The letter of application is intended to provide detailed information on why you are are a qualified candidate for the job. A character vector of the concatenated values. be applied over. FUN is found by a call to match.fun and typically MARGIN: A numeric vector indicating the dimension over which to traverse; 1 means rows and 2 means columns.. FUN: The function to apply (for example, sum or mean). You can create a function like this for any apply function, not just tapply. As you can see, this didn’t work because apply was expecting the data to have at least two dimensions. First, let’s go over the basic apply function. Here are the agruments for the three functions: In this case, X is a vector or list, and FUN is the function you want to use. If your data is a vector you need to use lapply, sapply, or vapply instead. Where X has named dimnames, it can be a character Count in R using the apply function Imagine you counted the birds in your backyard on three different days and stored the counts in a matrix like this: Why? In the case of functions like +, %*%, etc., the I’ve been on r/a2c since I was a freshman; this has probably affected my mental health in the long run, but I’ve always loved this community.. One thing, however, that I was not a fan of was … TL;DR at bottom. For example, using dataset t, I could divide one column by another column to create a new value. If you know me IRL: no, you don’t. There is a part 2 coming that will look at density plots with ggplot, but first I thought I would go on a tangent to give some examples of the apply family, as they come up a lot working with R. I am expecting each item in the list to return a single numeric value, so FUN.VALUE = numeric(1). Another use for mapply would be to create a new variable. Practical advice for writing a cover letter. Apply a Function over a List or Vector. I created a numeric vector of length 10 using the vector function. The articles on the left provide an introduction to R for people who are … This will be of length zero if all the objects are, unless collapse is non-NULL in which case it is a single empty string.. One thing, however, that I was not a fan of was the astronomically high GPAs around every corner. the arguments for mapply are mapply(FUN, …, MoreArgs = NULL, SIMPLIFY = TRUE, USE.NAMES = TRUE). The purpose of apply() is primarily to avoid explicit uses of loop constructs. The apply functions that this chapter will address are apply, lapply, sapply, vapply, tapply, and mapply. Use this form to apply for the Paycheck Protection Program (PPP) with an eligible lender for a First Draw loan The results of the mapply function are then saved into the vector. Then I saved them as objects that could be used later. Monster staff. The Basics of R (Ch 2 – 5) This section presents an orientation to using R. Chapter 2 introduces the R This post will show you how you can use the R apply() function, its variants such as mapply() and a few of apply()'s relatives, applied to different data structures. 586 Main St. Brighton, TX 45965. the apply function looks like this: apply(X, MARGIN, FUN). If your function were to return more than one numeric value, FUN.VALUE = numeric(1) will cause the function to return an error. Welcome. You can use the help section to get a description of this function. R has a large number of in-built functions and the user can create their own functions. If you set the MARGIN to 1:2 it will have the function operate on each cell. To call a function for each row in an R data frame, we shall use R apply function. The arguments for tapply are tapply(X, INDEX, FUN). dim set to MARGIN if this has length greater than one. It relies on forking and hence is not available on Windows unless mc.cores = 1. mcmapply is a parallelized version of mapply, and mcMap corresponds to Map. This is because lapply applies treats the vector like a list, and applies the function to each point in the vector. This last section will be a few examples of using apply functions on real data.This section will make use of the MASS package, which is a collection of publicly available datasets. sapply works just like lapply, but will simplify the output if possible. This may be useful if you want to have the function available to use later. a vector giving the subscripts which the function will Arguments are recycled if necessary. In the above function calls, the argument matching of formal argument to the actual arguments takes place in positional order. Now, let’s present a conceptual overview of the organization of the book. state.region is a factor with four levels: Northeast, South, North Central, and West. 2 indicates columns, c(1, 2) indicates rows and The apply command