Week_4 Data Dive
Anurag
2024-01-30
Lets create 5 sample from the Lahman Database -
(Each sample is 50% of the original data)
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## stamplahman_data = read.csv("/Users/anuragreddy/Desktop/Statistics with R/Lahmans Databse .csv")
proportion=0.5#lets find out the no.of rows in the lahman database.
set.seed(123)
Number_Rows <- nrow(lahman_data)
Lahman_Sample1 <- lahman_data[sample(nrow(lahman_data), size = proportion*Number_Rows, replace = TRUE), ]
Lahman_Sample2 <- lahman_data[sample(nrow(lahman_data), size = proportion*Number_Rows, replace = TRUE), ]
Lahman_Sample3 <- lahman_data[sample(nrow(lahman_data), size = proportion*Number_Rows, replace = TRUE), ]
Lahman_Sample4 <- lahman_data[sample(nrow(lahman_data), size = proportion*Number_Rows, replace = TRUE), ]
Lahman_Sample5 <- lahman_data[sample(nrow(lahman_data), size = proportion*Number_Rows, replace = TRUE), ]Lets find out how different they are from each other -
(The best way is to first find the summary of each sample)
It would be confusing if we try to look at the all the columns in these five samples. Lets decide some columns and try to question those columns. Find out how different they are from each sample.
For this data dive lets take three columns (Each from different category like Offensive, Defensive and Pitching) and try to work on them.
Runs Per Game- Offensive Statistic
ERA (Earned Runs Average) - Pitching Statistic
CS (Caught Stealing) - Defensive Statistic
Domain Knowledge:
- Runs Per Game: On average No.of runs scored by a team in one baseball game.
- ERA (Earned Runs Average): No.of runs scored by a offensive for which the pitcher is held responsible. Lower ERA means better performance by pitcher.
- CS (Caught Stealing): The number of times a baserunner is caught by the opposing team’s defense while attempting to steal a base.
Note: We will calculate above statistic by grouping team wise. So that we can analyse the team’s performance in above statistics and find out the difference between the samples.
Runs per game is the column which we have to integrate into the samples.
Calculate Runs per Game, ERA and CS and integrate that into the samples.
Sample1 <-
Lahman_Sample1 |>
group_by(franchID) |>
summarise(Runs_per_Game = round(sum(R)/sum(G),2),
Team_ERA = round((9*sum(ER))/(sum(IPouts)/3),2),
Team_CS = round(mean(CS),2))
Sample2 <-
Lahman_Sample2 |>
group_by(franchID) |>
summarise(Runs_per_Game = round(sum(R)/sum(G),2),
Team_ERA = round((9*sum(ER))/(sum(IPouts)/3),2),
Team_CS = round(mean(CS),2))
Sample3 <-
Lahman_Sample3 |>
group_by(franchID) |>
summarise(Runs_per_Game = round(sum(R)/sum(G),2),
Team_ERA = round((9*sum(ER))/(sum(IPouts)/3),2),
Team_CS = round(mean(CS),2))
Sample4 <-
Lahman_Sample4 |>
group_by(franchID) |>
summarise(Runs_per_Game = round(sum(R)/sum(G),2),
Team_ERA = round((9*sum(ER))/(sum(IPouts)/3),2),
Team_CS = round(mean(CS),2))
Sample5 <-
Lahman_Sample5 |>
group_by(franchID) |>
summarise(Runs_per_Game = round(sum(R)/sum(G),2),
Team_ERA = round((9*sum(ER))/(sum(IPouts)/3),2),
Team_CS = round(mean(CS),2))Now we have 5 samples (sample1, sample2, sample3, sample4, sample5) with the columns - Runs per game, Earned Runs Average, Caught Stealing.
lets start with the basic differentiation - descriptive statistics.
