Statistic Project

Scatterplot for all data (standardized) Single Variable statistic Home Runs per game (HR/G) Runs Batted In per game (RBI/G) Mean 0.110703 0.506843 Median 0.1 0.5 Correlation Coefficient (R) 0.7618 Coefficient of Determination (R 2 ) 0.5803 The line of best fit representing the data is y = 1.6189x + 0.3276. The slope (m) of the line of best fit is 1.6189, indicating the increase of 1.6189 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 0.3276. There is a strong, positive linear correlation since the R value is approximately 0.7618, which is between 0.67 and 1. y = 1.6189x + 0.3276 R² = 0.5803 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 RBI/G HR/G HR/G v.s RBI/G ( standardized)

Scatterplot for the 40s Single Variable statistic Home Runs per game (HR/G) Runs Batted In per game (RBI/G) Mean 0.072061 0.461848 Median 0.055944 0.462585 Correlation Coefficient (R) 0.7732 Coefficient of Determination (R 2 ) 0.5979 The line of best fit representing the data is y = 1.6928x + 0.3399. The slope (m) of the line of best fit is 1.6928, indicating the increase of 1.6928 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 0.3399. There is a strong, positive linear correlation since the R value is approximately 0.7732, which is between 0.67 and 1. Comparing to the previous decade, both mean and median for the runs batted in per game has dropped. The mean for the home run per game has increased slightly, but the median for the home run per game decreases. The correlation coefficient also decreased, showing a weakening linear correlation. y = 1.6928x + 0.3399 R² = 0.5979 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 RBI HR HR v.s RBI for 40s

median for the home run per game decreased slightly, as well as the mean and median for the runs batted in per game. However, the correlation coefficient has increased considerably, from 0.7951 to 0.8760, showing a strengthening linear correlation Scatterplot for the 70s Single Variable statistic Home Runs per game (HR/G) Runs Batted In per game (RBI/G) Mean 0.107814 0.480483 Median 0.106772 0.489508 Correlation Coefficient (R) 0.8679 Coefficient of Determination (R 2 ) 0.7532 The line of best fit representing the data is y = 1.9891x + 0.266. The slope (m) of the line of best fit is 1.9891, indicating the increase of 1.9891 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 0.266. There is a strong, positive linear correlation since the R value is approximately 0.8679, which is between 0.67 and 1. Comparing to the previous year, both the mean and median for the runs batted in per game and the median for home runs per game increased. However, the mean for the home runs per game dropped slightly. The correlation coefficient dropped. y = 1.9891x + 0.266 R² = 0.7532 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 RBI HR HR v.s RBI for 70s

Scatterplot for the 2000s Single Variable statistic Home Runs per game (HR/G) Runs Batted In per game (RBI/G) Mean 0.135794 0.541216 Median 0.131142 0.536839 Correlation Coefficient (R) 0.8233 Coefficient of Determination (R 2 ) 0.6779 The line of best fit representing the data is y = 1.7482x + 0.3038. The slope (m) of the line of best fit is 1.7482, indicating the increase of 1.7482 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 0.3038. There is a strong, positive linear correlation since the R value is approximately 0.8233, which is between 0.67 and 1. Comparing to the previous decade, the mean and median for both home runs per game and runs batted in per game has increased. However, the correlation coefficient dropped from 0.87 to 0.82, showing a slightly weaker linear correlation. y = 1.7482x + 0.3038 R² = 0.6779 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 RBI HR HR v.s RBI in 2000s

Single Variable statistic Home Runs per game (HR/G) Runs Batted In per game (RBI/G) Mean 0.139583 0.58019 Median 0.138158 0.569536 Correlation Coefficient (R) 0.7125 Coefficient of Determination (R 2 ) 0.5076 The line of best fit representing the data is y = 1.4358x + 0.3798. The slope (m) of the line of best fit is 1.4358, indicating the increase of 1.4358 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 0.3798. There is a strong, positive linear correlation since the R value is approximately 0.7125, which is between 0.67 and 1. Scatterplot for the outfielder y = 1.5758x + 0.331 R² = 0.5601 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 RBI HR HR v.s RBI for outfielder

