In-class activity 5

Let us continue getting started with R as we start discussing important statistical concepts in Sports Analytics.

Case-scenario 1

This is the fourth season of outfielder Luis Robert with the Chicago White Socks. If during the first three seasons he hit 11, 13, and 12 home runs, how many does he need on this season for his overall average to be at least 20?

#Given that x1=11,x2=13,x3=12

#We want to find x4 such that the mean (average) number of home-runs is x¯>=20

#Notice that in this case n=4

#According to the information above: 20×4=11+13+12+x4; so when x4=44,the home-runs average will be 20.

#Answer Here

# Home-runs so far
HR_before <- c(11, 13, 12)
# Average Number of Home-runs per season wanted
wanted_HR <- 20
# Number of seasons
n_seasons <- 4
# Needed Home-runs on season 4
x_4 <- n_seasons*wanted_HR - sum(HR_before)
# Minimum number of Home-runs needed by Robert
x_4

[1] 44

According to the calculations above, Robert must hit 44 home-runs or better on this season to get an average number of home-runs per season of at least 20.

We could confirm this, by using the function mean() in R

# Robert's performance
Robert_HRs <- c(11, 13, 12,44)
# Find mean
mean(Robert_HRs)

[1] 20

sd(Robert_HRs)

[1] 16.02082

max(Robert_HRs)

[1] 44

min(Robert_HRs)

[1] 11

summary(Robert_HRs)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  11.00   11.75   12.50   20.00   20.75   44.00 

# walks so far
walks_before <- c(79, 108,41,145, 135)
# Average Number of walks per season wanted
wanted_walks <- 100
# Number of seasons
n_seasons <- 6
# Needed number of walks in season 6
x_6 <- n_seasons*wanted_walks - sum(walks_before)
# Minimum number of walks needed by Juan Soto
x_6

[1] 92

Case-scenario 2 The average salary of 10 baseball players is 72,000 dollars a week and the average salary of 4 soccer players is 84,000. Find the mean salary of all 14 professional players.

Solution - We can easily find the joined mean by adding both mean and dividing by the total number of people.

Let n1=10 denote the number of baseball players, and y1=72000 their mean salary.Let n2=4 the number of soccer players and y2=84000 their mean salary. Then the mean salary of all 16 individuals is: (n1x1+n2x2)/(n1+n2)

n_1 <- 10
n_2 <- 4
y_1 <- 72000
y_2 <- 84000
# Mean salary overall
salary_ave <-  (n_1*y_1 + n_2*y_2)/(n_1+n_2)
salary_ave

[1] 75428.57

doubles_hit<-read.csv("doubles_hit.csv", header = TRUE, sep = ",")
doubles_hits<-doubles_hit$doubles_hit

View(doubles_hit)

# Mean 
doubles_mean  <- mean(doubles_hits)
doubles_mean

[1] 23.55

# Median 
doubles_median  <- median(doubles_hits)
doubles_median

[1] 23.5

# Find number of observations
doubles_n <- length(doubles_hits)
# Find standard deviation
doubles_sd <- sd(doubles_hits)
doubles_sd

[1] 13.37371

What percentage of the data lies within one standard deviation of the mean?

doubles_w1sd <- sum((doubles_hits - doubles_mean)/doubles_sd < 1)/ doubles_n
# Percentage of observation within one standard deviation of the mean
doubles_w1sd

[1] 0.79

## Difference from empirical 
doubles_w1sd - 0.68

[1] 0.11

What percentage of the data lies within two standard deviations of the mean?

## Within 2 sd
doubles_w2sd <- sum((doubles_hits - doubles_mean)/ doubles_sd < 2)/doubles_n
doubles_w2sd

[1] 1

## Difference from empirical 
doubles_w2sd - 0.95

[1] 0.05

What percent of the data lies within three standard deviations of the mean?

## Within 3 sd 
doubles_w3sd <- sum((doubles_hits - doubles_mean)/ doubles_sd < 3)/doubles_n
doubles_w3sd

[1] 1

## Difference from empirical 
doubles_w3sd - 0.9973

[1] 0.0027

?hist

# Create histogram
hist(doubles_hits,xlab = "Number of Doubles Hits",col = "green",border = "red", xlim = c(0,50), ylim = c(0,28),breaks = 5)