Runs_Per_Game_Descriptive_Statistics <- data.frame(Runs_Per_Game = c('Min','Mean','Median','Max','Normality (Shapiro)','Standard Deviation'),
Sample_1 = c(round(min(Sample1$Runs_per_Game),2),round(mean(Sample1$Runs_per_Game),2),round(median(Sample1$Runs_per_Game),2),round(max(Sample1$Runs_per_Game),2),round(shapiro.test(Sample1$Runs_per_Game)$p.value,2),round(sd(Sample1$Runs_per_Game),2)),
Sample_2 = c(round(min(Sample2$Runs_per_Game),2),round(mean(Sample2$Runs_per_Game),2),round(median(Sample2$Runs_per_Game),2),round(max(Sample2$Runs_per_Game),2),round(shapiro.test(Sample2$Runs_per_Game)$p.value,2),round(sd(Sample2$Runs_per_Game),2)),
Sample_3 = c(round(min(Sample3$Runs_per_Game),2),round(mean(Sample3$Runs_per_Game),2),round(median(Sample3$Runs_per_Game),2),round(max(Sample3$Runs_per_Game),2),round(shapiro.test(Sample3$Runs_per_Game)$p.value,2),round(sd(Sample3$Runs_per_Game),2)),
Sample_4 = c(round(min(Sample4$Runs_per_Game),2),round(mean(Sample4$Runs_per_Game),2),round(median(Sample4$Runs_per_Game),2),round(max(Sample4$Runs_per_Game),2),round(shapiro.test(Sample4$Runs_per_Game)$p.value,2),round(sd(Sample4$Runs_per_Game),2)),
Sample_5 = c(round(min(Sample5$Runs_per_Game),2),round(mean(Sample5$Runs_per_Game),2),round(median(Sample5$Runs_per_Game),2),round(max(Sample5$Runs_per_Game),2),round(shapiro.test(Sample5$Runs_per_Game)$p.value,2),round(sd(Sample5$Runs_per_Game),2)))
Runs_Per_Game_Descriptive_Statistics## Runs_Per_Game Sample_1 Sample_2 Sample_3 Sample_4 Sample_5
## 1 Min 4.08 4.00 4.09 4.15 3.89
## 2 Mean 4.56 4.52 4.58 4.59 4.58
## 3 Median 4.53 4.44 4.51 4.53 4.57
## 4 Max 5.15 5.34 5.31 5.30 5.21
## 5 Normality (Shapiro) 0.49 0.15 0.18 0.04 0.97
## 6 Standard Deviation 0.29 0.28 0.28 0.31 0.30ERA_Descriptive_Statistics <- data.frame(Earned_Runs_Average = c('Min','Mean','Median','Max','Normality (Shapiro)','Standard Deviation'),
Sample_1 = c(round(min(Sample1$Team_ERA),2),round(mean(Sample1$Team_ERA),2),round(median(Sample1$Team_ERA),2),round(max(Sample1$Team_ERA),2),round(shapiro.test(Sample1$Team_ERA)$p.value,2),round(sd(Sample1$Team_ERA),2)),
Sample_2 = c(round(min(Sample2$Team_ERA),2),round(mean(Sample2$Team_ERA),2),round(median(Sample2$Team_ERA),2),round(max(Sample2$Team_ERA),2),round(shapiro.test(Sample2$Team_ERA)$p.value,2),round(sd(Sample2$Team_ERA),2)),
Sample_3 = c(round(min(Sample3$Team_ERA),2),round(mean(Sample3$Team_ERA),2),round(median(Sample3$Team_ERA),2),round(max(Sample3$Team_ERA),2),round(shapiro.test(Sample3$Team_ERA)$p.value,2),round(sd(Sample3$Team_ERA),2)),
Sample_4 = c(round(min(Sample4$Team_ERA),2),round(mean(Sample4$Team_ERA),2),round(median(Sample4$Team_ERA),2),round(max(Sample4$Team_ERA),2),round(shapiro.test(Sample4$Team_ERA)$p.value,2),round(sd(Sample4$Team_ERA),2)),
Sample_5 = c(round(min(Sample5$Team_ERA),2),round(mean(Sample5$Team_ERA),2),round(median(Sample5$Team_ERA),2),round(max(Sample5$Team_ERA),2),round(shapiro.test(Sample5$Team_ERA)$p.value,2),round(sd(Sample5$Team_ERA),2)))
ERA_Descriptive_Statistics## Earned_Runs_Average Sample_1 Sample_2 Sample_3 Sample_4 Sample_5
## 1 Min 3.50 3.58 3.38 3.67 3.54
## 2 Mean 4.25 4.21 4.23 4.27 4.25
## 3 Median 4.27 4.20 4.20 4.24 4.22
## 4 Max 4.79 5.00 4.95 4.98 5.03
## 5 Normality (Shapiro) 0.82 0.07 0.90 0.39 0.24
## 6 Standard Deviation 0.29 0.33 0.37 0.32 0.31CS_Descriptive_Statistics <- data.frame(Caught_Stealing = c('Min','Mean','Median','Max','Normality (Shapiro)','Standard Deviation'),
Sample_1 = c(round(min(Sample1$Team_CS),2),round(mean(Sample1$Team_CS),2),round(median(Sample1$Team_CS),2),round(max(Sample1$Team_CS),2),round(shapiro.test(Sample1$Team_CS)$p.value,2),round(sd(Sample1$Team_CS),2)),