Analysis, Discovery, and justification of hypothesis by positions I chose the positions as one of my subcategories since I would like to analyze the linear correlation for each position. By comparing their correlation coefficient, mean, and median, I’m able to identify trends, similarities and differences, hidden variables, and make predictions. Furthermore, this would prove whether my hypothesis held true for different positions or if there were any exceptions.  After finding the correlation coefficient for three positions I chose, it has again proven that my hypothesis held truth. The correlation coefficient for first base and outfielder suggested that there’s a strong, positive correlation between home run per game and runs batted in per game. As the home run per game increased, the runs batted in per game increased. The correlation coefficient was slightly below 0.67, therefore considered as a moderated – strong correlation between home run per game and runs batted in per game. Moreover, among all three positions, the outfielder has the largest correlation coefficient, indicating that outfielder has the strongest correlation between home runs per game and runs batted in per game. T his implies that outfielders are particularly effective and consistent at driving in runs when they hit home runs. In contrast, shortstops has the small correlation coefficient, indicating that shortstops has the relatively weaker correlation between home runs per game and runs batted in per game Overall, my hypothesis is supported by the data. The runs batted in per game will increase as home runs increase, and there’s a strong positive correlation between home run per game and runs batted In per game. This applies to all three positions.

Analysis, Discovery, and justification of hypothesis by Hits I chose the hits as one of my subcategories since I would like to analyze the linear correlation for groups of players with different hits per game. A hit  occurs when a batter strikes the baseball into fair territory and reaches base without doing so via an error or a fielder's choice . In baseball statistic, hits are often categorized as player's offensive productivity and would possibly be related to runs batted in and the number of home run per game. By comparing their correlation coefficient, mean, and median, I’m able to identify trends, similarities and differences, and make predictions. Furthermore, this would prove whether my hypothesis held true for different positions or if there were any exceptions.  After finding the correlation coefficient for the 1 st quartile of hits and 3 rd quartile of hits, it has proven that my hypothesis held truth. The correlation coefficient for both groups suggested that there’s a strong, positive correlation between home run per game and runs batted in per game. As the home run per game increased, the runs batted in per game increased. Both 1 st quartile of hits and 3 rd quartile of hits has high correlation coefficient. Surprisingly, the 3 rd quartile of hits has relatively lower R value compared to the 1 st quartile. The mean and median for the runs batted in per game are way higher for the 3 rd quartile, proving that the hits per game (offensive productivity) may have impact on the runs batted in per game. Overall, my hypothesis is supported by the data. The runs batted in per game will increase as home runs increase, and there’s a strong positive correlation between home run per game and runs batted in per game. This applies to the different quartiles of hits. Scatterplot for the 1 st quartile of walks (BB) Single Variable statistic Home Runs per game Runs Batted In per game y = 1.7175x + 0.3101 R² = 0.5491 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 RBI/G HR/G HR/G v.s RBI/G in the 1st Quartile of walks

(HR/G) (RBI/G) Mean 0.084676 0.455575 Median 0.068442 0.447294 Correlation Coefficient (R) 0.7410 Coefficient of Determination (R 2 ) 0.5491 The line of best fit representing the data is y = 1.7175x + 0.3101. The slope (m) of the line of best fit is 1.7175, indicating the increase of 1.7175 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 03101. There is a strong, positive linear correlation since the R value is approximately 0.7410, which is between 0.67 and 1. Scatterplot for above 3 rd quartile of walks (BB) Single Variable statistic Home Runs per game (HR/G) Runs Batted In per game (RBI/G) Mean 0.154434 0.585833 Median 0.151079 0.590604 Correlation Coefficient (R) 0.7403 Coefficient of Determination (R 2 ) 0.548 The line of best fit representing the data is y = 1.4651x + 0.3596. The slope (m) of the line of best fit is 1.4651, indicating the increase of 1.4651 runs batted per game for every home run per game. The y-intercept, representing the number of runs batted in per game if there are no home runs, is 0.3596. There is a strong, positive linear correlation since the R value is approximately 0.7403, which is between 0.67 and 1. y = 1.4651x + 0.3596 R² = 0.548 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 RBI/G HR/G HR/G v.s RBI/G above 3rd Quartile of walks

