`seq(1, 11, length.out = 6)`

`[1] 1 3 5 7 9 11`

Appendix

More on R

- Functions in R
- More on Vectors
- Lists
- Data Frames
- Data Entry
- R Style Guide

Functions in R

- Basic way to interact with R is through
**expression evaluation**:- You enter an epression
- The system evaluates it and prints the result

- Expressions work on
**objects** **Object**: anything that can be assigned to a variable- Objects encountered so far are:
- Scalars
- Vectors

**Functions**are a class of objectsFormat of a function is

**name**followed by parentheses containing**arguments**Functions take

**arguments**and**return**a resultWe already encountered several built in functions:

`plot(x, y)`

`lines(x, y)`

`seq(x)`

`print("Stats is great!")`

`cat("R is great!")`

`mean(x)`

`sin(x)`

- Functions have
**actual arguments**and**formal arguments** - Example:
`plot(x, y)`

has formal arguments two vectors`x`

and`y`

`plot(height, weight)`

has actual arguments`height`

and`weight`

- When you write
`plot(height, weight)`

the arguments are matched:`height`

corresponds to x-variable`weight`

corresponds to y-variable- This is called
**positional matching**

If a function has a lot of arguments, positional matching is tedious

For example

`plot()`

accepts the following (and more!) arguments

Argument | Description |
---|---|

`x` |
x coordinate of points in the plot |

`y` |
y coordinate of points in the plot |

`type` |
Type of plot to be drawn |

`main` |
Title of the plot |

`xlab` |
Label of x axis |

`ylab` |
Label of y axis |

`pch` |
Shape of points |

Issue with having too many arguments is the following:

- We might want to specify
`pch = 2`

- But then we would have to match all the arguments preceding
`pch`

`x`

`y`

`type`

`xlab`

`ylab`

- Thankfully we can use
**named actual arguments**:- The name of a formal argument can be matched to an actual argument
- This is independent of position

- For example we can specify
`pch = 2`

by the call`plot(weight, height, pch = 2)`

- In the above:
`weight`

is implicitly matched to`x`

`height`

is implicitly matched to`y`

`pch`

is explicitly matched to`2`

- Note that the following call would give same output
`plot(x = weight, y = height, pch = 2)`

- Named actual arguments
**override**positional arguments - Example: The following commands yield the same plot
`plot(height, weight)`

`plot(x = height, y = weight)`

`plot(y = weight, x = height)`

We have already seen another example of **named actual arguments**

`seq(from = 1, to = 11, by = 2)`

`seq(1, 11, 2)`

- These yield the same output. Why?
- Because in this case
**named actual arguments**match**positional arguments**

If however we want to divide the interval [1, 11] in 5 equal parts:

- Have to use
`seq(1, 11, length.out = 6)`

- The above is different from
`seq(1, 11, 6)`

- They are different because:
- The 3rd positional argument of
`seq()`

is`by`

- Hence the command
`seq(1, 11, 6)`

assumes that`by = 6`

- The 3rd positional argument of

- You can call functions without specifying arguments
- However you have to use brackets
`()`

- Example:
`getwd()`

– which outputs current working directory`ls()`

– which outputs names of objects currently in memory

- You can define your own functions in R
- Syntax for definining custom function
`my_function`

is below - You can call your custom function by typing
`my_function(arguments)`

The R function

`mean(x)`

computes the sample mean of vector`x`

We want to define our own function to compute the mean

**Example:**The mean of`x`

could be computed via`sum(x) / length(x)`

We want to implement this code into the function

`my_mean(x)`

`my_mean`

takes vector`x`

as argument`my_mean`

returns a scalar – the mean of`x`

- Let us use our function
`my_mean`

on an example

```
# Generate a random vector of 1000 entries from N(0,1)
x <- rnorm(1000)
# Compute mean of x with my_mean
xbar <- my_mean(x)
# Compute mean of x with built in function mean
xbar_check <- mean(x)
cat("Mean of x computed with my_mean is:", xbar)
cat("Mean of x computed with R mean is:", xbar_check)
cat("They coincide!")
```

`Mean of x computed with my_mean is: 0.02337842`

`Mean of x computed with R mean is: 0.02337842`

`They coincide!`

More on Vectors

- We have seen vectors of numbers
- Further type of vectors are:
- Character vectors
- Logical vectors