or rather family of commands, pertains to the R base package. This function didn’t add up the values like we may have expected it to. There are so many different apply functions because they are meant to operate on different types of data. I’ve been on r/a2c since I was a freshman; this has probably affected my mental health in the long run, but I’ve always loved this community. Using the apply family makes sense only if you need that result. If you do not want your results to be simplified to a vector, lapply should be used. The only new argument is INDEX, which is the factor you want to use to separate the data. If X is not an array but an object of a class with a non-null my.matrx is a matrix with 1-10 in column 1, 11-20 in column 2, and 21-30 in column 3. my.matrx will be used to show some of the basic uses for the apply function. Many functions in R work in a vectorized way, so there’s often no need to use this. state.area and state.x77 are not from the same dataset, but that is fine as long as the vectors are the same length and the data is in the same order. If you do not have MASS installed, you can uncomment the code below. Slam the brakes! Spark objects are partitioned so they can be distributed across a cluster. spark_apply() applies an R function to a Spark object (typically, a Spark DataFrame). Usage mapply(FUN, ..., MoreArgs = NULL, SIMPLIFY = TRUE, USE.NAMES = TRUE) through: this both avoids partial matching to MARGIN If n is 0, the result has length 0 but not necessarily the ‘correct’ dimension.. Following is an example R Script to demonstrate how to apply a function for each row in an R Data Frame. character string specifying a function to be searched for from the A letter of application, also known as a cover letter, is a document sent with your resume to provide additional information about your skills and experience to an employer. Apply is the head of the family. If you are trying to decide which of these three functions to use, because it is the simplest, I would suggest to use sapply if possible. tapply(X, INDEX, FUN = NULL,..., simplify = TRUE) This example uses the builtin dataset CO2, sum up the uptake grouped by different plants. Named Arguments. Here are the available R apply functions: apply, lapply, sapply, vapply, mapply, rapply and tapply. They want a cover letter. Otherwise x can be any R object for which length and subsetting by integers make sense: S3 or S4 methods for these operations will be dispatched as appropriate. columns. In this example, the apply function is used to transform the values in each cell. This is an important idiom for writing code in R, and it usually goes by the name Split, Apply, and Combine (SAC). 24. Acme Investments, Inc. Attn: Thomas Burgin. In this example, I want to find the population density for each state. Pay attention to the MARGIN argument. … The apply() collection is bundled with r essential package if you install R with Anaconda. Dataset t will be created by adding a factor to matrix m and converting it to a dataframe. Mr. Burgin, I write to apply for the Office Manager position at Acme Investments, Inc. mapply is a multivariate version of sapply. sparklyr provides support to run arbitrary R code at scale within your Spark Cluster through spark_apply(). Meet three of the members. This means that instead of returning a list like lapply, it will return a vector instead if the data is simplifiable. Sometimes you may want to perform the apply function on some data, but have it separated by factor. MARGIN or FUN. lapply is probably a better choice than apply here, as apply first coerces your data.frame to an array which means all the columns must have the same type. function to margins of an array or matrix. You can use tapply to do some quick summary statistics on a variable split by condition. Say hello to apply(), sapply(), and lapply(), the most used members of the apply family. What is a Job Application Letter? The Apply Functions As Alternatives To Loops. of the basic vector types before the dimensions are set, so that (for mapply applies FUN to the first elements of each ... argument, the second elements, the third elements, and so on. So what is the apply function in R? Now let’s use column 1 as the index and find the mean of column 2. If you run this function it will return the error: Error in apply(v, 1, sum) : dim(X) must have a positive length. 