Analysis, Discovery, and justification of hypothesis by walk I chose the walks as one of my subcategories since I would like to analyze the linear correlation for groups of players with different numbers of walks. A walk (or base on balls)  occurs when a pitcher throws four pitches out of the strike zone, none of which are swung at by the hitter . After refrain ing from swinging at four pitches out of the zone, the batter is awarded first base. It appears that the number of walks could have direct impact on the runs batted in since players also receive an RBI for a bases-loaded walk or hit by pitch. By comparing their correlation coefficient, mean, and median, I’m able to identify trends, similarities and differences, and make predictions. Furthermore, this would prove whether my hypothesis held true for different numbers of walks or if there were any exceptions. After finding the correlation coefficient for the 1 st quartile of hits and 3 rd quartile of walks, both values were within 0.69 ~ 1, with the 1 st quartile slightly higher than the 3 rd . the values suggest that there’s a strong, positive correlation between home run per game and runs batted in per game. As the home run per game increased, the runs batted in per game increased. The mean and median of the runs in batted per game for the 3 rd quartile of walks is significantly higher than the 1 st quartile, suggesting that number of walks could have direct impact on the runs batted of a player and further affect the correlation between home runs per game and runs in batted per game slightly. I also realized that the mean and median value of the home run for the 3 rd quartile of walks is considerably higher than the 1 st quartile of walks. This indicates that the number of walks may have impact on the home run as well, which I was not expecting. Overall, the number of walks of an individual influence player’s performance, whether it’s the stats for home runs per game or runs in batted per game. However, both groups shows that there’s a strong, positive correlation between home runs per game and runs batted in per game, despite the difference between number of walks. This has proven that my hypothesis is correct. Critical analysis

Outliers An outlier is  an observation that lies an abnormal distance from other values in a random sample from a population . Outliers are defined as any point that is 1.5 interquartile ranges (IQR) below the quadrant 1, and any points above quadrant 3, of the x and y values. Outlier may possibly influence correlation coefficient. Therefore, it is vital to remove outliers in each decade subcategory and compare the new correlation coefficient of all data to the former one which outlier were presented. In addition, outliers may allow me to better determine the hidden variables later. 20s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.01714184 0.0630478 7 0 .04590603 0 .0688590 45 -0.05171725 0.1319069 15 X ≤- 0.05171725 X ≥ 0.13190691 5 Runs batted in per game (RBI/G) 0.40641858 0.6392197 7 0 .23280119 0 .3492017 85 0 .05721695 0.9884215 55 y ≤ 0 .05721695 y ≥ 0.98842155 5 Outliers for the 20s are the points where the x value is greater than or equal to 0.131906915 or less than or equal to - 0.05171725 and where the y value is greater than or equal to 0.988421555 or less than or equal to 0 .05721695. Points that have an outlier value: (0.148148 , 0.651852), (0.165414 , 0.766917), (0.301471 , 0.838235), (0.197279, 0.619048), (0.269737, 0.861842), (0.133333 , 0.806667), (0.229008 , 0.748092), (0.205479, 0.883562), (0.303226, 1.129032 ), (0.397351, 1.086093 ). Outliers: (0.303226, 1.129032 ), (0.397351, 1.086093 ). 30s

Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1. 5 IQR Range for Outliers Home Runs per game (HR/G) 0.027777 78 0.1005244 8 0.0727467 0.10912 005 - 0.08134227 0.2096 4453 x ≤ -0.08134227 x ≥ 0.20964453 Runs batted in per game (RBI/G) 0.382027 83 0.640271 0.25824317 0.38736 4755 - 0.05336925 0.8985 1417 y ≤ -0.05336925 y ≥ 0.89851417 Outliers for the 30s are the points where the x value is greater than or equal to 0.20964453 or less than or equal to -0.08134227 and where the y value is greater than or equal to 0.89851417 or less than or equal to -0.05336925 . Points that have an outlier value: ( 0.210526 , 0.914474), (0.322148 , 1.09396), (0.248175 , 0.751825), (0.198718, 0.987179 ), (0.304636 , 1.10596), (0.24 , 0.846667), (0.25974, 1.188312 ), (0.235669 , 1.012739 ) Outliers: ( 0.210526 , 0.914474), (0.322148 , 1.09396), (0.304636 , 1.10596), (0.25974, 1.188312 ), 0.235669 , 1.012739 ) 40s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1. 5 IQR Range for Outliers Home Runs per game (HR/G) 0.02409812 0.09878049 0.07463 237 0.112023 555 - o.0879254 35 0.1734 1286 x ≤ -0.087925435 x ≥ 0.17341286 Runs batted in per game (RBI/G) 0.35125899 0.55555556 0.20429 657 0.306444 855 0.0448141 35 0.8620 00415 y ≤ 0.044814135 y ≥ 0.862000415

Outliers for the 40s are the points where the x value is greater than or equal to 0.17341286 or less than or equal to -0.087925435 and where the y value is greater than or equal to 0.862000415 or less than or equal to 0.044814135 . Points that have an outlier value: ( 0.188312 , 0.831169), (0.219355 , 0.76129), (0.219858 , 0.609929), (0.210145 , 0.615942), (0.232258 , 0.690323), (0.331169 , 0.896104 ), (0.175676 , 0.506757), (0.191489 , 0.602837), (0.335526 , 0.835526), (0.205128 , 0.730769) Outlier: (0.331169 , 0.896104 ) 50s Q1 Q3 IQR 1.5 IQR Q1- 1.5IQR Q3+1. 5 IQR Range for Outliers Home Runs per game (HR/G) 0.06 0.16535433 0.1053543 3 0.158031 495 - o.o98031 495 0.3233 85825 x ≤ -0.098031495 x ≥ 0.323385825 Runs batted in per game (RBI/G) 0.40140 845 0.65384615 0.2524377 0.378656 55 0.022751 9 0.9062 8385 y ≤ 0.0227519 y ≥ 0.90628385 Outliers for the 50s are the points where the x value is greater than or equal to 0.323385825 or less than or equal to -0.098031495 and where the y value is greater than or equal to 0.90628385 or less than or equal to 0.0227519 . Points that have an outlier value: no points Outlier: no outliers  

60s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.05140363 0.1562454 8 0.10484 185 0.157262 775 - 0.1058591 45 0.31350 8255 x ≤ -0.105859145 x ≥ 0.313508255 Runs batted in per game (RBI/G) 0.32751992 0.5454143 8 0.21789 446 0.326841 69 0.0006782 8 0.87225 607 y ≤ 0.00067828 y ≥ 0.87225607 Outliers for the 60s are the points where the x value is greater than or equal to 0.313508255 or less than or equal to -0.105859145 and where the y value is greater than or equal to 0.87225607or less than or equal to 0.00067828 . Points that have an outlier value: ( 0.316901, 0.676056) Outliers: no outliers. 70s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.0511416 8 0.1502451 0.09910 342 0.148655 13 - 0.0975134 5 0.298900 23 x ≤ -0.09751345 x ≥ 0.29890023 Runs batted in per game (RBI/G) 0.3525093 0.59046864 0.23795 934 0.356939 01 - 0.0044297 1 0.947407 65 y ≤ -0.00442971 y ≥ 0.94740765 Outliers for the 70s are the points where the x value is greater than or equal to 0.29890023 or less than or equal to -0.09751345 and where the y value is greater than or equal to 0.94740765or less than or equal to -0.00442971 . Points that have an outlier value:

