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| -rw-r--r-- | r.html.markdown | 523 |
1 files changed, 343 insertions, 180 deletions
diff --git a/r.html.markdown b/r.html.markdown index ea94ae42..cd09e8da 100644 --- a/r.html.markdown +++ b/r.html.markdown @@ -6,34 +6,42 @@ contributors: filename: learnr.r --- -R is a statistical computing language. It has lots of libraries for uploading and cleaning data sets, running statistical procedures, and making graphs. You can also run `R`commands within a LaTeX document. +R is a statistical computing language. It has lots of libraries for uploading and cleaning data sets, running statistical procedures, and making graphs. You can also run `R` commands within a LaTeX document. ```python # Comments start with number symbols. -# You can't make a multi-line comment per se, +# You can't make multi-line comments, # but you can stack multiple comments like so. -# in Windows, hit COMMAND-ENTER to execute a line +# in Windows or Mac, hit COMMAND-ENTER to execute a line -################################################################### + +############################################################################# # Stuff you can do without understanding anything about programming -################################################################### +############################################################################# + +# In this section, we show off some of the cool stuff you can do in +# R without understanding anything about programming. Do not worry +# about understanding everything the code does. Just enjoy! -data() # Browse pre-loaded data sets -data(rivers) # Lengths of Major North American Rivers -ls() # Notice that "rivers" appears in the workspace -head(rivers) # peek at the dataset +data() # browse pre-loaded data sets +data(rivers) # get this one: "Lengths of Major North American Rivers" +ls() # notice that "rivers" now appears in the workspace +head(rivers) # peek at the data set # 735 320 325 392 524 450 + length(rivers) # how many rivers were measured? # 141 -summary(rivers) +summary(rivers) # what are some summary statistics? # Min. 1st Qu. Median Mean 3rd Qu. Max. # 135.0 310.0 425.0 591.2 680.0 3710.0 -stem(rivers) #stem-and-leaf plot (like a histogram) -# + +# make a stem-and-leaf plot (a histogram-like data visualization) +stem(rivers) + # The decimal point is 2 digit(s) to the right of the | # # 0 | 4 @@ -56,8 +64,8 @@ stem(rivers) #stem-and-leaf plot (like a histogram) # 34 | # 36 | 1 - -stem(log(rivers)) #Notice that the data are neither normal nor log-normal! Take that, Bell Curve fundamentalists. +stem(log(rivers)) # Notice that the data are neither normal nor log-normal! +# Take that, Bell curve fundamentalists. # The decimal point is 1 digit(s) to the left of the | # @@ -80,17 +88,19 @@ stem(log(rivers)) #Notice that the data are neither normal nor log-normal! Take # 80 | # 82 | 2 +# make a histogram: +hist(rivers, col="#333333", border="white", breaks=25) # play around with these parameters +hist(log(rivers), col="#333333", border="white", breaks=25) # you'll do more plotting later -hist(rivers, col="#333333", border="white", breaks=25) #play around with these parameters -hist(log(rivers), col="#333333", border="white", breaks=25) #you'll do more plotting later - -#Here's another neat data set that comes pre-loaded. R has tons of these. data() +# Here's another neat data set that comes pre-loaded. R has tons of these. data(discoveries) -plot(discoveries, col="#333333", lwd=3, xlab="Year", main="Number of important discoveries per year") -plot(discoveries, col="#333333", lwd=3, type = "h", xlab="Year", main="Number of important discoveries per year") +plot(discoveries, col="#333333", lwd=3, xlab="Year", + main="Number of important discoveries per year") +plot(discoveries, col="#333333", lwd=3, type = "h", xlab="Year", + main="Number of important discoveries per year") - -#rather than leaving the default ordering (by year) we could also sort to see what's typical +# Rather than leaving the default ordering (by year), +# we could also sort to see what's typical: sort(discoveries) # [1] 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 # [26] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 @@ -117,231 +127,250 @@ stem(discoveries, scale=2) max(discoveries) # 12 - summary(discoveries) # Min. 1st Qu. Median Mean 3rd Qu. Max. # 0.0 2.0 3.0 3.1 4.0 12.0 - - - -#Basic statistical operations don't require any programming knowledge either - -#roll a die a few times +# Roll a die a few times round(runif(7, min=.5, max=6.5)) # 1 4 6 1 4 6 4 +# Your numbers will differ from mine unless we set the same random.seed(31337) -#your numbers will differ from mine unless we set the same random.seed(31337) - - -#draw from a standard Gaussian 9 times +# Draw from a standard Gaussian 9 times rnorm(9) # [1] 0.07528471 1.03499859 1.34809556 -0.82356087 0.61638975 -1.88757271 # [7] -0.59975593 0.57629164 1.08455362 - - - - - - -######################### -# Basic programming stuff -######################### - -# NUMBERS - -# "numeric" means double-precision floating-point numbers -5 # 5 -class(5) # "numeric" -5e4 # 50000 #handy when dealing with large,small,or variable orders of magnitude -6.02e23 # Avogadro's number -1.6e-35 # Planck length - -# long-storage integers are written with L -5L # 5 -class(5L) # "integer" - -# Try ?class for more information on the class() function -# In fact, you can look up the documentation on `xyz` with ?xyz -# or see the source for `xyz` by evaluating xyz - -# Arithmetic -10 + 66 # 76 -53.2 - 4 # 49.2 -2 * 2.0 # 4 -3L / 4 # 0.75 -3 %% 2 # 1 - -# Weird number types -class(NaN) # "numeric" +################################################## +# Data types and basic arithmetic +################################################## + +# Now for the programming-oriented part of the tutorial. +# In this section you will meet the important data types of R: +# integers, numerics, characters, logicals, and factors. +# There are others, but these are the bare minimum you need to +# get started. + +# INTEGERS +# Long-storage integers are written with L +5L # 5 +class(5L) # "integer" +# (Try ?class for more information on the class() function.) +# In R, every single value, like 5L, is considered a vector of length 1 +length(5L) # 1 +# You can have an integer vector with length > 1 too: +c(4L, 5L, 8L, 3L) # 4 5 8 3 +length(c(4L, 5L, 8L, 3L)) # 4 +class(c(4L, 5L, 8L, 3L)) # "integer" + +# NUMERICS +# A "numeric" is a double-precision floating-point number +5 # 5 +class(5) # "numeric" +# Again, everything in R is a vector; +# you can make a numeric vector with more than one element +c(3,3,3,2,2,1) # 3 3 3 2 2 1 +# You can use scientific notation too +5e4 # 50000 +6.02e23 # Avogadro's number +1.6e-35 # Planck length +# You can also have infinitely large or small numbers class(Inf) # "numeric" -class(-Inf) # "numeric" #used in for example integrate( dnorm(x), 3, Inf ) -- which obviates Z-score tables - -# but beware, NaN isn't the only weird type... -class(NA) # see below -class(NULL) # NULL - - -# SIMPLE LISTS -c(6, 8, 7, 5, 3, 0, 9) # 6 8 7 5 3 0 9 -c('alef', 'bet', 'gimmel', 'dalet', 'he') # "alef" "bet" "gimmel" "dalet" "he" -c('Z', 'o', 'r', 'o') == "Zoro" # FALSE FALSE FALSE FALSE - -#some more nice built-ins -5:15 # 5 6 7 8 9 10 11 12 13 14 15 - -seq(from=0, to=31337, by=1337) -# [1] 0 1337 2674 4011 5348 6685 8022 9359 10696 12033 13370 14707 -# [13] 16044 17381 18718 20055 21392 22729 24066 25403 26740 28077 29414 30751 - -letters -# [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j" "k" "l" "m" "n" "o" "p" "q" "r" "s" -# [20] "t" "u" "v" "w" "x" "y" "z" - -month.abb # "Jan" "Feb" "Mar" "Apr" "May" "Jun" "Jul" "Aug" "Sep" "Oct" "Nov" "Dec" - - -# Access the n'th element of a list with list.name[n] or sometimes list.name[[n]] -letters[18] # "r" -LETTERS[13] # "M" -month.name[9] # "September" -c(6, 8, 7, 5, 3, 0, 9)[3] # 7 - - +class(-Inf) # "numeric" +# You might use "Inf", for example, in integrate( dnorm(x), 3, Inf); +# this obviates Z-score tables. + +# BASIC ARITHMETIC +# You can do arithmetic with numbers +# Doing arithmetic on a mix of integers and numerics gives you another numeric +10L + 66L # 76 # integer plus integer gives integer +53.2 - 4 # 49.2 # numeric minus numeric gives numeric +2.0 * 2L # 4 # numeric times integer gives numeric +3L / 4 # 0.75 # integer over numeric gives numeric +3 %% 2 # 1 # the remainder of two numerics is another numeric +# Illegal arithmetic yeilds you a "not-a-number": +0 / 0 # NaN +class(NaN) # "numeric" +# You can do arithmetic on two vectors with length greater than 1, +# so long as the larger vector's length is an integer multiple of the smaller +c(1,2,3) + c(1,2,3) # 2 4 6 # CHARACTERS - # There's no difference between strings and characters in R - -"Horatio" # "Horatio" +"Horatio" # "Horatio" class("Horatio") # "character" -substr("Fortuna multis dat nimis, nulli satis.", 9, 15) # "multis " -gsub('u', 'ø', "Fortuna multis dat nimis, nulli satis.") # "Fortøna møltis dat nimis, nølli satis." - - +class('H') # "character" +# Those were both character vectors of length 1 +# Here is a longer one: +c('alef', 'bet', 'gimmel', 'dalet', 'he') +# => +# "alef" "bet" "gimmel" "dalet" "he" +length(c("Call","me","Ishmael")) # 3 +# You can do regex operations on character vectors: +substr("Fortuna multis dat nimis, nulli satis.", 9, 15) # "multis " +gsub('u', 'ø', "Fortuna multis dat nimis, nulli satis.") # "Fortøna møltis dat nimis, nølli satis." +# R has several built-in character vectors: +letters +# => +# [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j" "k" "l" "m" "n" "o" "p" "q" "r" "s" +# [20] "t" "u" "v" "w" "x" "y" "z" +month.abb # "Jan" "Feb" "Mar" "Apr" "May" "Jun" "Jul" "Aug" "Sep" "Oct" "Nov" "Dec" # LOGICALS - -# booleans +# In R, a "logical" is a boolean class(TRUE) # "logical" class(FALSE) # "logical" -# Behavior is normal +# Their behavior is normal TRUE == TRUE # TRUE TRUE == FALSE # FALSE FALSE != FALSE # FALSE FALSE != TRUE # TRUE # Missing data (NA) is logical, too class(NA) # "logical" - - +# Here we get a logical vector with many elements: +c('Z', 'o', 'r', 'r', 'o') == "Zorro" # FALSE FALSE FALSE FALSE FALSE +c('Z', 'o', 'r', 'r', 'o') == "Z" # TRUE FALSE FALSE FALSE FALSE # FACTORS - # The factor class is for categorical data -# which can be ordered (like childrens' grade levels) -# or unordered (like gender) -levels(factor(c("female", "male", "male", "female", "NA", "female"))) # "female" "male" "NA" - +# Factors can be ordered (like childrens' grade levels) or unordered (like gender) factor(c("female", "female", "male", "NA", "female")) # female female male NA female # Levels: female male NA +# The "levels" are the values the categorical data can take +levels(factor(c("male", "male", "female", "NA", "female"))) # "female" "male" "NA" +# If a factor vector has length 1, its levels will have length 1, too +length(factor("male")) # 1 +length(levels(factor("male"))) # 1 +# Factors are commonly seen in data frames, a data structure we will cover later +data(infert) # "Infertility after Spontaneous and Induced Abortion" +levels(infert$education) # "0-5yrs" "6-11yrs" "12+ yrs" + +# NULL +# "NULL" is a weird one; use it to "blank out" a vector +class(NULL) # NULL +parakeet +# => +# [1] "beak" "feathers" "wings" "eyes" +parakeet <- NULL +parakeet +# => +# NULL + +# TYPE COERCION +# Type-coercion is when you force a value to take on a different type +as.character(c(6, 