paired t test to determine whether or not the number of home runs hit per year has risen from the period to the period
So we ran a hypothesis test on the two groups with the following hypotheses:
H0 : HR (‘76 - ‘87) = HR(‘88-’99)
HA : HR(‘76 - ‘87) not= HR(‘88-’99)

12. Results
Paired t test
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
hr1 |
hr2 |
diff |
Ho: mean(hr1 - hr2) = mean(diff) = 0
Ha: mean(diff) < Ha: mean(diff) ~= Ha: mean(diff) > 0
t = t = t =
P < t = P > |t| = P > t =

13. Interpretation
The mean for the years from 1976 to 1987 was HR/game vs HR/game from 1988 to 1999.
We can reject our null hypothesis because we found t = (much less than the critical value -1.96).
The the probability of Type I error is only
Therefore, the mean number of home runs per game from 1988 to 1999 was significantly greater than the mean number from ‘76 to ‘87.
So, the number of home runs per year does seem to be increasing over time.

14. Home Runs by Position
First we looked at last year’s home runs by position for each team.
The following is a sample of the data we accumulated...
Team SS HR 1B HR 2B HR 3B HR C HR LF HR CF HR RF HR TOT HR
Anaheim
NY Mets
San Fran
Next we calculated the total number of home runs and at bats as well as the average
number of home runs per at bat from each position for the whole league
(in order of performance)...
Position HR/AB HRs ABs
First Base
Left Field
Right Field
Center Field
Third Base
Catcher
Shortstop
Second Base

15. Do some positions hit significantly more than the average?
The league average of home runs per at bat is
For each position, we used binomial hypothesis tests to test whether or not the number of home runs per at bat from that position differs significantly from the mean.
For each position,
Ho : HR/AB = .0384
HA : HR/AB not= .0384
(Reject if |z| > 1.96)

16. Results SIGNIFICANTLY BETTER (reject null) ABOUT AVERAGE (accept null)
First Base: z = 7.978
Left Field: z = 5.731
Right Field: z = 5.104
ABOUT AVERAGE (accept null)
Center Field: z = 1.127
Third Base: z = 0.812
Catcher: z =
BELOW AVERAGE (reject null)
Shortstop: z =
Second Base: z =

17. Interpretation
So, we’ve proven that first basemen, left fielders and right fielders are significantly above the mean in home run hitting.
Shortstop and second basemen are significantly below the mean in home run hitting.
Center fielders, third basemen and catchers are about average.
This makes sense - the players at positions that require the most mobility (shortstop, second base) would obviously not be as powerful as those who play positions require less speed and agility.
It is interesting that center fielders are significantly different from the other outfielders - they do have to have a lot more flexibility and speed.

18. Does salary affect performance?
We looked at team salary vs. number of wins to see if the amount of money paid to the players has a significant affect on a team’s performance. Below is some of the data we used.

19. Wins vs. Payroll for 2000

20. Wins vs. Payroll for 1999

21. Wins vs. Payroll for 1998

22. Results for 2000

23. Results for 1999

24. Results for 1998

25. Interpretation
The R2 value for the year 2000 (.1952) did not reflect a significant correlation, however years 1998 (.5442) and 1999 (.4691) reflect a relationship between total payroll and number of wins
Because the coefficient of the number of wins is roughly 1 for all three years, we can conclude that an additional win costs about a million dollars.

26. Salary and home run hitting
Finally, we thought we’d combine these two studies of salary and home run hitting and analyze how the changes in average salary have been resulted in changes in the number of home runs hit per person. Exactly how many more home runs are we getting per $1?
We looked at data from 1969 to 2000.
We found average salary but we could not find average number of home runs/player. However we thought the leader in home run percentage might give some kind of portrayal of the number of home runs being hit.

27. Salary vs. Home Runs
Year Salary(thousands) HR Pct Leader Year Salary(thousands) HR Pct Leader

28. Regression

29. Results Source | SS df MS Number of obs = 32
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
hrpct | Coef. Std. Err t P>|t| [95% Conf. Interval]
sal |
_cons |

30. Interpretation
The coefficient of salary is and the entire confidence interval for this value is positive. So it seems that an increase in salary may produce an increase in home run hitting.
For every additional hundred thousand dollars in average salary, the leading home run hitter would hit home runs .2% more.
We found an R2 value of .4247, which is fairly significant. However, from 1969 to 1976, salary stayed fairly standard (compared to the inflation today), so this may have hurt our regression since home runs were increasing at the time, although not as rapidly as recently.
This suggests that home runs and salary may be increasing independently through time. There may not be an actual relationship between the two. Further study would be needed to determine if they are related.