( 0.329114 , 0.943038) Outlier: no outliers 80s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.0621118 0.17763158 0.1155 1978 0.17327 967 - o.1111678 7 0.3509112 5 x ≤ -0.11116787 x ≥ 0.35091125 Runs batted in per game (RBI/G) 0.4014598 5 0.61589404 0.2144 3419 0.32165 1285 0.0798085 65 0.9375453 25 y ≤ 0.079808565 y ≥ 0.937545325 Outliers for the 80s are the points where the x value is greater than or equal to 0.35091125 or less than or equal to - 0.11116787 and where the y value is greater than or equal to 0.937545325 or less than or equal to 0.079808565 . Points that have an outlier value: no points Outliers: no outliers 90s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.0625982 7 0.176184 11 0.11358 584 0.1703 7876 - 0.10778049 0.3465628 7 x ≤ -0.10778049 x ≥ 0.34656287 Runs batted in per game (RBI/G) 0.3926644 5 0.640202 28 0.24753 783 0.3713 06745 0.02135770 5 1.0115090 25 y ≤ 0.021357705 y ≥ 1.011509025

Outliers for the 90s are the points where the x value is greater than or equal to 0.34656287 or less than or equal to -0.10778049 and where the y value is greater than or equal to 1.011509025 or less than or equal to 0.021357705 . Points that have an outlier value: ( 0.356688 , 0.936306), (0.371795, 0.788462) Outliers: no outliers 2000s Q1 Q3 IQR 1.5 IQR Q1-1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.08426704 0.17987179 0.09 5604 75 0.14340 7125 - 0.0591400 85 0.3232789 15 x ≤ -0.059140085 x ≥ 0.323278915 Runs batted in per game (RBI/G ) 0.43312212 0.64675245 0.21 3630 33 0.32044 5495 0.1126766 25 0.9671979 45 y ≤ 0.112676625 y ≥ 0.967197945 Outliers for the 2000s are the points where the x value is greater than or equal to 0.323278915 or less than or equal to -0.059140085 and where the y value is greater than or equal to 0.967197945 or less than or equal to 0.112676625 . Points that have an outlier value: ( 0.346154 , 0.692308), (0.326389 , 0.944444) Outlier: no outliers

2010s Q1 Q3 IQR 1.5 IQR Q1- 1.5IQR Q3+1.5 IQR Range for Outliers Home Runs per game (HR/G) 0.09655 172 0.1838235 3 0.08830 633 0.1324594 95 - 0.03590 7775 0.316283 025 x ≤ -0.035907775 x ≥ 0.316283025 Runs batted in per game (RBI/G ) 0.4 0.5960264 9 0.19602 649 0.2940397 35 0.10596 0265 0.890066 225 y ≤ 0.105960265 y ≥ 0.890066225 Outliers for the 2010s are the points where the x value is greater than or equal to 0.316283025 or less than or equal to -0.035907775 and where the y value is greater than or equal to 0.890066225 or less than or equal to 0.105960265 . Points that have an outlier value: (0.371069, 0.830189) Outliers: no outliers

Outlier analysis After removing all the outliers in each decade, the correlation coefficient is approximately 0.7543, a decrease from 0.7618 when outliers exist. This seems surprising to me as I believed that the removal of outliers will enhance data’s overall correlation. The new correlation coefficient still supports my hypothesis that there’s a linear, strong, positive correlation between home run per game and runs in batted per game. I did a bit of research on what I discovered. “In most practical circumstances an outlier decreases the value of a correlation coefficient and weakens the regression relationship, but it’s also possible that in some circumstances an outlier may increase a correlation value and improve regression. This is an example of influential   outlier. Influential outliers are points in a data set that influence the regression equation and improve correlation.” In another word, the outliers helped to maintain and enhance the linear form of the data points and their removal negatively impacted the correlation present. As a result, even though the correlation coefficient only decreased by a little (0.075), I decided stick with my original graphs with the existence of outliers to do further analysis later, since it has slightly stronger correlation between home run per game and runs batted in per game. When determine the outliers, I also realized that the outliers in each decade decrease as the time pass on. This explained the reason why the coefficient correlation improved decades after decades. y = 1.5792x + 0.3307 R² = 0.5689 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 HR/G RBI/G HR/G v.s RBI/G (outliers removed)