Sample_2 = c(round(min(Sample2$Team_CS),2),round(mean(Sample2$Team_CS),2),round(median(Sample2$Team_CS),2),round(max(Sample2$Team_CS),2),round(shapiro.test(Sample2$Team_CS)$p.value,2),round(sd(Sample2$Team_CS),2)),
Sample_3 = c(round(min(Sample3$Team_CS),2),round(mean(Sample3$Team_CS),2),round(median(Sample3$Team_CS),2),round(max(Sample3$Team_CS),2),round(shapiro.test(Sample3$Team_CS)$p.value,2),round(sd(Sample3$Team_CS),2)),
Sample_4 = c(round(min(Sample4$Team_CS),2),round(mean(Sample4$Team_CS),2),round(median(Sample4$Team_CS),2),round(max(Sample4$Team_CS),2),round(shapiro.test(Sample4$Team_CS)$p.value,2),round(sd(Sample4$Team_CS),2)),
Sample_5 = c(round(min(Sample5$Team_CS),2),round(mean(Sample5$Team_CS),2),round(median(Sample5$Team_CS),2),round(max(Sample5$Team_CS),2),round(shapiro.test(Sample5$Team_CS)$p.value,2),round(sd(Sample5$Team_CS),2)))
CS_Descriptive_Statistics## Caught_Stealing Sample_1 Sample_2 Sample_3 Sample_4 Sample_5
## 1 Min 22.56 24.42 26.07 24.86 24.14
## 2 Mean 35.40 35.16 35.31 34.46 34.02
## 3 Median 34.89 35.12 35.08 34.05 34.00
## 4 Max 50.43 56.00 44.91 45.70 49.40
## 5 Normality (Shapiro) 0.46 0.13 0.42 0.58 0.51
## 6 Standard Deviation 6.25 6.61 5.20 5.35 5.791. How Different they are and what are the common aspects?
Answer: To answer this question lets look at the above three descriptive statistics data frames.
| Statistic | Runs Per Game | Earned Runs Average | Caught Stealing |
|---|---|---|---|
| Min | Almost same in 5 samples. It lies between (4,4.1). Sample 4 has the highest minimum value among the samples. | Ranging between 3.5-3.7. Sample 3 has highest min value and sample 1 has the lowest. | All the samples show variation in the caught stealing minimum value. It ranges from (22,26). Sample 3 has the highest min value. |
| Mean | Mean is quite similar in all the samples ranges between (4.5,4.6) Runs Per Game in 22 years. | Mean is quite similar in all the samples. approx ~4.2 Earned Runs Average in last 22 years. | Mean of caught stealing is ranging between (34,35) caught steals in last 22 years. |
| Median | The middle most value of all the samples lies between (4.5,4.6). First 4 samples have their mean and median almost same. | The middle most value is quite similar in all the 5 samples. approx 4.2 same as mean. | First 3 samples have approx 35 ha their median and remaining two are at 34. |
| Max | When it comes to maximum value - there is little variation of at least 0.2 between samples | The max ERA in all the samples is approximately equals to ~5. In sample 1 it is 4.7. The least max value in all samples. | Max of CS has a high variation compared to other 2 stats. Sample 2 hsa 56 as the max remaining are close to 50. |
| Normality | We have used shapiro test for testing the normality of the data. It is found that apart from 4th sample all other samples are normally distributed (sig>0.05). | We have used shapiro test for testing the normality of the data. It is found all ERA samples are normally distributed (sig>0.05) | We have used shapiro test for testing the normality of the data. It is found all CS samples are normally distributed (sig>0.05). |
| SD | SD is almost same in all the 5 sample approx ~0.3 | SD is almost same in all the 5 samples, approx ~0.3. except in sample2 it is 0.4. | There is a variation in the SD of caught stealing samples. Samples 1 and 2 are highly dispersed compared to other samples. |
Lets look at the distribution of these samples and try to plot the normal distribution curve over to verify the samples are normal or not.
