Mat 150 Introductory Statistics

How Do We Construct Frequency Distribution Tables Using R?

Part 2 (Relative Frequeancy Distribution Table, Cumulative Frequency Distribution, and Cumulative Relative Frequency Distribution Table)

We will use States Visited, (StatesV) variable included in first day class survey, Sur1.

1. Upload Sur1.csv data set from your desktop to R

Sur1 <-read.csv("~/Desktop/Sur1.csv")

NOTE: If case you did not save Sur1.csv file correctly to your desktop, execute the following code
Sur1 <-read.csv(read.csv(file.choose(),header = TRUE))

1. Examinne the strucure of Sur1 data set

str(Sur1)

## 'data.frame':    20 obs. of  6 variables:
##  $ SEX      : Factor w/ 2 levels "Female","Male": 1 1 1 1 2 2 2 1 1 1 ...
##  $ POLITICS : Factor w/ 3 levels "Conservative",..: 3 3 3 3 3 3 2 2 1 3 ...
##  $ NSiblings: int  4 4 2 2 2 4 5 4 4 1 ...
##  $ StatesV  : int  5 11 6 3 7 3 4 4 6 3 ...
##  $ ShoeS    : num  6 9 8 7.5 13 10.5 9 8 6 8.5 ...
##  $ UsedExcel: int  2 2 2 1 1 3 3 2 2 1 ...

1. View first six row of the data in spreadsheet format

head(Sur1, 6)

##      SEX POLITICS NSiblings StatesV ShoeS UsedExcel
## 1 Female Moderate         4       5   6.0         2
## 2 Female Moderate         4      11   9.0         2
## 3 Female Moderate         2       6   8.0         2
## 4 Female Moderate         2       3   7.5         1
## 5   Male Moderate         2       7  13.0         1
## 6   Male Moderate         4       3  10.5         3

1. Attach Sur1 to R in order to work with data variables without using $ symbol (remember to detach(Sur1) after completion of analysis)

attach(Sur1)

1. Summarize StatesV variable

summary(StatesV)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    3.00    4.00    6.00    7.15    9.50   18.00

1. Construct a frequency distribution table of States Visited using breaks to list class limits

breaks <-seq(0,18, by=3)

1. Group StatesV variable data into bins

StatesV.cut <-cut(StatesV,breaks)

1. Construct frequency table for StatesV variable in horizontal display

 table(StatesV.cut)

## StatesV.cut
##   (0,3]   (3,6]   (6,9]  (9,12] (12,15] (15,18] 
##       3       8       4       3       1       1

1. Transform this table to 2-column display

transform(table(StatesV.cut))

##   StatesV.cut Freq
## 1       (0,3]    3
## 2       (3,6]    8
## 3       (6,9]    4
## 4      (9,12]    3
## 5     (12,15]    1
## 6     (15,18]    1

1. Convert frequency table to relative frequency table by dividing frequencies by total number of StatesV data, in our case by 20

transform(table(StatesV.cut)/20)

##   StatesV.cut Freq
## 1       (0,3] 0.15
## 2       (3,6] 0.40
## 3       (6,9] 0.20
## 4      (9,12] 0.15
## 5     (12,15] 0.05
## 6     (15,18] 0.05

1. Convert frequency table to cumulative frequency table by summing frequencies of StatesV data cumulatively

transform(table(StatesV.cut),Cum_Freq=cumsum(Freq))

##   StatesV.cut Freq Cum_Freq
## 1       (0,3]    3        3
## 2       (3,6]    8       11
## 3       (6,9]    4       15
## 4      (9,12]    3       18
## 5     (12,15]    1       19
## 6     (15,18]    1       20

1. Convert frequency table to cumulative relative frequency table by dividing cumulative frequencies by total number of StatesV data, in our case by 20

transform(table(StatesV.cut),RelCum_Freq=cumsum(Freq)/20)

##   StatesV.cut Freq RelCum_Freq
## 1       (0,3]    3        0.15
## 2       (3,6]    8        0.55
## 3       (6,9]    4        0.75
## 4      (9,12]    3        0.90
## 5     (12,15]    1        0.95
## 6     (15,18]    1        1.00

How Do We Construct A Frequency Distribution Table Using R? Part 1

We will use States Visited, (StatesV) variables included in the first-day class survey, Sur1.

