Chapter 1 Introdcution to Statistics and R

1.1 Statistics

1.1.1 What is Statistics?

The term has got three different meanings.

• Plural of the term statistic , which refers to any function of sample values, for example, $$\bar x = \frac {\sum_i^n x_i} n$$
• Table of values
Table 1.1: A Subset from Iris Data Set
Sepal.Length Sepal.Width Petal.Length Petal.Width Species
5.1 3.5 1.4 0.2 setosa
4.9 3.0 1.4 0.2 setosa
4.7 3.2 1.3 0.2 setosa
4.6 3.1 1.5 0.2 setosa
5.0 3.6 1.4 0.2 setosa
5.4 3.9 1.7 0.4 setosa
• Technique of dealing with data
• Collection
• Organization
• Analysis (such as regression analysis)
• Interpretation
• Presentation

1.1.2 Statistics vs Data Science

Statistics deals mainly with analyzing and interpreting data, while data science deals more with predictive analytics. (more)

1.1.3 Statistics vs Mathematics

Consider the following equations

• $$Y = X + 0.2 \times X$$

Say, $$X$$ = Basic Salary of an employee in a company, while $$Y$$ is computed from $$X$$, with adding to $$X$$ 20% of $$X$$. In this scenario, given the values of $$X$$, we can always tell what the gross salary would be.

What if we have an equation such as the following:

$$Y = X + 0.1 \times R$$

Where, $$R$$ is the revenue the company earns through the employee. In this case, the salary of the employee would vary in each month.

The salaries from month to month would unpredictably vary, which is where statistics comes in. Statistics deals with randomness, situations where we cannot exactly tell which outcome we might get. We may (or may not) know the possible outcomes (like when tossing a coin, we know the possible outcome, but which will happen)

1.1.4 Other Statistical Concepts

They will be briefly explained later when the relevant R codes are mentioned. They include various charts, concepts of central tendency and dispersion, correlation, regression, test of hypothesis, among others.

1.2 Introdcution to R

1.2.1 Why R?

R is the most popular programming language for statistical analysis, second most popular for machine learning.

Reasons at a glance

• Free and Open Source Software (FOSS)
• Big Community
• Made by statisticians for statisticians
• Easy to use codes
• Stunning graphics, esp. with ggplot2
• Reproducibility
• Runs on a wide array of platforms, including but not limited to Windows, Linux, and Mac OS X.

1.2.2 Who Use R?

R is both used in academia and industry.

• Good analyses for theses are now accomplished using R.
• Industries heavily rely on R for statistical analysis, predictive analytics, and machine learning.

Some of the renowned companies using R are:

• Microsoft
• Uber
• Airbnb
• IBM
• ANZ
• HP
• Ford

1.2.3 Who Developed R?

R was developed by Ross Ihaka and Robert Gentleman (statistician from New Zealand and Canada, respectively).

1.2.4 Other Languages and Packages

Some other languages for data analysis are:

• Python
• Julia
• Java
• Scala

Packages

• SPSS
• STATA
• Eviews

1.2.6 Start Writing R Code (Windows, Linux, and Command Line)

• Using R Console directly: write code and press enter (NOT a good method)
• Using Rstudio Console: Equivalent to using R console
• Using R Script from Rstudio: to run, press Ctrl + Enter

It is best to use Rstudio.

1.2.7 Effectively Using Rstudio

• Keep things organized
• Make a project Put all codes, data, and output inside that project directory.
• Use View function to view data tables, for example View(iris), which displays the iris data set.

1.2.8 R Script

An R script is a convenient tool to organize a work. A project may consist of several or many such scripts. They can be easily shared with others. An R script has the extension .R or .r.

Tips

• Use line gaps often to separate code segments

1.2.8.1 Quoting R codes from another R file.

source('r_file.R')

Thus, you can use functions, data, variables etc. defined or saved elsewhere.