Note: According to shapiro test the sample are normal (except runs per game of sample5).
Runs Per Game:
RPG_1 <-
ggplot(Sample1,aes(x=Runs_per_Game))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_density(color='blue',linewidth=2)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample1$Runs_per_Game),sd=sd(Sample1$Runs_per_Game)))+
ggtitle("Sample 1")+
theme_classic()
RPG_2 <-
ggplot(Sample2,aes(x=Runs_per_Game))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample2$Runs_per_Game),sd=sd(Sample2$Runs_per_Game)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 2")+
theme_classic()
RPG_3 <-
ggplot(Sample3,aes(x=Runs_per_Game))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample3$Runs_per_Game),sd=sd(Sample3$Runs_per_Game)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 3")+
theme_classic()
RPG_4 <-
ggplot(Sample4,aes(x=Runs_per_Game))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample4$Runs_per_Game),sd=sd(Sample4$Runs_per_Game)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 4")+
theme_classic()
RPG_5 <-
ggplot(Sample5,aes(x=Runs_per_Game))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample5$Runs_per_Game),sd=sd(Sample5$Runs_per_Game)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 5")+
theme_classic()
plot_grid(RPG_1,RPG_2,RPG_3,RPG_4,RPG_5,ncol = 2)## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
Interpretation: By looking at the 5 graphs we can interpret that all the samples have the distribution almost equal to the normal distribution. But when it comes to the sample 4 though both the graphs coincides one can interpret this as the data is normal but when we look at shapiro test’s p value was <0.05, meaning data is not normal.
NOTE: It's important to note that visual inspection alone is not a definitive method for assessing normality
Shapiro Test Null Hypothesis: It states that data is normally distributed.
(Similarly lets visualize the distribution for other 2 statistics (ERA and CS)
Earned Runs Average:
ERA_1 <-
ggplot(Sample1,aes(x=Team_ERA))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_density(color='blue',linewidth=2)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample1$Team_ERA),sd=sd(Sample1$Team_ERA)))+
ggtitle("Sample 1")+
theme_classic()
ERA_2 <-
ggplot(Sample2,aes(x=Team_ERA))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample2$Team_ERA),sd=sd(Sample2$Team_ERA)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 2")+
theme_classic()
ERA_3 <-
ggplot(Sample3,aes(x=Team_ERA))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample3$Team_ERA),sd=sd(Sample3$Team_ERA)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 3")+
theme_classic()
ERA_4 <-
ggplot(Sample4,aes(x=Team_ERA))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample4$Team_ERA),sd=sd(Sample4$Team_ERA)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 4")+
theme_classic()
ERA_5 <-
ggplot(Sample5,aes(x=Team_ERA))+
geom_histogram(color='black',aes(y=..density..),binwidth=0.1)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample5$Team_ERA),sd=sd(Sample5$Team_ERA)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 5")+
theme_classic()
plot_grid(ERA_1,ERA_2,ERA_3,ERA_4,ERA_5,ncol = 2)
Interpretation: By looking at the 5 graphs we can interpret that the all the 5 samples of ERA have the distribution almost equal to the normal distribution. Shapiro test’s sig or p value >0.05. (Fail to reject the null hypothesis).
Caught Stealing:
CS_1 <-
ggplot(Sample1,aes(x=Team_CS))+
geom_histogram(color='black',aes(y=..density..),binwidth=1.5)+
geom_density(color='blue',linewidth=2)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample1$Team_CS),sd=sd(Sample1$Team_CS)))+
ggtitle("Sample 1")+
theme_classic()
CS_2 <-
ggplot(Sample2,aes(x=Team_CS))+
geom_histogram(color='black',aes(y=..density..),binwidth=1.5)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample2$Team_CS),sd=sd(Sample2$Team_CS)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 2")+
theme_classic()
CS_3 <-
ggplot(Sample3,aes(x=Team_CS))+
geom_histogram(color='black',aes(y=..density..),binwidth=1.5)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample3$Team_CS),sd=sd(Sample3$Team_CS)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 3")+
theme_classic()
CS_4 <-
ggplot(Sample4,aes(x=Team_CS))+
geom_histogram(color='black',aes(y=..density..),binwidth=1.5)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample4$Team_CS),sd=sd(Sample4$Team_CS)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 4")+
theme_classic()
CS_5 <-
ggplot(Sample5,aes(x=Team_CS))+
geom_histogram(color='black',aes(y=..density..),binwidth=1.5)+
geom_function(fun=dnorm,color='red', linewidth=2, args = list(mean=mean(Sample5$Team_CS),sd=sd(Sample5$Team_CS)))+
geom_density(color='blue',linewidth=2)+
ggtitle("Sample 5")+
theme_classic()
plot_grid(CS_1,CS_2,CS_3,CS_4,CS_5,ncol = 2)
Interpretation: By looking at the 5 graphs we can interpret that the all the 5 samples of CS have the distribution almost equal to the normal distribution. Shapiro test’s sig or p value >0.05. (Fail to reject the null hypothesis).