1. Upload Sur1.csv data set from your desktop to R

Sur1 <-read.csv("~/Desktop/Sur1.csv")

NOTE: In case you did not save Sur1.csv file correctly to your desktop, execute the following code
Sur1 <-read.csv(read.csv(file.choose(),header = TRUE))

1. Examinne the strucure of Sur1 data set

str(Sur1)

## 'data.frame':    20 obs. of  6 variables:
##  $ SEX      : Factor w/ 2 levels "Female","Male": 1 1 1 1 2 2 2 1 1 1 ...
##  $ POLITICS : Factor w/ 3 levels "Conservative",..: 3 3 3 3 3 3 2 2 1 3 ...
##  $ NSiblings: int  4 4 2 2 2 4 5 4 4 1 ...
##  $ StatesV  : int  5 11 6 3 7 3 4 4 6 3 ...
##  $ ShoeS    : num  6 9 8 7.5 13 10.5 9 8 6 8.5 ...
##  $ UsedExcel: int  2 2 2 1 1 3 3 2 2 1 ...

1. View first six row of the data in spreadsheet format

head(Sur1, 6)

##      SEX POLITICS NSiblings StatesV ShoeS UsedExcel
## 1 Female Moderate         4       5   6.0         2
## 2 Female Moderate         4      11   9.0         2
## 3 Female Moderate         2       6   8.0         2
## 4 Female Moderate         2       3   7.5         1
## 5   Male Moderate         2       7  13.0         1
## 6   Male Moderate         4       3  10.5         3

1. Attach Sur1 to R in order to work with data variables without using $ symbol (remember to detach(Sur1) after completion of analysis)

attach(Sur1)

1. Summarize StatesV variable

summary(StatesV)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    3.00    4.00    6.00    7.15    9.50   18.00

1. Construct a frequency distribution table of States Visited using breaks to list class limits

breaks <-seq(0,18, by=3)

1. Group StatesV variable data into bins

StatesV.cut <-cut(StatesV,breaks)

1. Construct frequency table for StatesV variable in horizontal display

 table(StatesV.cut)

## StatesV.cut
##   (0,3]   (3,6]   (6,9]  (9,12] (12,15] (15,18] 
##       3       8       4       3       1       1

1. Transform this table to 2-column display

transform(table(StatesV.cut))

##   StatesV.cut Freq
## 1       (0,3]    3
## 2       (3,6]    8
## 3       (6,9]    4
## 4      (9,12]    3
## 5     (12,15]    1
## 6     (15,18]    1

1. Construct a histogram of the frequency distribution of StatesV variable

hist(StatesV,breaks,col="red",border = "blue",xlab="Number of States Visited",xlim = c(0,21),ylab="Frequency",main="Distribution of States Visited by Mat150-1701 Students")

Day 1: Using R as Calculator (We will not save files in R format for now!) 

Objective: How to upload a .csv file into R?

Step 1 Prepare your workspace

Whenever you begin/complete your work with R execute these two commands: rm(list=ls()) and ls() .

Launch R and type into console panel:

rm(list=ls()) 

press ENTER key, then check if your workspace is cleared

 by using:

ls() 

Step 2 Upload survey data, Sur1.csv by using following syntax

Sur1 <-read.csv(file.choose(),header=TRUE)

Note: You must have your .csv file already downloaded to your computer, (e.g. to your desktop).