- A character vector is a vector of text strings
- Elements are specified and printed in quotes

- You can use single- or double-quote symbols to specify strings
- This is as long as the left quote is the same as the right quote

`Print`

and `cat`

produce different output on character vectors:

`print(x)`

prints all the strings in`x`

separately`cat(x)`

concatenates strings. There is no way to tell how many were there

- Logical vectors can take the values
`TRUE`

,`FALSE`

or`NA`

`TRUE`

and`FALSE`

can be abbreviated with`T`

and`F`

`NA`

stands for**not available**

Logical vectors are extremely useful to evaluate conditions

Example:

- given a numerical vector
`x`

- we want to count how many entries are above a value
`t`

- given a numerical vector

```
# Generate a vector containing sequence 1 to 8
x <- seq(from = 1 , to = 8, by = 1)
# Generate vector of flags for entries strictly above 5
y <- ( x > 5 )
cat("Vector x is: (", x, ")")
cat("Entries above 5 are: (", y, ")")
```

`Vector x is: ( 1 2 3 4 5 6 7 8 )`

`Entries above 5 are: ( FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE )`

- Generate a vector of 1000 numbers from N(0,1)
- Count how many entries are above the mean 0
- Since there are many (1000) entries, we expect a result close to 500
- This is because
*sample mean*converges to*true mean*0

- This is because

**Question**: How to do this?

**Hint**: `T/F`

are interpreted as `1/0`

in arithmetic operations

- The function
`sum(x)`

sums the entries of a vector`x`

- We can use
`sum(x)`

to count the number of`T`

entries in a logical vector`x`

```
x <- rnorm(1000) # Generates vector with 1000 normal entries
y <- (x > 0) # Generates logical vector of entries above 0
above_zero <- sum(y) # Counts entries above zero
cat("Number of entries which are above the average 0 is", above_zero)
cat("This is pretty close to 500!")
```

`Number of entries which are above the average 0 is 472`

`This is pretty close to 500!`

- In practical data analysis, a data point is frequently unavailable
- Statistical software needs ways to deal with this
- R allows vectors to contain a special
`NA`

value - Not Available `NA`

is carried through in computations: operations on`NA`

yield`NA`

as the result

Components of a vector can be retrieved by

**indexing**`vector[k]`

returns k-th component of`vector`

To modify an element of a vector use the following:

`vector[k] <- value`

stores`value`

in k-th component of`vector`

Returning multiple items of a vactor is known as **slicing**

`vector[c(k1, ..., kn)]`

returns components`k1, ..., kn`

`vector[k1:k2]`

returns components`k1`

to`k2`

- Elements of a vector
`x`

can be deleted by using`x[ -c(k1, ..., kn) ]`

which deletes entries`k1, ..., kn`

```
# Create a vector x
x <- c(11, 22, 33, 44, 55, 66, 77, 88, 99, 100)
# Print vector x
cat("Vector x is:", x)
# Delete 2nd, 3rd and 7th entries of x
x <- x[ -c(2, 3, 7) ]
# Print x again
cat("Vector x with 2nd, 3rd and 7th entries removed:", x)
```

`Vector x is: 11 22 33 44 55 66 77 88 99 100`

`Vector x with 2nd, 3rd and 7th entries removed: 11 44 55 66 88 99 100`

- You can index or slice vectors by entering explicit indices
- You can also index vectors, or
**subset**, by using**logical flag vectors**:- Element is extracted if corresponding entry in the flag vector is TRUE
- Logical flag vectors should be the same length as vector to subset

**Code**: Suppose given a vector `x`

Create a flag vector by using

`flag <- condition(x)`

`condition()`

is any function which returns`T/F`

vector of same length as`x`

Subset

`x`

by using`x[flag]`

- The following code extracts negative components from a numeric vector
- This can be done by using
`x[ x < 0 ]`

```
# Create numeric vector x
x <- c(5, -2.3, 4, 4, 4, 6, 8, 10, 40221, -8)
# Get negative components from x and store them in neg_x
neg_x <- x[ x < 0 ]
cat("Vector x is:", x)
cat("Negative components of x are:", neg_x)
```