NOTE: To visualize the distribution of sample with the distribution of normal data - Q-Q plots (Quantile - Quantile) are best graphs. Lets try to implement the Q-Q plots for the any one of the statistic samples.
Lets use caught stealing statistic and visualize the Q-Q plots for the all 5 samples.
CS_QQ1 <-
ggplot(Sample1, aes(sample=Team_CS))+
geom_qq()+
geom_qq_line()+
labs(x='Theoretical Qunatiles', y='Team_CS Quantiles')+
ggtitle("Q-Q Plot for Sample 1")+
theme_classic()
CS_QQ2 <-
ggplot(Sample2, aes(sample=Team_CS))+
geom_qq()+
geom_qq_line()+
labs(x='Theoretical Qunatiles', y='Team_CS Quantiles')+
ggtitle("Q-Q Plot for Sample 2")+
theme_classic()
CS_QQ3 <-
ggplot(Sample3, aes(sample=Team_CS))+
geom_qq()+
geom_qq_line()+
labs(x='Theoretical Qunatiles', y='Team_CS Quantiles')+
ggtitle("Q-Q Plot for Sample 3")+
theme_classic()
CS_QQ4 <-
ggplot(Sample4, aes(sample=Team_CS))+
geom_qq()+
geom_qq_line()+
labs(x='Theoretical Qunatiles', y='Team_CS Quantiles')+
ggtitle("Q-Q Plot for Sample 4")+
theme_classic()
CS_QQ5 <-
ggplot(Sample5, aes(sample=Team_CS))+
geom_qq()+
geom_qq_line()+
labs(x='Theoretical Qunatiles', y='Team_CS Quantiles')+
ggtitle("Q-Q Plot for Sample 5")+
theme_classic()
plot_grid(CS_QQ1,CS_QQ2,CS_QQ3,CS_QQ4,CS_QQ5,ncol = 3)
Interpretation: If the points closely follow a straight line, it indicates that the data points are consistent with what would be expected if the data were normally distributed. In the above grid of graphs the points are almost following the line, indicating the data is expected to follow normal distribution but just thick tails.
2. Anomalies in one sample won’t be same in another sample.
The best way to identify the anomalies or outliers is by looking at the box plots.
Runs Per Game
RPG_BP1 <-
ggplot(Sample1,aes(x='Sample1',y=Runs_per_Game))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
RPG_BP2 <-
ggplot(Sample2,aes(x='Sample2',y=Runs_per_Game))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
RPG_BP3 <-
ggplot(Sample3,aes(x='Sample3',y=Runs_per_Game))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
RPG_BP4 <-
ggplot(Sample4,aes(x='Sample4',y=Runs_per_Game))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
RPG_BP5 <-
ggplot(Sample5,aes(x='Sample5',y=Runs_per_Game))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
plot_grid(RPG_BP1,RPG_BP2,RPG_BP3,RPG_BP4,RPG_BP5)
Interpretation: As from the above grid of graphs we can say that the sample function has really did a great job in sampling the data from the data set. If we look at the sample 2,3 and 4 the ouliers of these samples are not actually outliers in other samples.
For example: An outlier, 5.2 Runs per game scored in sample 2,3,4 is actually not an outlier in the samples 1,2.
This illustrates an anomalies or outliers in one sample may not be in the other samples. Similarly we can identify for the other two statistics (ERA and CS).