1.2.9 R Documentation (Help)

To get help, type ?keyword or help(keyword)

For example, ?mean would show options and examples for the mean function.

1.2.10 Handling Error

• If the code is not run, and shows a + sign, it means the code is not complete yet; complete it or press esc to start over.
• If the error message shows could not find function ..., correct the function name.
• If you do not understand the error message, copy and paste it to your browser search bar, and see what help the community has to offer.

1.2.11 R Packages

R packages are extensions of base R, providing some very useful tasks. Many R packages made R more popular and useful, such as ggplot2, karet, and rmarkdown.

To install a package, run install.packages("package_name"), for example install.packages("tidyverse") installs the package tidyverse. When installing, the package name must be enclosed within quotation marks (" ").

Before being able to make use of a package, one must load the package, by running library(package_name), for example, to load ggplot2, run library(ggplot2), this time without quotation marks (" ").

1.2.12 R Mathematical Operations

• Make a table: Purpose, code, example, output

1.2.12.1 Arithmetic Operators

Purpose Operator Example Output
Addition + 2+3 5
Subtraction - 10-9 1
Multiplication * 10*8 80
Division / 10/5 2
Exponent ^ or ** 10^2 100
Modulus (Remainder) %% 10%%4 2
Integer Division %/% 12%/%5 2

1.2.12.2 Relational Operators

Purpose Operator Example Output
Less than < 2<3 5
Greater than > 10>11 1
Less than or equal to <= 10<=8 80
Greater than or equal to >= 10>=5 2
Equal to == 10^2==100 100
Not equal to != 100!=99 2

1.2.13 Assigning Values

Variables make it easy to assign values and use them later.

• To assign values to variables, you can use either = or <-, but in R, <- is preferred. In Rstudio, pressing alt + - is a very good shortcut for correctly typing <-.
• Comments start with hash (#)

Example

x <-  3
y <-  4
x+y
## [1] 7
x*y
## [1] 12
x+10-y
## [1] 9
x^y
## [1] 81
x**y
## [1] 81
log(x)
## [1] 1.098612
round(log(x), 3)
## [1] 1.099

1.2.13.1 Round, Floor, and Ceiling

Suppose, we have a number 3.9856

• round rounds the number;
x <- 3.9856
round(x,3) # (up to 3 digits)
## [1] 3.986
• celing switches the number to the next integer;
ceiling(x)
## [1] 4

floor gives the previous integer.

floor(x)
## [1] 3
• celing and floor always give integer output.

1.2.14 Generating Multiple Numbers

x <- 1:10
x
##  [1]  1  2  3  4  5  6  7  8  9 10
seq(1,20, 2) # Keeping fixed gap between the numbers
##  [1]  1  3  5  7  9 11 13 15 17 19
seq(1,50, length.out = 5) # Generating specific amount of numbers.
## [1]  1.00 13.25 25.50 37.75 50.00

1.2.15 Data Types

• Logical
• Numeric
• integer
• Double
• Character

• Stat Mania artciles and link to contents
• Books
• Coursera, Edx, and other MOOCs.

1.2.17 Vector

A vector is set of similar items. In Linear Algebra, it is defined as a matrix with only one column or one row. It could contain numbers of different types, strings, or logical values.

A vector makes it easy to simultaneously operate on multiple items.

• We make a vector when we are dealing with only one variable.
• A vector can contain only one type of values, such ac numeric, logical etc.

A vector in R is usually made using c, which stands for concatenate. A vector can also be made using seq command shown earlier, or by using a colon (:) sign, if the values are successive integers.

x <- c(4, 5, 7)
a <- 10:12
y <- c("red", "green", "blue", "black", "orange")
z <- c(TRUE, FALSE, TRUE, TRUE, FALSE)

• If a scalar (a single value) is added to a vector, it would be added to values.

• If two (or more) vectors with equal lengths are added together, corresponding values would be added; the same goes for almost any other mathematical operation (such as subtraction or division).