`Vector x is: 5 -2.3 4 4 4 6 8 10 40221 -8`

`Negative components of x are: -2.3 -8`

- The following code extracts components falling between
`a`

and`b`

- This can be done by using logical operator and
`&`

`x[ (x > a) & (x < b) ]`

```
# Create numeric vector
x <- c(5, -2.3, 4, 4, 4, 6, 8, 10, 40221, -8)
# Get components between 0 and 100
range_x <- x[ (x > 0) & (x < 100) ]
cat("Vector x is:", x)
cat("Components of x between 0 and 100 are:", range_x)
```

`Vector x is: 5 -2.3 4 4 4 6 8 10 40221 -8`

`Components of x between 0 and 100 are: 5 4 4 4 6 8 10`

`which()`

allows to convert a logical vector`flag`

into a numeric index vector`which(flag)`

is vector of indices of`flag`

which correspond to`TRUE`

```
# Create a logical flag vector
flag <- c(T, F, F, T, F)
# Indices for flag which
true_flag <- which(flag)
cat("Flag vector is:", flag)
cat("Positions for which Flag is TRUE are:", true_flag)
```

`Flag vector is: TRUE FALSE FALSE TRUE FALSE`

`Positions for which Flag is TRUE are: 1 4`

`which()`

can be used to delete certain entries from a vector `x`

Create a flag vector by using

`flag <- condition(x)`

`condition()`

is any function which returns`T/F`

vector of same length as`x`

Delete entries flagged by

`condition`

using the code`x[ -which(flag) ]`

```
# Create numeric vector x
x <- c(5, -2.3, 4, 4, 4, 6, 8, 10, 40221, -8)
# Print x
cat("Vector x is:", x)
# Flag positive components of x
flag_pos_x <- (x > 0)
# Remove positive components from x
x <- x[ -which(flag_pos_x) ]
# Print x again
cat("Vector x with positive components removed:", x)
```

`Vector x is: 5 -2.3 4 4 4 6 8 10 40221 -8`

`Vector x with positive components removed: -2.3 -8`

The main functions to generate vectors are

`c()`

concatenate`seq()`

sequence`rep()`

replicate

We have already met `c()`

and `seq()`

but there are more details to discuss

**Recall:** `c()`

generates a vector containing the input values

`c()`

can also concatenate vectors- This was you can
**add**entries to an**existing vector**

You can assign names to vector elements

This modiﬁes the way the vector is printed

Given a named vector `x`

- Names can be extracted with
`names(x)`

- Values can be extracted with
`unname(x)`

```
# Create named vector
x <- c(first = "Red", second = "Green", third = "Blue")
# Access names of x via names(x)
names_x <- names(x)
# Access values of x via unname(x)
values_x <- unname(x)
cat("Names of x are:", names(x))
cat("Values of x are:", unname(x))
```

`Names of x are: first second third`

`Values of x are: Red Green Blue`

- All elements of a vector have the
**same**type - Concatenating vectors of different types leads to conversion

- Recall the syntax of
`seq`

is`seq(from =, to =, by =, length.out =)`

- Omitting the third argument assumes that
`by = 1`

`seq(x1, x2)`

is equivalent to`x1:x2`

- Syntax
`x1:x2`

is**preferred**to`seq(x1, x2)`

```
# Generate two vectors of integers from 1 to 6
x <- seq(1, 6)
y <- 1:6
cat("Vector x is:", x)
cat("Vector y is:", y)
cat("They are the same!")
```

`Vector x is: 1 2 3 4 5 6`

`Vector y is: 1 2 3 4 5 6`

`They are the same!`

`rep`

generates repeated values from a vector:

`x`

vector`n`

integer`rep(x, n)`

repeats`n`

times the vector`x`

```
# Create a vector with 3 components
x <- c(2, 1, 3)
# Repeats 4 times the vector x
y <- rep(x, 4)
cat("Original vector is:", x)
cat("Original vector repeated 4 times:", y)
```

`Original vector is: 2 1 3`

`Original vector repeated 4 times: 2 1 3 2 1 3 2 1 3 2 1 3`

The second argument of `rep()`

can also be a vector:

- Given
`x`

and`y`

vectors `rep(x, y)`

repeats entries of`x`

as many times as corresponding entries of`y`

```
x <- c(2, 1, 3) # Vector to replicate
y <- c(1, 2, 3) # Vector saying how to replicate
z <- rep(x, y) # 1st entry of x is replicated 1 time
# 2nd entry of x is replicated 2 times
# 3rd entry of x is replicated 3 times
cat("Original vector is:", x)
cat("Original vector repeated is:", z)
```