Hidden Variables: Dead Ball era The  Dead ball Era   was a period in the early 20th Century characterized by low scoring and an emphasis on  pitching   and  defense . L eague   batting averages   dropped as low as .239 in 1908, producing the lowest league   run   average in history, with teams averaging only 3.4 runs per game. The possible causes were that teams played in spacious ballparks that limited hitting for power. As a further hindrance to scoring, the   ball   used then, compared to modern baseballs, was "dead" both by design and from overuse ( ball scuffing).   For pitchers, it was the era of the "spit ball" completely legal at the time. Many pitchers relied on the spit ball and other trickery to keep batters on their toes. Some of the most skilled pitchers of all time developed in baseball's Dead ball Era, Batters used heavy bats, choked up on the handle and didn't attack the pitch aggressively.   Also, the foul strike rule was a major rule change that, in just a few years, sent baseball from a high-scoring game to a game where scoring any runs was a struggle. Under the foul strike rule, a batter who   fouls off   is charged with a   strike   unless he already has two strikes against him. Some players and fans complained about the low-scoring games, and league officials sought to remedy the situation.   The dead-ball era ended suddenly. By 1921, offenses were scoring 40% more runs and hitting four times as many home runs as they had in 1918. Solution were the cleaner baseballs, change in baseball construction ( Ben Shibe   invented the   cork -centered ball, The change in the ball dramatically   increased the batting average), banned of spitball, ballpark( changes in the dimensions of the ballparks.), Baby Ruth( theory that the prolific success of Babe Ruth at hitting home runs led players around the league to forsake their old methods of hitting (described above) and adopt a "free-swinging" style designed to hit the ball hard and with an uppercut stroke, with the intention of hitting more home runs). Due to the dead ball era, there are less home run per game and runs batted in per game. As a result, dead ball era is a hidden variable and causes the R value to decrease. The correlation coefficient of the 20s were significantly lower than the 30s, which explain the impact of dead ball era. The steroid era "The steroids era" refers to a period of time in Major League Baseball when a number of players were believed to have used performance-enhancing drugs, resulting in increased offensive output throughout the game. It is generally considered to have run from the late '80s through the late 2000s. Though steroids have been banned in MLB since 1991, the league did not implement leaguewide PED testing until 2003. Anabolic steroids help build muscle tissue and increase body mass by acting like the body's natural male hormone, testosterone. During the 1990s, Major League Baseball experienced an increase in offensive output that resulted in some unprecedented home run totals for the power hitters of the decade. While just three players reached the 50-home run mark in any season between 1961 and 1994, many sluggers would start to surpass that number in the mid-90s. The average number of players who hit more than 40 HR in a single season significantly increased during the steroid era . Former MLB player  Jose Canseco has estimated that around 85 percent   of MLB players use steroids. Ken Carminati, another former player, estimated approximately 50 percent. This could possibly explain the dramatic increase of mean and median for home run per game from 70s to 80s. The steroids era causes the R value to