Step 3 Check the structure of data uploaded by using the following code

Str(Sur1)

Step 4 Optional: Verify names of variables included in data by using the following syntax

names(Sur1)

Step 5 View the first five rows of Sur1 data using the following syntax:

head(Sur1, 5)

Step 6

 Attach Sur1 data in order to use names of respected variables without $ sign

attach(Sur1)

Step 7 Get a quick look at the summary of variables of Sur1

• summary(Sur1)

 

Sampling Distribution of Sample Mean

Example 1 (Data Set 21)

1. Upload Data Set 21 to R in previously saved .csv format.
2. Use Data Set 21 to construct a histogram of DEPTHS of 600 earthquakes.
3. Select 10000 random samples of size=50 from the DEPTHS variable, calculate the mean of each and construct a histogram of the sampling distribution of the sample means.
   SOLUTION

EarthQ <- read.csv("~/Desktop/csv/21 - Earthquakes.csv")

attach(EarthQ)

head(EarthQ)

##   MAGNITUDE DEPTH
## 1      2.45   0.7
## 2      3.62   6.0
## 3      3.06   7.0
## 4      3.30   5.4
## 5      1.09   0.5
## 6      3.10   0.0

2.  

breaks <-seq(0,45,by=5)
hist(DEPTH,breaks, col="red",xlab = "Depth [km]",ylab = "Frequency",main = "Histogram of Earthquakes Depths")

mean(DEPTH)

## [1] 5.822

sd(DEPTH)

## [1] 4.927049

fifty.depths <- function() {
    depth.S <- sample(DEPTH,
    size = 50,replace = TRUE)
    return(mean(depth.S))
}

 sim1 <-replicate(n=10000,expr=fifty.depths())

head(sim1)

## [1] 5.540 6.576 5.224 6.800 5.262 4.952

summary(sim1)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   3.650   5.338   5.798   5.825   6.270   8.742

sd(sim1)

## [1] 0.695623

breaks <-seq(2.5,10.5,by=0.5)
hist(sim1,breaks,xlab ="Mean[km]",ylab="Frequency",col="red",border = "green",main="Sampling Distribution of the Sample Mean")

Things to ponder:

1. the shape of DEPTH histogram
2. values of the mean(DEPTH) and sd(DEPTH)
3. shape of means(DEPTH) sampling distribution
4. values of the mean of the sampling distribution and standard deviation of the sampling distribution

Five fair dice were rolled once and the proportion of ODD outcomes was observed.

Example 1

x <-sample(1:6,5,replace = TRUE); x

## [1] 5 2 2 5 4

What is the proportion of ODD outcomes?

y <- x %% 2;y

## [1] 1 0 0 1 0

prop <-sum(y)/length(x);prop

## [1] 0.4

Example 2 (Simulation)

Replicate 100,000 five-dice rolls and record the proportion of ODD numbers for each roll.

five.dice <- function() {
  dice <-sample(1:6,5, replace = TRUE)
  return(sum(dice %% 2)/length(dice))
}
sim1 <-replicate(100000, five.dice())

Here are the first six recorded proportions of ODD outcomes.

head(sim1)

## [1] 0.2 0.6 0.4 0.6 0.2 1.0

Table of Proportions of ODD outcomes from 100,000 simulated rolls

transform(table(sim1))

##   sim1  Freq
## 1    0  3065
## 2  0.2 15541
## 3  0.4 31303
## 4  0.6 31434
## 5  0.8 15626
## 6    1  3031

Histogram of Proportions of ODD Outcomes

breaks <-seq(-0.1,1.1,by=0.2)
hist(sim1,breaks, col="red", border = "green",xlab="Proportion of ODD Numbers",ylab="Frequency", main = "Sampling Distribution of Sample Proportion")

Finally, the mean proportion of simulated proportions is equal to:

mean(sim1)

## [1] 0.500216

Ponder on the following: a) shape of the sampling distribution of the sample proportion
b) the mean proportion of the sample proportion
c) proportion of ODD outcomes when rolling a fair die once
d) writing a simulation like the one above regarding the proportion of EVEN outcomes in 100,000 rolls of five-fair dice.