• If, however, the lengths are unequal, the values of the smaller vector would be repeated from the beginning.

x + 3 # Adds 3 all values of x.
## [1]  7  8 10
x + a # Corresponding values are added.
## [1] 14 16 19
b <- 6:7
x + b # Values of b are repeated. 
## Warning in x + b: longer object length is not a multiple of shorter object
## length
## [1] 10 12 13

1.2.17.2 Indexing Vectors

• Using []:

x
## [1] 4 5 7
x[2] # Extracts the second value.
## [1] 5
x[2:3] # Extracts second through third values.
## [1] 5 7
x[c(1,3)] # Extracts the first and third values.
## [1] 4 7
x[-1] # Extracts all except the first value.
## [1] 5 7
x[-c(1,3)] # Extracts all except the first and third values.
## [1] 5
• Using Logical

x[c(TRUE, TRUE, FALSE)] # Extracts the first and second values.
## [1] 4 5
y
## [1] "red"    "green"  "blue"   "black"  "orange"
z
## [1]  TRUE FALSE  TRUE  TRUE FALSE
y[z] # Using variables already stored. Does not extract values corresponding to FALSE items. 
## [1] "red"   "blue"  "black"

1.2.18 Matrix

A matrix a rectangular array of similar items. Although it has more than two rows and columns, it can only contain items of a single type.

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1.2.19 Data Frame

A Data frame contains many variables; each variable can be different type. Distinct variables are placed in columns and values/observations are in rows.

Example

Table 1.2: A Subset from mtcars Data Set
mpg cyl disp hp drat wt
Mazda RX4 21.0 6 160.0 110 3.90 2.620
Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875
Datsun 710 22.8 4 108.0 93 3.85 2.320
Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215
Hornet Sportabout 18.7 8 360.0 175 3.15 3.440
Valiant 18.1 6 225.0 105 2.76 3.460
Duster 360 14.3 8 360.0 245 3.21 3.570
Merc 240D 24.4 4 146.7 62 3.69 3.190
Merc 230 22.8 4 140.8 95 3.92 3.150
Merc 280 19.2 6 167.6 123 3.92 3.440

1.2.19.1 Making A New Data Frame

data.frame command is used to produce a data frame.

• Length of each variable must be equal.
df <- data.frame(x=c(10, 12, 15),
y=c("Dhaka", "Cumilla", "Rajshahi"),
w=sample(100, 3),
v=20:22)

1.2.19.2 Indexing A Data Frame

• Methods used for matrices apply.
Table 1.3: An Example Data Frame
x y w v
10 Dhaka 17 20
12 Cumilla 91 21
15 Rajshahi 77 22
df[2,3] # Extracts value from the third column in the second row. 
## [1] 91

1.2.20 List

A list can contain scalars, vectors, matrices, data frames, as well as other lists!

1.2.21 Functions

A function is used to

• avoid repetitive tasks and mistakes therefrom
• find values from a complicated formula

A function to compute Harmonic Mean (HM)

Formula: Reciprocal of Mean of $$\frac{1}{x_i}$$

Reciprocal of $$\frac{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}{n}$$

Thus, $$HM = \frac{n}{\sum \frac{1}{x_i}} =\frac 1 {\text{Mean of 1/x}}$$

hm <- function(x) {
1/sum(1/x)
}

We have, x = 4, 5, 7

Therefore,

hm(x)
## [1] 1.686747

Since this function is actually a one-liner, we can write it as

hm <- function(x) 1/sum(1/x)

1.2.22 Loops (Alternatives and Comparison with Other Languages)

In R, loops are rarely used.

1.2.22.1 For loop example

A for loop to add numbers 1 through 10.

sum <- 0
for (i in 1:10){
sum <- sum + i
}
sum
## [1] 55

Values to loop through can also be called from a variable.

x <- c(10, 12, 8, 19, 23, 25)
sum <- 0
for (i in x){
sum <- sum + i
}
sum
## [1] 97

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