increase as the home run per game increased. As the PED testing were introduced worldwide in the 2000s, the R value decreased significantly. This may be a result of players being more aware of the use of steroids. World War II   World War II   was a   world war   that lasted from 1939 to 1945. It involved the   vast majority of the world's countries —including the United States. Baseball was far and away the favorite sport in the country in the 1940s. Fans watched as many players left to join the military in the conflict. in all, 500 Major League players and more than 2,000 minor league players left to join the military,  according to   the American Veterans Center. While the statistical numbers were down during these years with so many talented players gone, the game itself grew in popularity as the war went on. The significant reduction of players during World War II causes the lower of sample size, which leads to the occurrence of sampling bias. The mean and median for both home run per game and runs batted in per game were considerably lower, comparing to the post war statistic. This explained that during war, the level of game decreased, as well as the statistics. As a result, World War II is a hidden variable potentially affecting the correlation between home run per game and runs batted in per game. Great depression The Great Depression was period of worldwide economic depression  between 1929 and 1939 . The major effect of the Great Depression on baseball was a decrease in attendance at professional baseball games. Because of the Depression, people had less money available for leisure activities. Baseball games were a luxury that could no longer be afforded by the common American. Many teams, strong and weak ones alike, kept costs down by reducing the number of coaches, or by eliminating them and employing player-managers. Owners also reduce the number of rosters. Even the best players — Babe Ruth among them — took pay cuts. Connie Mack sold many of the stars from the pennant-winning Philadelphia Athletics teams of 1929, 1930 and 1931. Since less player was playing during great depression, the sampling size were reduced, leading to the occurrence of sampling bias. Covid 19 The   COVID-19 pandemic   has caused disruption to major leage baseball. Leagues across the world experienced delayed starts, cancelled seasons, limited or no fan attendance, game postponements, and other restrictions. Players who had covid 19 may experience a decline in performance. In a study that included 71 hitters and 61 pitchers who were confirmed to have had Covid,   The Athletic   found that performance was well down from expectations in the first two weeks back off the injured list. In the sample population, hitters’ median OPS went down 63 points compared to pre-season projections; Pitchers’ median ERA went up 11 points compared to pre-season projections; 69 percent of the pitchers lost velocity compared to the 15 days before, with a median velocity loss of 0.4 mph; 54 percent of the hitters lost exit velocity compared to the 15 days before, with a median exit velocity loss of 0.6 mph. It is clear that covid 19 has tremendously impacted player’s overall performance. As a result, I would consider covid 19 as a hidden varible contributing to the lowering of correlation coefficient.  

Conclusion Based on the data, graphs, and analysis, I believe that my hypothesis held true. The data shows that there’s a strong, positive, linear correlation between home runs per game and runs batted in per game. All the subcategories reveal the same results: the runs batted in per game increases as the home run per game increased. This correlation corresponds and applies to every decades investigated, different positions chosen, different quartiles of hits, and different quartile of walks. The result is very consistent without an exception. All the correlation coefficient value are within 0.67~1, suggest a strong positive correlation. Therefore, I can conclude that there’s a strong, positive, linear correlation between home runs per game and runs batted in per game, however, not necessarily a cause-and-effect relationship. I also create another data excluding the outliers data. The correlation coefficient is surprisingly lower, indicating that the removal of outliers has weaken the overall correlation of data. This is found to be an example of influential   outlier, which outliers are points in a data set that influence the regression equation and improve correlation. In addition, I include mean and median for every graph, allowing me to better identify the hidden variables, trends, and its effect on data’s correlation. I’ve also investigated five possible hidden variables that may have impact on the correlation coefficient. This is proven by the data as well. For example, during the steroid era, the number of home runs increased dramatically. The dramatic increase in mean and median for home run per game from 70s to 80s explained this situation. The steroids era causes the R value to increase as the home run per game increased. As the PED testing were introduced worldwide in the 2000s, the R value decreased significantly and may be a result of players being more aware of the use of steroids. Sampling technique I collected the data from Major league baseball website for the regular season hitting statistic. Using systematic sampling, I randomly select two numbers from one to nine, to determine the ending of year for each decade. I selected every ending with three and seven. This way, I would have two random years that were systematically chosen for each decade, minimizing the possibility of sampling bias. Using the data collected for each decade, I created scatterplots that shows the correlation between home runs per game and runs batted in per game. I’ve chosen to use linear model since the data weren’t increasing exponentially, and this model worked out well for my data. I graphed the scatterplot for all the subcategories discussed. For the graph that exclude outliers, I first separate all the data by decades. In each decades, I determined the first quartile, third quartile, and interquartile range. Then, I eliminated all the outliers within a decade by finding and excluding values that are 1.5 interquartile range above third quadrant and 1.5 interquartile range below first quadrant. I repeated this process for all the decades, from 1920s to 2010s. Lastly, I combine all the decades and create a scatterplot that reveals the new correlation coefficient with the absence of outliers. Improvement I believe that there is a lot of improvement that could be made to enhance the overall analysis for this project. First, I could increase the sample size for decades. Instead of selecting two years for each decade, I should select four years for each decades in order to maximize the accuracy and better determine the trends. Also, instead of choosing three positions, I should have chosen all hitting positions in order to better analyze the correlation in each position and the relationships between each position. Also, when selecting the ending year for each decade, I could have use