8)) # "6" "8" +as.logical(c(1,0,1,1)) # TRUE FALSE TRUE TRUE +# If you put elements of different types into a vector, weird coercions happen: +c(TRUE, 4) # 1 4 +c("dog", TRUE, 4) # "dog" "TRUE" "4" +as.numeric("Bilbo") +# => +# [1] NA +# Warning message: +# NAs introduced by coercion -data(infert) #Infertility after Spontaneous and Induced Abortion -levels(infert$education) # "0-5yrs" "6-11yrs" "12+ yrs" +# Also note: those were just the basic data types +# There are many more data types, such as for dates, time series, etc. -# VARIABLES +################################################## +# Variables, loops, if/else +################################################## -# Lots of way to assign stuff +# A variable is like a box you store a value in for later use. +# We call this "assigning" the value to the variable. +# Having variables lets us write loops, functions, and if/else statements + +# VARIABLES +# Lots of way to assign stuff: x = 5 # this is possible y <- "1" # this is preferred TRUE -> z # this works but is weird -# We can use coerce variables to different classes -as.numeric(y) # 1 -as.character(x) # "5" - # LOOPS - # We've got for loops for (i in 1:4) { print(i) } - # We've got while loops a <- 10 while (a > 4) { cat(a, "...", sep = "") a <- a - 1 } - # Keep in mind that for and while loops run slowly in R # Operations on entire vectors (i.e. a whole row, a whole column) # or apply()-type functions (we'll discuss later) are preferred # IF/ELSE - # Again, pretty standard if (4 > 3) { - print("Huzzah! It worked!") + print("4 is greater than 3") } else { - print("Noooo! This is blatantly illogical!") + print("4 is not greater than 3") } # => -# [1] "Huzzah! It worked!" +# [1] "4 is greater than 3" # FUNCTIONS - # Defined like so: jiggle <- function(x) { x = x + rnorm(1, sd=.1) #add in a bit of (controlled) noise return(x) } - # Called like any other R function: jiggle(5) # 5±ε. After set.seed(2716057), jiggle(5)==5.005043 -######################### -# Fun with data: vectors, matrices, data frames, and arrays -######################### + + +########################################################################### +# Data structures: Vectors, matrices, data frames, and arrays +########################################################################### # ONE-DIMENSIONAL -# You can vectorize anything, so long as all components have the same type +# Let's start from the very beginning, and with something you already know: vectors. vec <- c(8, 9, 10, 11) vec # 8 9 10 11 -# The class of a vector is the class of its components -class(vec) # "numeric" -# If you vectorize items of different classes, weird coercions happen -c(TRUE, 4) # 1 4 -c("dog", TRUE, 4) # "dog" "TRUE" "4" - -# We ask for specific components like so (R starts counting from 1) -vec[1] # 8 +# We ask for specific elements by subsetting with square brackets +# (Note that R starts counting from 1) +vec[1] # 8 +letters[18] # "r" +LETTERS[13] # "M" +month.name[9] # "September" +c(6, 8, 7, 5, 3, 0, 9)[3] # 7 # We can also search for the indices of specific components, which(vec %% 2 == 0) # 1 3 -# or grab just the first or last entry in the vector +# grab just the first or last few entries in the vector, head(vec, 1) # 8 -tail(vec, 1) # 11 +tail(vec, w) # 10 11 +# or figure out if a certain value is in the vector +any(vec == 10) # TRUE # If an index "goes over" you'll get NA: vec[6] # NA # You can find the length of your vector with length() length(vec) # 4 - # You can perform operations on entire vectors or subsets of vectors vec * 4 # 16 20 24 28 vec[2:3] * 5 # 25 30 -# and there are many built-in functions to summarize vectors +any(vec[2:3] == 8) # FALSE +# and R has many built-in functions to summarize vectors mean(vec) # 9.5 var(vec) # 1.666667 -sd(vec) # 1.290994 +sd(vec) # 1.290994 max(vec) # 11 min(vec) # 8 