4634 W. Industrial Dr., Ste. If you want to print messages to the console with print() or cat() for example, using the apply family is unnecessary. The articles on the left provide an introduction to R for people who are … I can use the length function to do this. tapply, and convenience functions lapply returns a list of the same length as X, each element of which is the result of applying FUN to the corresponding element of X.. sapply is a user-friendly version and wrapper of lapply by default returning a vector, matrix or, if simplify = "array", an array if appropriate, by applying simplify2array(). environment of the call to apply. If n is 0, the result has length 0 but not necessarily Apply functions are a family of functions in base R which allow you to repetitively perform an action on multiple chunks of data. load the state dataset. In the arguments I created a function that returns length - 1. The apply() function can be feed with many functions to perform redundant application on a collection of object (data frame, list, vector, etc.). What if I wanted to summarize the data in matrix m by finding the sum of each row? Count in R using the apply function. First, let’s create data with an factor for indexing. This is an introductory post about using apply, sapply and lapply, best suited for people relatively new to R or unfamiliar with these functions. apply returns a list of length prod(dim(X)[MARGIN]) with First, try looking up lapply in the help section to see a description of all three function. What if instead, I wanted to find n-1 for each column? mclapply is a parallelized version of lapply, it returns a list of the same length as X, each element of which is the result of applying FUN to the corresponding element of X.. Example: “I will update my resume with relevant qualifications, so I can apply to three open positions for the manager of a development team at a tech startup.” R = Relevant When setting goals for yourself, consider whether or not they are relevant. This order is based on the order of arguments in the rep function itself. What if I wanted to be able to find how many datapoints (n) are in each column of m? Earlier, we created the vector v. Let’s use that vector to test out the lapply function. See how these two examples gave the same answers, but returned a vector instead? This is especially useful where there is a need to use functionality available only in R or R packages that is not available in Apache Spark nor Spark Packages. Because we are using columns, MARGIN = 2. Phone: (555) 555-1212. Arguments are recycled if necessary. I have a function f(var1, var2) in R. Suppose we set var2 = 1 and now I want to apply the function f() to the list L. Basically I want to get a new list L* with the outputs [f(L[1],1),f(L[2],1),.... Stack Overflow. You can use spark_apply with the default partitions or you can define your own partitions with the group… to coerce it to an array via as.matrix if it is two-dimensional sapply and vapply have extra arguments, but most of them have default values, so you don’t need to worry about them. Will the apply function work? R Examples. Wadsworth & Brooks/Cole. More Examples How to run the code Finding data sources. Returns a vector or array or list of values obtained by applying a In order to do this, I want to divide population by area. It contains information about all 50 states, Let’s look at the data we will be using. through …. If you want to print messages to the console with print() or cat() for example, using the apply family is unnecessary. (m = matrix (1: 6, nrow = 2)) apply (m, 1, sum) apply (m, 1: 2, sqrt) # "sweep" returns an array obtained from an input array by sweeping out # a summary statistic. the ‘correct’ dimension. This presents some very handy opportunities. or FUN and ensures that a sensible error message is given if If each call to FUN returns a vector of length n, then apply returns an array of dimension c(n, dim(X)[MARGIN]) if n > 1.If n equals 1, apply returns a vector if MARGIN has length 1 and an array of dimension dim(X)[MARGIN] otherwise. Houston, TX 45987. Apply a Function to Multiple List or Vector Arguments. example) factor results will be coerced to a character array. In the previous examples, apply was used to summarize over a row or column. Email: hwaybird@email.com. R apply function with multiple parameters - Stack Overflow. In R, you can use the apply () function to apply a function over every row or column of a matrix or data frame. The arguments for the vector function are vector(mode, length). Please install MASS if you do not already have it. Welcome. The arguments are X = m, MARGIN = 1 (for row), and FUN = sum. An apply function is essentially a loop, but run faster than loops and often require less code. Basics Functions Countdown User input Random number game Lists Reading data Filtering data. Say you wanted to simulate rolls of a die, and you want to get ten results. Of course, using the with() function, you can write your line of … I read Data from a csv file. Of course, not all the variants can be discussed, but when possible, you will be introduced to the use of these functions in cooperation, via a couple of slightly more beefy examples. Inside mapply I created a function to multiple two variables together. Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) Here are some sources I used to help me create this chapter: Datacamp tutorial on apply functions: https://www.datacamp.com/community/tutorials/r-tutorial-apply-family, r-bloggers: Using apply, sapply, and lapply in R: https://www.r-bloggers.com/using-apply-sapply-lapply-in-r/, stackoverflow: Why is vapply safer than sapply? Depending on your context, this could have unintended consequences. arguments named X, MARGIN or FUN are passed To do so, you make use of sample(), which takes a vector as input; then you tell it how many samples to draw from that list. For sample the default for size is the number of items inferred from the first argument, so that sample(x) generates a random permutation of the elements of x (or 1:x). tapply()applies a function to each cell of a ragged array, that is to each (non-empty) group of values given by a unique combination of the levels of certain factors. Harold Waybird. If I see this file in R, I have: V1 V2 V3 V4 V5 V6 V7 1 14 25 83 64 987 45 78 2 15 65 789 32 14 NA NA 3 14 67 89 14 NA NA NA If I want the maximum value in each column, I use this: apply(df,2,max) and this is the result: V1 V2 V3 V4 V5 V6 V7 15 67 789 64 NA NA NA In this example, 1:9 is specifying the value to repeat, and 9:1 is specifying how many times to repeat. In this example, a function to find standard error was created, then passed into an apply function. Consider an example: If a data frame has 4 columns out of which the first one belongs to the character class, then use below code: apply(Data.df[,2:4],2,func_name) In this example, I want to find out some information about the population of states split by region. dim(X)[MARGIN] otherwise. If each call to FUN returns a vector of length n, then Sample Letter of Application. Consider the following basic example: > sapply (c ('a','b'), switch, a='Hello', b='Goodbye') a b "Hello" "Goodbye". Imagine you counted the birds in your backyard on three different days and stored the counts in a matrix like this: > counts <- matrix (c (3,2,4,6,5,1,8,6,1), ncol=3) > colnames (counts) <- c ('sparrow','dove','crow') > counts sparrow dove crow [1,] 3 6 8 [2,] 2 5 6 [3,] 4 1 1. If you don’t want to write a function inside of the arguments, you can define the function outside of apply, and then use that function in apply later. First you list the function, followed by the vectors you are using the rest of the arguments have default values so they don’t need to be changed for now. In this case, you split a vector into groups, apply a function to each group, and then combine the result into a vector. Value. Apply a Function to Multiple List or Vector Arguments Description. > tapply(CO2$uptake,CO2$Plant, sum) function name must be backquoted or quoted. It can also be used to repeat a function on cells within a matrix. E.g., for a matrix 1 indicates rows, Using the apply family makes sense only if you need that result. Apply functions are a family of functions in base R which allow you to repetitively perform an action on multiple chunks of data. As you can see, the function correctly returned a vector of n-1 for each column. The apply() function takes four arguments:. Like apply, these functions can also be used for transforming data inside the list. lapply and there, simplify2array; Say hello to apply(), sapply(), and lapply(), the most used members of the apply family. Hi guys! In a previous post, you covered part of the R language control flow, the cycles or loop structures.In a subsequent one, you learned more about how to avoid looping by using the apply() family of functions, which act on compound data in repetitive ways. You can use apply to find measures of central tendency and dispersion. (e.g., a data frame) or via as.array. Apply operates on arrays: apply(X, MARGIN, FUN, …). Welcome. mapply is a multivariate version of sapply. When using an apply family function to create a new variable, one option is to create a new vector ahead of time with the size of the vector pre-allocated. : http://stackoverflow.com/questions/12339650/why-is-vapply-safer-than-sapply, ---
title: 'Chapter 4: apply Functions'
author: "Erin Sovansky Winter"
output:
  html_document:
    theme: cerulean
    highlight: textmate
    fontsize: 8pt
    toc: true
    number_sections: true
    code_download: true
    toc_float:
      collapsed: false
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