`Original vector is: 2 1 3`

`Original vector repeated is: 2 1 1 3 3 3`

`rep()`

can be useful to create vectors of labels**Example**: Suppose we want to collect some numeric data on 3 Cats and 4 Dogs

Lists

Vectors can contain only one data type (number, character, boolean)

Lists are data structures that can contain

**any R object**Lists can be created similarly to vectors, with the command

`list()`

Elements of a list can be retrieved by **indexing**

`my_list[[k]]`

returns k-th element of`my_list`

You can return multiple items of a list via **slicing**

`my_list[c(k1, ..., kn)]`

returns elements in positions`k1, ..., kn`

`my_list[k1:k2]`

returns elements`k1`

to`k2`

- Components of a list can be
**named**. Names can be assigned with`names(my_list) <- c("name_1", ..., "name_k")`

```
# Create list with 3 elements
my_list <- list(2, c(T,F,T,T), "hello")
# Name each of the 3 elements
names(my_list) <- c("number", "TF_vector", "string")
# Print the named list: the list is printed along with element names
print(my_list)
```

```
$number
[1] 2
$TF_vector
[1] TRUE FALSE TRUE TRUE
$string
[1] "hello"
```

- A component of
`my_list`

named`my_name`

can be accessed with dollar operator`my_list$my_name`

```
# Create list with 3 elements and name them
my_list <- list(2, c(T,F,T,T), "hello")
names(my_list) <- c("number", "TF_vector", "string")
# Access 2nd element using dollar operator and store it in variable
second_component <- my_list$TF_vector
# Print 2nd element
print(second_component)
```

`[1] TRUE FALSE TRUE TRUE`

Data Frames

Data Frames are the best way of presenting a data set in R:

- Each variable has assigned a collection of recorded observations

Data frames can contain any R object

Data Frames are similar to Lists, with the difference that:

- Members of a Data Frame must all be vectors of equal length

Data frames are constructed similarly to lists, using

`data.frame()`

**Important**: Elements of data frame must be vectors of the same length**Example**: We construct the**Family Guy**data frame. Variables are`person`

– Name of character`age`

– Age of character`sex`

– Sex of character

- R prints data frames like matrices
- First row contains vector names

- First column contains row names
- Data are paired: e.g. Peter is 42 and Male

Think of a data frame as a

*matrix*You can extract element in position

`(m,n)`

by using`my_data[m, n]`

**Example**: Peter is in 1st row. We can extract Peter’s name as follows

`[1] "Peter"`

To extract multiple elements on the same row or column type

`my_data[c(k1,...,kn), m]`

\quad or \quad`my_data[k1:k2, m]`

`my_data[n, c(k1,...,km)]`

\quad or \quad`my_data[n, k1:k2]`

**Example**: Meg is listed in 3rd row. We extract her *age* and *sex*

```
age sex
3 17 F
```

To extract entire rows or columns type

`my_data[c(k1,...,kn), ]`

\quad or \quad`my_data[k1:k2, ]`

`my_data[, c(k1,...,km)]`

\quad or \quad`my_data[, k1:k2]`

```
peter_data <- family[1, ] # Extracts first row - Peter
sex_age <- family[, c(3,2)] # Extracts third and second columns:
# sex and age
print(peter_data)
print(sex_age)
```

```
person age sex
1 Peter 42 M
```

```
sex age
1 M 42
2 F 40
3 F 17
4 M 14
5 M 1
```

Use dollar operator to access data frame columns

- Suppose data set
`my_data`

contains a variable called`my_variable`

`my_data$my_variable`

accesses column`my_variable`

`my_data$my_variable`

is a vector

**Example**: To access *age* in the `family`

data frame type

```
ages <- family$age # Stores ages in a vector
cat("Ages of the Family Guy characters are", ages)
cat("Meg's age is", ages[3])
```

`Ages of the Family Guy characters are 42 40 17 14 1`

`Meg's age is 17`

The size of a data frame can be discovered using:

`nrow(my_data)`

\quad number of rows`ncol(my_data)`

\quad number of columns`dim(my_data)`

\quad \quad vector containing number of rows and columns

```
family_dim <- dim(family) # Stores dimensions of family in a vector
cat("The Family Guy data frame has", family_dim[1],
"rows and", family_dim[2], "columns")
```