Earned Runs Average
ERA_BP1 <-
ggplot(Sample1,aes(x='Sample1',y=Team_ERA))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
ERA_BP2 <-
ggplot(Sample2,aes(x='Sample2',y=Team_ERA))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
ERA_BP3 <-
ggplot(Sample3,aes(x='Sample3',y=Team_ERA))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
ERA_BP4 <-
ggplot(Sample4,aes(x='Sample4',y=Team_ERA))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
ERA_BP5 <-
ggplot(Sample5,aes(x='Sample5',y=Team_ERA))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
plot_grid(ERA_BP1,ERA_BP2,ERA_BP3,ERA_BP4,ERA_BP5)
Interpretation: As from the above grid of graphs we can say that the sample function has really did a great job in sampling the data from the data set. If we look at the sample 2 the ouliers of these samples are not actually outliers in other samples.
For example: An outlier, 5.2 ERA in sample 2,4,5 is actually not an outlier in the samples 1,3.
Caught Stealing
CS_BP1 <-
ggplot(Sample1,aes(x='Sample1',y=Team_CS))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
CS_BP2 <-
ggplot(Sample2,aes(x='Sample2',y=Team_CS))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
CS_BP3 <-
ggplot(Sample3,aes(x='Sample3',y=Team_CS))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
CS_BP4 <-
ggplot(Sample4,aes(x='Sample4',y=Team_CS))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
CS_BP5 <-
ggplot(Sample5,aes(x='Sample5',y=Team_CS))+
geom_boxplot(fill='lightblue',width=0.5,color='black')+
theme_classic()
plot_grid(CS_BP1,CS_BP2,CS_BP3,CS_BP4,CS_BP5)
Interpretation: As from the above grid of graphs we can say that the sample function has really did a great job in sampling the data from the data set. If we look at the sample 1 the ouliers of these samples are not actually outliers in other samples.
For example: An outlier, 52 CS in sample 1,2 is actually not an outlier in the samples 3,4,5.
Monte Carlo Simulations
Lets try to simulate no.of home runs made by Los Angeles Angels in MLB.
set.seed(345)
Home_Runs <- lahman_data|>
filter(franchID == 'ANA') |>
select(franchID, HR)
#Define no.of simulations
Home_Runs <- na.omit(Home_Runs)
num_simulations <- 1000
HR_Simulated_Mean <- numeric(1000)
for(i in 1:num_simulations){
Simulated_HR <- Home_Runs$HR[sample(length(Home_Runs$HR),replace = TRUE)]
HR_Simulated_Mean[i] <- mean(Simulated_HR)
}
#These are the simulated mean of home runs of Los Angeles Angels.
head(HR_Simulated_Mean)## [1] 174.8636 169.7273 161.1818 177.8182 157.6818 146.7273Now lets visualize the above simulated mean Home Runs in histogram.
ANA_HR_Simulated <- data.frame(Home_Runs = HR_Simulated_Mean)
ggplot(ANA_HR_Simulated,aes(x=Home_Runs)) +
geom_histogram(fill='lightblue',color='black',aes(y = after_stat(density)))+
geom_density(color='red')+
labs(title = "Monte Carlo Simulation of Home Run Totals",
x = "Home Run Totals", y = "Frequency")+
theme_classic()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
As we can see, the simulated mean of team home runs follows normal distribution. Now we can query this graph and find out some interesting insights.
Question: What is the probability of a team hitting Home Runs greater than 150?
HR_Lessthan_150 <- mean(HR_Simulated_Mean < 150)
HR_Lessthan_150## [1] 0.01ggplot(ANA_HR_Simulated,aes(x=Home_Runs)) +
geom_histogram(fill='lightblue',color='black',aes(y = ..density..))+
geom_density(color='red')+
geom_vline(xintercept = 150,linetype='solid',color='black')+
annotate('rect',xmin=min(HR_Simulated_Mean),xmax=150,ymin=0,ymax=HR_Lessthan_150,color='black',fill='white')+
annotate('text',label = as.character(HR_Lessthan_150),x=min(HR_Simulated_Mean)+1,y=HR_Lessthan_150+0.005,color='black')+
labs(title = "Monte Carlo Simulation of Home Run Totals",
x = "Home Run Totals", y = "Frequency")+
theme_classic()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Interpretation: There is 1% of probability a team end up scoring at most 150 Home Runs in a season. The shaded region is the histogram is the probability of team scoring less than 150 home runs.
Question: Lets try to find out the 95% confidence interval of team’s Home Runs?
To answer this we will be using quantile() function. It is an in-built function in R.
Confidence_Interval <- quantile(HR_Simulated_Mean, c(0.025,0.975))
Confidence_Interval## 2.5% 97.5%
## 151.9989 177.6386