#  What are apply functions?
Apply functions are a family of functions in base R which allow you to repetitively perform an
action on multiple chunks of data. An apply function is essentially a loop, but run faster than 
loops and often require less code. 

The apply functions that this chapter will address are apply, lapply, sapply, vapply, tapply, and
mapply. There are so many different apply functions because they are meant to operate on different
types of data. 

#  The apply function
First, let's go over the basic apply function. You can use the help section to get a description
of this function.
```{r, eval=FALSE}
?apply
```
the apply function looks like this: apply(X, MARGIN, FUN). 

* X is an array or matrix (this is the data that you will be performing the function on)
* Margin specifies whether you want to apply the function across rows (1) or columns (2)
* FUN is the function you want to use

## apply examples
my.matrx is a matrix with 1-10 in column 1, 11-20 in column 2, and 21-30 in column 3. 
my.matrx will be used to show some of the basic uses for the apply function.
```{r}
my.matrx <- matrix(c(1:10, 11:20, 21:30), nrow = 10, ncol = 3)
my.matrx
```

### Example 1: Using apply to find row sums
What if I wanted to summarize the data in matrix m by finding the sum of each row? The arguments 
are X = m, MARGIN = 1 (for row), and FUN = sum

```{r}
apply(my.matrx, 1, sum)
```
The apply function returned a vector containing the sums for each row.

### Example 2: Creating a function in the arguments
What if I wanted to be able to find how many datapoints (n) are in each column of m? I can use 
the length function to do this. Because we are using columns, MARGIN = 2.
```{r}
apply(my.matrx, 2, length)
```
What if instead, I wanted to find n-1 for each column? There isn't a function in R to do this
automatically, so I can create my own function. If the function is simple, you can create it
right inside the arguments for apply. In the arguments I created a function that returns
length - 1.
```{r}
apply(my.matrx, 2, function (x) length(x)-1)
```
As you can see, the function correctly returned a vector of n-1 for each column.
 
### Example 3: Using a function defined outside of apply
If you don't want to write a function inside of the arguments, you can define the function 
outside of apply, and then use that function in apply later. This may be useful if you want to 
have the function available to use later. In this example, a function to find standard error was
created, then passed into an apply function.
```{r}
st.err <- function(x){
  sd(x)/sqrt(length(x))
}
apply(my.matrx,2, st.err)
```

### Example 4: Transforming data
Now for something a little different. In the previous examples, apply was used to summarize
over a row or column. It can also be used to repeat a function on cells within a matrix. In this
example, the apply function is used to transform the values in each cell. Pay attention to the
MARGIN argument. If you set the MARGIN to 1:2 it will have the function operate on each cell.
```{r}
my.matrx2 <- apply(my.matrx,1:2, function(x) x+3)
my.matrx2
```

### Example 5: Vectors?
The previous examples showed several ways to use the apply function on a matrix. But what if I 
wanted to loop through a vector instead? Will the apply function work?

```{r, }
vec <- c(1:10)
vec
```
```{r, eval=FALSE}
apply(vec, 1, sum)
```
If you run this function it will return the error: Error in apply(v, 1, sum) : dim(X) must have a positive length. 
As you can see, this didn't work because apply was expecting the data to have at least two dimensions. If your data is a vector you need to use lapply, sapply, or vapply instead.

# lapply, sapply, and vapply
lapply, sapply, and vapply are all functions that will loop a function through data in a list or
vector. First, try looking up lapply in the help section to see a description of all three 
function.

```{r, eval=FALSE}
?lapply
```

Here are the agruments for the three functions:

* lapply(X, FUN, ...)
* sapply(X, FUN, ..., simplify = TRUE, USE.NAMES = TRUE)
* vapply(X, FUN, FUN.VALUE, ..., USE.NAMES = TRUE)

In this case, X is a vector or list, and FUN is the function you want to use. sapply and vapply have extra arguments, but most of them have default values, so you don't need to worry about
them. However, vapply requires another agrument called FUN.VALUE, which we will look at later.