`The Family Guy data frame has 5 rows and 3 columns`

Adding data to an existing data frame `my_data`

- Add more records (adding to rows)
- Create single row data frame
`new_record`

`new_record`

must match the structure of`my_data`

- Add to
`my_data`

with`my_data <- rbind(my_data, new_record)`

- Create single row data frame
- Add a set of observations for a new variable (adding to columns)
- Create a vector
`new_variable`

`new_variable`

must have as many components as rows in`my_data`

- Add to
`my_data`

with`my_data <- cbind(my_data, new_variable)`

- Create a vector

- Consider the usual
**Family Guy**data frame`family`

- Suppose we want to add data for
**Brian** - Create a new record: a single row data frame with columns
*person*,*age*,*sex*

- Now we add
`new_record`

to`family`

```
person age sex
1 Peter 42 M
2 Lois 40 F
3 Meg 17 F
4 Chris 14 M
5 Stewie 1 M
6 Brian 7 M
```

- We want to add a new variable to the
**Family Guy**data frame`family`

- This variable is called
`funny`

- It records how funny each character is, with levels
*Low*,*Med*,*High*

- Create a vector
`funny`

with entries matching each character (including**Brian**)

- Add
`funny`

to the**Family Guy**data frame`family`

```
person age sex funny
1 Peter 42 M High
2 Lois 40 F High
3 Meg 17 F Low
4 Chris 14 M Med
5 Stewie 1 M High
6 Brian 7 M Med
```

Instead of using `cbind`

we can add a new varibale using dollar operator:

- We want to add a variable called
`new_variable`

- Create a vector
`v`

containing values for the new variable `v`

must have as many components as rows in`my_data`

- Add to
`my_data`

with`my_data$new_variable <- v`

**Example**:

- We add age expressed in months to the
**Family Guy**data frame`family`

- Age in months can be computed by multiplying vector
`family$age`

by 12

```
v <- family$age * 12 # Computes vector of ages in months
family$age.months <- v # Stores vector as new column in family
print(family)
```

```
person age sex funny age.months
1 Peter 42 M High 504
2 Lois 40 F High 480
3 Meg 17 F Low 204
4 Chris 14 M Med 168
5 Stewie 1 M High 12
6 Brian 7 M Med 84
```

We saw how to use

**logical flag vectors**to subset vectorsWe can use logical flag vectors to subset data frames as well

Suppose to have data frame

`my_data`

containing a variable`my_variable`

Want to subset records in

`my_data`

for which`my_variable`

satisfies a conditionUse commands

`flag <- condition(my_data$my_variable)`

`my_data[flag, ]`

**Example**:

- Consider again the
**Family Guy**data frame`family`

- We subset Male characters using flag
`family$sex == "M"`

```
# Create flag vector for male Family Guy characters
flag <- (family$sex == "M")
# Subset data frame "family" and store in data frame "subset"
subset <- family[flag, ]
# Print subset
print(subset)
```

```
person age sex funny age.months
1 Peter 42 M High 504
4 Chris 14 M Med 168
5 Stewie 1 M High 12
6 Brian 7 M Med 84
```

Data Entry

R has a many functions for reading

**characters**from stored**files**We will see how to read

**Table-Format**files**Table-Formats**are just tables stored in plain-text filesTypical file estensions are:

`.txt`

for plain-text files`.csv`

for comma-separated values

**Table-Formats**can be read into R with the command`read.table()`

**Header**:- If present, header should be the first line of the file
- Header is used to provide names for each column of data
- If a header is present, you need to tell this to R when importing
- If not, R cannot tell if first line is a header or observed data values

**Delimiter**:- A character used to separate the entries in each line
- Delimiter character cannot be used for anything else in the file
- Delimiter tells R when a specific entry begins and ends
- Default delimiter is
**whitespace**

**Missing value**:- Character string used exclusively to denote a missing value
- When reading the file, R will turn these entries into
`NA`

**Comments**:- Table files can include comments
- Comment lines start with \quad
`#`

- R ignores such comments

- Table-Format for
**Family Guy**characters can be downloaded here family_guy.txt

- The text file looks like this

- Remarks:
- Header is present
- Delimiter is
**whitespace** - Missing values denoted by
`*`

