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plot(cars)

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mean(cars$speed)
## [1] 15.4
mean(cars$dist)
## [1] 42.98
max(cars$dist)
## [1] 120
max(cars$speed)
## [1] 25
4+1
## [1] 5
5-2
## [1] 3
2^2
## [1] 4
sqrt(25)
## [1] 5
2^5
## [1] 32
log(2.72)
## [1] 1.000632
log10(5)
## [1] 0.69897
log10(10)
## [1] 1
log10(100)
## [1] 2
#Here we are computing log base ten
log(10,base=5)
## [1] 1.430677
log(10,base=2)
## [1] 3.321928
log(1000,base=10)
## [1] 3

Question_1: Compute the log base 5 of 10 and the log of 10.

log(10,base=5)
## [1] 1.430677
log(10,base=10)
## [1] 1

Computing some offensive metrics in Baseball

BA=(29)/(112)
BA
## [1] 0.2589286
Batting_Average=round(BA,digits = 3)
Batting_Average
## [1] 0.259

Question_2:What is the batting average of a player that bats 42 hits in 212 at bats?

#On Base Percentage
#OBP=(H+BB+HBP)/(At Bats+H+BB+HBP+SF)
# H=Hits , BB= Base on balls (walks) , HBP = Hit By Pitch
# AB=At Bats, SF=Sacrifice flies

#Let us compute the OBP for a player with the following general stats
#AB=515,H=172,BB=84,HBP=5,SF=6
OBP=(172+84+5)/(515+172+84+5+6)
OBP
## [1] 0.3337596
On_Base_Percentage=round(OBP,digits = 3)
On_Base_Percentage
## [1] 0.334

Question_3:Compute the OBP for a player with the following general stats: AB=565,H=156,BB=65,HBP=3,SF=7

#OBP=(H+BB+HBP)/(AB+H+BB+HBP+SF)
OBP=(156+65+3)/(565+156+65+3+7)
OBP
## [1] 0.281407

Extra math exercises

3 == 8
## [1] FALSE
# Does 3 equals 8?
3 != 8
## [1] TRUE
# Is 3 different from 8?
3 <= 8
## [1] TRUE
# Is 3 less than or equal to 8?
3>4
## [1] FALSE

EXAMPLES: The logical operators are & for logical AND, | for logical OR, and ! for NOT.

# Logical Disjunction (or)
FALSE | FALSE # False OR False
## [1] FALSE
# Logical Conjunction (and)
TRUE & FALSE #True AND False
## [1] FALSE
# Negation
! FALSE # Not False
## [1] TRUE
# Combination of statements
2 < 3 | 1 == 5 # 2<3 is True, 1==5 is False, True OR False is True
## [1] TRUE

Assigning Values to Variables

#In R, we create a variable and assign it a value using <- as follows
Total_Bases <- 6 + 5
Total_Bases*3
## [1] 33
#To see the variables that are currently defined, use ls (as in “list”)

ls()
## [1] "BA"                 "Batting_Average"    "OBP"               
## [4] "On_Base_Percentage" "Total_Bases"
#To delete a variable, use rm (as in “remove”)

rm(Total_Bases)
#Either <- or = can be used to assign a value to a variable, but I prefer <- because is less likely to be confused with the logical operator ==

Vectors

#The basic type of object in R is a vector, which is an ordered list of values of the same type. You can create a vector using the c() function (as in “concatenate”).
pitches_by_innings <- c(12, 15, 10, 20, 10) 
pitches_by_innings
## [1] 12 15 10 20 10
#"Pitches by innings" refers to the number of pitches a pitcher throws in each specific inning of a game. 
strikes_by_innings <- c(9, 12, 6, 14, 9)
strikes_by_innings
## [1]  9 12  6 14  9
#"strikes by innings" refers to a detailed record of the number of strikes thrown by a pitcher in each specific inning they pitched.

Question_4: Define two vectors,runs_per_9innings and hits_per_9innings, each with five elements. 

runs_per_9innings <- c(1.5, 2.5, 3.5, 4.5, 5.5)
hits_per_9innings <- c(7.2, 8.3, 9.4, 9.6, 8.8)
#Stands for combine or concatenate
runs_per_9innings
## [1] 1.5 2.5 3.5 4.5 5.5
hits_per_9innings
## [1] 7.2 8.3 9.4 9.6 8.8

Functions that will create vectors with regular patterns, like repeated elements:

# replicate function
rep(2, 5)
## [1] 2 2 2 2 2
rep(1,4)
## [1] 1 1 1 1
# consecutive numbers
1:5
## [1] 1 2 3 4 5
2:10
## [1]  2  3  4  5  6  7  8  9 10
# sequence from 1 to 10 with a step of 2
seq(1, 10, by=2)
## [1] 1 3 5 7 9
seq(2,13,by=3)
## [1]  2  5  8 11

Many functions and operators like + or - will work on all elements of the vector.