### Example 1: Getting started with lapply
Earlier, we created the vector v. Let's use that vector to test out the lapply function.
```{r}
lapply(vec, sum)
```
This function didn't add up the values like we may have expected it to. This is because lapply
applies treats the vector like a list, and applies the function to each point in the vector.

Let's try using a list instead
```{r}
A<-c(1:9)
B<-c(1:12)
C<-c(1:15)
my.lst<-list(A,B,C)
lapply(my.lst, sum)
```
This time, the lapply function seemed to work better. The function summed each vector in the list
and returned a list of the 3 sums. 

### Example 2: sapply
sapply works just like lapply, but will simplify the output if possible. This means that instead
of returning a list like lapply, it will return a vector instead if the data is simplifiable.

```{r}
sapply(vec, sum)
```

```{r}
sapply(my.lst, sum)
```
See how these two examples gave the same answers, but returned a vector instead?

### Example 3: vapply
vapply is similar to sapply, but it requires you to specify what type of data you are expecting
the arguments for vapply are vapply(X, FUN, FUN.VALUE).
FUN.VALUE is where you specify the type of data you are expecting.
I am expecting each item in the list to return a single numeric value, so FUN.VALUE = numeric(1).

```{r}
vapply(vec, sum, numeric(1))
```

```{r}
vapply(my.lst, sum, numeric(1))
```

If your function were to return more than one numeric value, FUN.VALUE = numeric(1) will cause the function to return an error. This could be useful if you are expecting only one result per subject. 
```{r}
#vapply(my.lst, function(x) x+2, numeric(1))
```

### Example 4: Transforming data with sapply
Like apply, these functions can also be used for transforming data inside the list
```{r}
my.lst2 <- sapply(my.lst, function(x) x*2)
my.lst2
```

### Which function should I use, lapply, sapply, or vapply?

If you are trying to decide which of these three functions to use, because it is the simplest, I would suggest to use sapply if possible. If you do not want your results to be simplified to a vector, lapply should be used. If you want to specify the type of result you are expecting, use vapply.


# tapply

Sometimes you may want to perform the apply function on some data, but have it separated by 
factor. In that case, you should use tapply. Let's take a look at the information for tapply.

```{r, eval=FALSE}
?tapply
```
The arguments for tapply are tapply(X, INDEX, FUN). The only new argument is INDEX, which is the 
factor you want to use to separate the data.

### Example 1: Means split by condition
First, let's create data with an factor for indexing. Dataset t will be created by adding a factor to matrix m and converting it to a dataframe. 

```{r}
tdata <- as.data.frame(cbind(c(1,1,1,1,1,2,2,2,2,2), my.matrx))
colnames(tdata)
```
Now let's use column 1 as the index and find the mean of column 2

```{r}
tapply(tdata$V2, tdata$V1, mean)
```

### Example 2: Combining functions
You can use tapply to do some quick summary statistics on a variable split by condition. In this 
example, I created a function that returns a vector ofboth the mean and standard deviation. You 
can create a function like this for any apply function, not just tapply.
```{r}
summary <- tapply(tdata$V2, tdata$V1, function(x) c(mean(x), sd(x)))
summary
```

# mapply
the last apply function I will cover is mapply.
```{r, eval=FALSE}
?mapply
```
the arguments for mapply are mapply(FUN, ..., MoreArgs = NULL, SIMPLIFY = TRUE, USE.NAMES = TRUE).
First you list the function, followed by the vectors you are using
the rest of the arguments have default values so they don't need to be changed for now. 
When you have a function that takes 2 arguments, the first vector goes into the first argument
and the second vector goes into the second argument.