sum(vec) # 38 +# Some more nice built-ins: +5:15 # 5 6 7 8 9 10 11 12 13 14 15 +seq(from=0, to=31337, by=1337) +# => +# [1] 0 1337 2674 4011 5348 6685 8022 9359 10696 12033 13370 14707 +# [13] 16044 17381 18718 20055 21392 22729 24066 25403 26740 28077 29414 30751 # TWO-DIMENSIONAL (ALL ONE CLASS) @@ -361,6 +390,7 @@ mat[1,] # 1 4 3 * mat[,1] # 3 6 9 # Ask for a specific cell mat[3,2] # 6 + # Transpose the whole matrix t(mat) # => @@ -368,6 +398,14 @@ t(mat) # [1,] 1 2 3 # [2,] 4 5 6 +# Matrix multiplication +mat %*% t(mat) +# => +# [,1] [,2] [,3] +# [1,] 17 22 27 +# [2,] 22 29 36 +# [3,] 27 36 45 + # cbind() sticks vectors together column-wise to make a matrix mat2 <- cbind(1:4, c("dog", "cat", "bird", "dog")) mat2 @@ -390,33 +428,138 @@ mat3 # [,1] [,2] [,3] [,4] # [1,] 1 2 4 5 # [2,] 6 7 0 4 -# Aah, everything of the same class. No coercions. Much better. +# Ah, everything of the same class. No coercions. Much better. # TWO-DIMENSIONAL (DIFFERENT CLASSES) -# For columns of different classes, use the data frame -dat <- data.frame(c(5,2,1,4), c("dog", "cat", "bird", "dog")) -names(dat) <- c("number", "species") # name the columns -class(dat) # "data.frame" -dat +# For columns of different types, use a data frame +# This data structure is so useful for statistical programming, +# a version of it was added to Python in the package "pandas". + +students <- data.frame(c("Cedric","Fred","George","Cho","Draco","Ginny"), + c(3,2,2,1,0,-1), + c("H", "G", "G", "R", "S", "G")) +names(students) <- c("name", "year", "house") # name the columns +class(students) # "data.frame" +students # => -# number species -# 1 5 dog -# 2 2 cat -# 3 1 bird -# 4 4 dog -class(dat$number) # "numeric" -class(dat[,2]) # "factor" +# name year house +# 1 Cedric 3 H +# 2 Fred 2 G +# 3 George 2 G +# 4 Cho 1 R +# 5 Draco 0 S +# 6 Ginny -1 G +class(students$year) # "numeric" +class(students[,3]) # "factor" +# find the dimensions +nrow(students) # 6 +ncol(students) # 3 +dim(students) # 6 3 # The data.frame() function converts character vectors to factor vectors +# by default; turn this off by setting stringsAsFactors = FALSE when +# you create the data.frame +?data.frame # There are many twisty ways to subset data frames, all subtly unalike -dat$number # 5 2 1 4 -dat[,1] # 5 2 1 4 -dat[,"number"] # 5 2 1 4 +students$year # 3 2 2 1 0 -1 +students[,2] # 3 2 2 1 0 -1 +students[,"year"] # 3 2 2 1 0 -1 + +# An augmented version of the data.frame structure is the data.table +# If you're working with huge or panel data, or need to merge a few data +# sets, data.table can be a good choice. Here's a whirlwind tour: +install.packages("data.table") # download the package from CRAN +require(data.table) # load it +students <- as.data.table(students) +students # note the slightly different print-out +# => +# name year house +# 1: Cedric 3 H +# 2: Fred 2 G +# 3: George 2 G +# 4: Cho 1 R +# 5: Draco 0 S +# 6: Ginny -1 G +students[name=="Ginny"] # get rows with name == "Ginny" +# => +# name year house +# 1: Ginny -1 G +students[year==2] # get rows with year == 2 +# => +# name year house +# 1: Fred 2 G +# 2: George 2 G +# data.table makes merging two data sets easy +# let's make another data.table to merge with students +founders <- data.table(house=c("G","H","R","S"), + founder=c("Godric","Helga","Rowena","Salazar")) +founders +# => +# house founder +# 1: G Godric +# 2: H Helga +# 3: R Rowena +# 4: S Salazar +setkey(students, house) +setkey(founders, house) +students <- founders[students] # merge the two data sets by matching "house" +setnames(students, c("house","houseFounderName","studentName","year")) +students[,order(c("name","year","house","houseFounderName")), with=F] +# => +# studentName year house houseFounderName +# 1: Fred 2 G Godric +# 2: George 2 G Godric +# 3: Ginny -1 G Godric +# 4: Cedric 3 H Helga +# 5: Cho 1 R Rowena +# 6: Draco 0 S Salazar + +# data.table makes summary tables easy +# => +# students[,sum(year),by=house] +# house V1 +# 1: G 3 +# 2: H 3 +# 3: R 1 +# 4: S 0 + +# To drop a column from a data.frame or data.table, +# assign it the NULL value +students$houseFounderName <- NULL +students +# => +# studentName year house +# 1: Fred 2 G +# 2: George 2 G +# 3: Ginny -1 G +# 4: Cedric 3 H +# 5: Cho 1 R +# 6: Draco 0 S + +# Drop a row by subsetting +# Using data.table: +students[studentName != "Draco"] +# => +# house studentName year +# 1: G Fred 2 +# 2: G George 2 +# 3: G Ginny -1 +# 4: H Cedric 3 +# 5: R Cho 1 +# Using data.frame: +students <- as.data.frame(students) +students[students$house != "G",] +# => +# house houseFounderName studentName year +# 4 H Helga Cedric 3 +# 5 R Rowena Cho 1 +# 6 S Salazar Draco 0 -# MULTI-DIMENSIONAL (ALL OF ONE CLASS) +# MULTI-DIMENSIONAL (ALL ELEMENTS OF ONE TYPE) # Arrays creates n-dimensional tables +# All elements must be of the same type # You can make a two-dimensional table (sort of like a matrix) array(c(c(1,2,4,5),c(8,9,3,6)), dim=c(2,4)) # => @@ -446,15 +589,23 @@ array(c(c(c(2,300,4),c(8,9,0)),c(c(5,60,0),c(66,7,847))), dim=c(3,2,2)) list1 <- list(time = 1:40) list1$price = c(rnorm(40,.5*list1$time,4)) # random list1 - # You can get items in the list like so -list1$time -# You can subset list items like vectors +list1$time # one way +list1[["time"]] # another way +list1[[1]] # yet another way +# => +# [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 +# [34] 34 35 36 37 38 39 40 +# You can subset list items like any other vector list1$price[4] -######################### +# Lists are not the most efficient data structure to work with in R; +# unless you have a very good reason, you should stick to data.frames +# Lists are often returned by functions that perform linear regressions + +################################################## # The apply() family of functions -######################### +################################################## # Remember mat? mat @@ -467,7 +618,7 @@ mat # over rows (MAR = 1) or columns (MAR = 2) # That is, R does FUN to each row (or column) of X, much faster than a # for or while loop would do -apply(mat, MAR = 2, myFunc) +apply(mat, MAR = 2, jiggle) # => # [,1] [,2] # [1,] 3 15 @@ -478,16 +629,18 @@ apply(mat, MAR = 2, myFunc) # Don't feel too intimidated; everyone agrees they are rather confusing # The plyr package aims to replace (and improve upon!) the *apply() family. - install.packages("plyr") require(plyr) ?plyr + + ######################### # Loading data ######################### # "pets.csv" is a file on the internet +# (but it could just as easily be be a file on your own computer) pets <- read.csv("http://learnxinyminutes.com/docs/pets.csv") pets head(pets, 2) # first two rows @@ -499,10 +652,13 @@ write.csv(pets, "pets2.csv") # to make a new .csv file # Try ?read.csv and ?write.csv for more information + + ######################### # Plots ######################### +# BUILT-IN PLOTTING FUNCTIONS # Scatterplots! plot(list1$time, list1$price, main = "fake data") # Regressions! @@ -512,18 +668,25 @@ linearModel # outputs result of regression abline(linearModel, col = "red") # Get a variety of nice diagnostics plot(linearModel) - # Histograms! hist(rpois(n = 10000, lambda = 5), col = "thistle") - # Barplots! barplot(c(1,4,5,1,2), names.arg = c("red","blue","purple","green","yellow")) +# GGPLOT2 +# But these are not even the prettiest of R's plots # Try the ggplot2 package for more and better graphics - install.packages("ggplot2") require(ggplot2) ?ggplot2 +pp <- ggplot(students, aes(x=house)) +pp + geom_histogram() +ll <- as.data.table(list1) +pp <- ggplot(ll, aes(x=time,price)) +pp + geom_point() +# ggplot2 has excellent documentation (available http://docs.ggplot2.org/current/) + + ``` |