- Table-Formats can be read via
`read.table()`

- This reads a
`.txt`

or`.csv`

file and outputs a data frame

- This reads a
- Options of
`read.table()`

`header = T/F`

– Tells R if a header is present`na.strings = "string"`

– Tells R that`"string"`

means`NA`

To read `family_guy.txt`

into R proceed as follows:

Download family_guy.txt and move file to Desktop

Open the R Console and change working directory to

**Desktop**

- Read
`family_guy.txt`

into R and store it in data frame`family`

with code

- Note that we are telling
`read.table()`

that`family_guy.txt`

has a header- Missing values are denoted by
`*`

- Print data frame
`family`

to screen

```
person age sex funny age.mon
1 Peter NA M High 504
2 Lois 40 F <NA> 480
3 Meg 17 F Low 204
4 Chris 14 M Med 168
5 Stewie 1 M High NA
6 Brian NA M Med NA
```

- For comparison this is the
`.txt`

file

**Example**: Analysis of Consumer Confidence Index for 2008 crisis from Lecture 4

- We imported data into R using
`c()`

- This is ok for small datasets
- Suppose the CCI data is stored in a
`.txt`

file instead

**Goal**: Perform t-test on CCI difference for mean difference \mu = 0

- By reading CCI data into R using
`read.table()`

- By manipulating CCI data using
**data frames**

The CCI dataset can be downloaded here 2008_crisis.txt

The text file looks like this

To perform the t-test on data `2008_crisis.txt`

we proceed as follows:

Download dataset 2008_crisis.txt and move file to Desktop

Open the R Console and change working directory to

**Desktop**

- Read
`2008_crisis.txt`

into R and store it in data frame`scores`

with code

- Store 2nd and 3rd columns of
`scores`

into 2 vectors

```
# CCI from 2007 is stored in 2nd column
score_2007 <- scores[, 2]
# CCI from 2009 is stored in 3nd column
score_2009 <- scores[, 3]
```

- Now the t-test can be performed as done in Lecture 4

- We obtain the same result of Lecture 4
- p-value is p < 0.05
- Reject H_0: The mean difference is not 0
- In details, the output of
`t.test`

is below

- For convenience you can download the full code 2008_crisis_code.R

```
One Sample t-test
data: difference
t = 38.144, df = 11, p-value = 4.861e-13
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
68.15960 76.50706
sample estimates:
mean of x
72.33333
```

R Style Guide

They should be meaningful and end in .R

- Objects names shoulde be lowercase
- Use an underscore (
`_`

) to separate words within a name - Variable names should be nouns, not verbs
- Come up with names that are concise and meaningful

- Name functions with
`BigCamelCase`

(link) - This is to clearly distinguish functions from other objects
- Functions names should be verbs
- Come up with names that are concise and meaningful

If possible avoid using names of existing functions and variables

Use `<-`

and not `=`

for assignment

- Spacing is
**really**something you should be careful about - Place spaces around all infix operators (
`=`

,`+`

,`-`

,`<-`

, etc.) - Place spaces around
`=`

when calling a function - Always put a space after a comma, never before (like in regular English)

- Do not place spaces around code in parentheses or square brackets
- Unless there is a comma

- Symbols
`:`

,`::`

and`:::`

do not need spacing

- Place a space before left parentheses, except in a function call

Extra spacing is ok if it improves alignment of `=`

or `<-`

- An opening curly brace should never go on its own line
- An opening curly brace should always be followed by a new line
- Always
**indent**the code inside curly braces

- Limit code to 80 characters per line
- This fits comfortably on a printed page
- If you run out of room, encapsulate some of the work in separate function

- When indenting your code, use two spaces
- Never use tabs or mix tabs and spaces
- Indentation should be used for
**functions**,**if**,**for**, etc.

If a function definition runs over multiple lines, indent the second line to where the definition starts

- Functions can return objects
- R has an implicit return feature
- Do not rely on this feature, but explicitly mention
`return(object)`

Often you can call a function without explicitly naming arguments:

`plot(height, weight)`

`mean(weight)`

This might be fine for

`plot()`

or`mean`

However for less common functions:

- One might struggle to remember the meaning of
**arguments positions** - It is therefore good practice to
**name arguments**

- One might struggle to remember the meaning of

## Comments

Most importantly: Comment your code`#`

and a single space`-`

and`=`

to break up code into easily readable chunksHomepage License Contact