# add vectors
pitches_by_innings+strikes_by_innings
## [1] 21 27 16 34 19
# compare vectors
pitches_by_innings == strikes_by_innings
## [1] FALSE FALSE FALSE FALSE FALSE
# find length of vector
length(pitches_by_innings)
## [1] 5
# find minimum value in vector
min(pitches_by_innings)
## [1] 10
# find average value in vector
mean(pitches_by_innings)
## [1] 13.4
#You can access parts of a vector by using [. Recall what the value is of the vector pitches_by_innings.

pitches_by_innings
## [1] 12 15 10 20 10
# If you want to get the first element:
pitches_by_innings[1]
## [1] 12

Question_5: Get the first element of hits_per_9innings.

hits_per_9innings[1]
## [1] 7.2
#If you want to get the last element of pitches_by_innings without explicitly typing the number of elements of pitches_by_innings, make use of the length function, which calculates the length of a vector:

pitches_by_innings[length(pitches_by_innings)]
## [1] 10

Question_6: Get the last element of hits_per_9innings.

hits_per_9innings[length(hits_per_9innings)]
## [1] 8.8
#We can also extract multiple values from a vector. For instance to get the 2nd through 4th values use:

pitches_by_innings[c(2, 3, 4)]
## [1] 15 10 20
#Vectors can also be strings or logical values

player_positions <- c("catcher", "pitcher", "infielders", "outfielders")

Data Frames

In statistical applications, data is often stored as a data frame, which is like a spreadsheet, with rows as observations and columns as variables.

To manually create a data frame, use the data.frame() function.

data.frame(bonus = c(2, 3, 1),#in millions 
           active_roster = c("yes", "no", "yes"), 
           salary = c(1.5, 2.5, 1))#in millions

Most often we will be using data frames loaded from a file. For example, load the results of a fan’s survey. The function load or read.table can be used for this.

How to Make a Random Sample

To randomly select a sample we use the function sample(). The following code selects 5 numbers between 1 and 10 at random (without duplication)

sample(1:10, size=5)
## [1]  7  1  4  9 10

The first argument gives the vector of data to select elements from. The second argument (size=) gives the size of the sample to select. Taking a simple random sample from a data frame is only slightly more complicated, having two steps:

Use sample() to select a sample of size n from a vector of the row numbers of the data frame. Use the index operator [ to select those rows from the data frame. Consider the following example with fake data. First, make up a data frame with two columns. (LETTERS is a character vector of length 26 with capital letters, LETTERS is automatically defined and pre-loaded in R)

bar <- data.frame(var1 = LETTERS[1:10], var2 = 1:10)
# Check data frame
bar
#Suppose we want to select a random sample of size 5. First, define a variable n with the size of the sample, 5

n <- 5

Now, select a sample of size 5 from the vector with 1 to 10 (the number of rows in bar). Use the function nrow() to find the number of rows in bar instead of manually entering that number.

Use : to create a vector with all the integers between 1 and the number of rows in bar.

samplerows <- sample(1:nrow(bar), size=n) 
# print sample rows
samplerows
## [1] 2 4 8 3 1

The variable samplerows contains the rows of bar which make a random sample from all the rows in bar. Extract those rows from bar with:

# extract rows
barsample <- bar[samplerows, ]
# print sample
print(barsample)
##   var1 var2
## 2    B    2
## 4    D    4
## 8    H    8
## 3    C    3
## 1    A    1

The code above creates a new data frame called barsample with a random sample of rows from bar.

#In a single line of code:

bar[sample(1:nrow(bar), n), ]

Using Tables

The table() command allows us to look at tables. Its simplest usage looks like table(x) where x is a categorical variable.

For example, a survey asks people if they support the home team or not. The data is

Yes, No, No, Yes, Yes

We can enter this into R with the c() command, and summarize with the table() command as follows:

x <- c("Yes","No","No","Yes","Yes") 
table(x)
## x
##  No Yes 
##   2   3

Numerical measures of center and spread

Suppose, MLB Teams’ CEOs yearly compensations are sampled and the following are found (in millions)

12 .4 5 2 50 8 3 1 4 0.25

sals <- c(12, .4, 5, 2, 50, 8, 3, 1, 4, 0.25)
# the average
mean(sals) 
## [1] 8.565
# the variance
var(sals)
## [1] 225.5145
# the standard deviation
sd(sals)
## [1] 15.01714
# the median
median(sals)
## [1] 3.5
# Tukey's five number summary, usefull for boxplots
# five numbers: min, lower hinge, median, upper hinge, max
fivenum(sals)
## [1]  0.25  1.00  3.50  8.00 50.00
# summary statistics
summary(sals)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.250   1.250   3.500   8.565   7.250  50.000

How about the mode?

In R we can write our own functions, and a first example of a function is shown below in order to compute the mode of a vector of observations x

# Function to find the mode, i.e. most frequent value
getMode <- function(x) {
     ux <- unique(x)
     ux[which.max(tabulate(match(x, ux)))]
 }

As an example, we can use the function defined above to find the most frequent value of the number of pitches_by_innings

# Most frequent value in pitches_by_innings
getMode(pitches_by_innings)
## [1] 10

Question_7: Find the most frequent value of hits_per_9innings.

getMode(hits_per_9innings)
## [1] 7.2
#Question_8: Summarize the following survey with the `table()` command:
#What is your favorite day of the week to watch baseball? A total of 10 fans submitted this survey.
#Saturday, Saturday, Sunday, Monday, Saturday,Tuesday, Sunday, Friday, Friday, Monday
game_day<-c("Saturday", "Saturday", "Sunday", "Monday", "Saturday","Tuesday", "Sunday", "Friday", "Friday", "Monday")

table(game_day)
## game_day
##   Friday   Monday Saturday   Sunday  Tuesday 
##        2        2        3        2        1

Question_9: What is the most frequent answer recorded in the survey? Use the getMode function to compute results. 

# Most frequent answer in the survey
getMode(game_day)
## [1] "Saturday"