### Example 1: Understanding mapply
In this example, 1:9 is specifying the value to repeat, and 9:1 is specifying how many times
to repeat. This order is based on the order of arguments in the rep function itself.
```{r}
mapply(rep, 1:9, 9:1)
```

### Example 2: Creating a new variable
Another use for mapply would be to create a new variable. For example, using dataset t, I could
divide one column by another column to create a new value. This would be useful for creating a 
ratio of two variables as shown in the example below. 

```{r}
tdata$V5 <- mapply(function(x, y) x/y, tdata$V2, tdata$V4)
tdata$V5
```

### Example 3: Saving data into a premade vector
When using an apply family function to create a new variable, one option is to create a new vector ahead of time with the size of the vector pre-allocated. I created a numeric vector of length 10 using the vector function. The arguments for the vector function are vector(mode, length). Inside mapply I created a function to multiple two variables together. The results of the mapply function are then saved into the vector.

```{r}
new.vec <- vector(mode = "numeric", length = 10)
new.vec <- mapply(function(x, y) x*y, tdata$V3, tdata$V4)
new.vec
```

# Using apply functions on real datasets
This last section will be a few examples of using apply functions on real data.This section will
make use of the MASS package, which is a collection of publicly available datasets. Please
install MASS if you do not already have it. If you do not have MASS installed, you can uncomment
the code below.

```{r}
#install.packages("MASS")
library(MASS)
```

load the state dataset. It contains information about all 50 states
```{r}
data(state)
```
Let's look at the data we will be using. We will be using the state.x77 dataset
```{r}
head(state.x77)
str(state.x77)
```
All the data in the dataset happens to be numeric, which is necessary when the function inside the apply function requires numeric data.

### Example 1: using apply to get summary data
You can use apply to find measures of central tendency and dispersion
```{r}
apply(state.x77, 2, mean)
apply(state.x77, 2, median)
apply(state.x77, 2, sd)
```

### Example 2: Saving the results of apply

In this, I created one function that gives the mean and SD, and another that give min, median, and max. Then I saved them as objects that could be used later.
```{r}
state.summary<- apply(state.x77, 2, function(x) c(mean(x), sd(x))) 
state.summary
state.range <- apply(state.x77, 2, function(x) c(min(x), median(x), max(x)))
state.range
```

### Example 3: Using mapply to compute a new variable
In this example, I want to find the population density for each state. In order to do this, I 
want to divide population by area. state.area and state.x77 are not from the same dataset, but 
that is fine as long as the vectors are the same length and the data is in the same order. Both
vectors are alphabetically by state, so mapply can be used.
```{r}
population <- state.x77[1:50]
area <- state.area
pop.dens <- mapply(function(x, y) x/y, population, area)
pop.dens
```

### Example 4: Using tapply  to explore population by region
In this example, I want to find out some information about the population of states split by
region. state.region is a factor with four levels: Northeast, South, North Central, and West.
For each region, I want the minimum, median, and maximum populations.

```{r}
region.info <- tapply(population, state.region, function(x) c(min(x), median(x), max(x)))
region.info
```

# References
Here are some sources I used to help me create this chapter:

Datacamp tutorial on apply functions: https://www.datacamp.com/community/tutorials/r-tutorial-apply-family

r-bloggers: Using apply, sapply, and lapply in R: https://www.r-bloggers.com/using-apply-sapply-lapply-in-r/

stackoverflow: Why is vapply safer than sapply?: http://stackoverflow.com/questions/12339650/why-is-vapply-safer-than-sapply


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, A Language, not a Letter: Learning Statistics in R, https://www.datacamp.com/community/tutorials/r-tutorial-apply-family, https://www.r-bloggers.com/using-apply-sapply-lapply-in-r/, http://stackoverflow.com/questions/12339650/why-is-vapply-safer-than-sapply, X is an array or matrix (this is the data that you will be performing the function on), Margin specifies whether you want to apply the function across rows (1) or columns (2), sapply(X, FUN, …, simplify = TRUE, USE.NAMES = TRUE), vapply(X, FUN, FUN.VALUE, …, USE.NAMES = TRUE).