nine pokers, each containing different numbers. In this case, the sampling bias will be minimized since it’s purely random, instead of personal preference. Prediction To predict the future increases of runs batted in for every home run, I created a graph that shows the relationship between decade and the slopes of all the line of best fit for the decade subcategory. I put the slope as the dependent variable and decades as independent variable. Decades 20s 30s 40s 50s 60s 70s 80s 90s 2000s 2010s Slope 2.0022 2.3454 1.6928 1.7267 1.7704 1.9891 1.757 4 1.8322 1.7482 1.5754   The equation of the line of best fit for this relation is y=-0.0043x+10.197. To make prediction for specific decade, simply substitute the decade value into x, to solve for the slope, y. E.g., during 1960s, the number of runs batted in will increase by 1.769 for every home run made.

References Wikimedia Foundation. (2022, December 6).  Dead-Ball Era . Wikipedia. Retrieved December 19, 2022, from https://en.wikipedia.org/wiki/Dead-ball_era Team, I. S. E. (2018, July 5).  What was baseball like during World War 2?: Historic girls baseball . Imagine Sports. Retrieved December 19, 2022, from https://imaginesports.com/news/baseball-during-world-war-2 Corso, J. (2017, October 3).  Major League Baseball's popularity during WWII . Bleacher Report. Retrieved December 19, 2022, from https://bleacherreport.com/articles/161265-major- league-baseballs-popularity-during-wwii Sarris, E. (n.d.).  How does Covid Impact MLB Players' Performance? what athletes, trainers and the stats say . The Athletic. Retrieved December 19, 2022, from https://theathletic.com/3488516/2022/08/26/mlb-players-covid-return-effects/ Ramy Elitzur Associate Professor. (2022, July 20).  Pandemic moneyball: How covid-19 has affected baseball odds . The Conversation. Retrieved December 19, 2022, from https://theconversation.com/pandemic-moneyball-how-covid-19-has-affected-baseball- odds-157203 How did the Great Depression affect baseball in the 1930s . Cram. (n.d.). Retrieved December 19, 2022, from https://www.cram.com/essay/How-Did-The-Great-Depression-Affect- Baseball/FCC6LWLQWV Cautions about correlation and regression: STAT 800 . PennState: Statistics Online Courses. (n.d.). Retrieved December 19, 2022, from https://online.stat.psu.edu/stat800/lesson/cautions-about-correlation-and-regression ESPN Internet Ventures. (n.d.).  The steroids era . ESPN. Retrieved December 19, 2022, from https://www.espn.com/mlb/topics/_/page/the-steroids-era Wikimedia Foundation. (2022, October 28).  Base on balls . Wikipedia. Retrieved December 19, 2022, from https://en.wikipedia.org/wiki/Base_on_balls Walk (BB): Glossary . MLB.com. (n.d.). Retrieved December 19, 2022, from https://www.mlb.com/glossary/standard-stats/walk The official site of Major League Baseball . MLB.com. (n.d.). Retrieved December 19, 2022, from https://www.mlb.com/