Lab 04: Simple Data Collection and Data visualization

library(tidyverse)
library(openintro)

ex1_1a=read_csv("data/ex1_1a.csv", show_col_types = FALSE)
ex1_1b=read_csv("data/ex1_1b.csv", show_col_types = FALSE)
ex1_2a=read_csv("data/ex1_2a.csv", show_col_types = FALSE)
ex1_2b=read_csv("data/ex1_2b.csv", show_col_types = FALSE)

ex2_1a=read_csv("data/ex2_1a.csv", show_col_types = FALSE)
ex2_1b=read_csv("data/ex2_1b.csv", show_col_types = FALSE)
ex2_2a=read_csv("data/ex2_2a.csv", show_col_types = FALSE)
ex2_2b=read_csv("data/ex2_2b.csv", show_col_types = FALSE)

ex3_1a=read_csv("data/ex3_1a.csv", show_col_types = FALSE)
ex3_1b=read_csv("data/ex3_1b.csv", show_col_types = FALSE)
ex3_2a=read_csv("data/ex3_2a.csv", show_col_types = FALSE)
ex3_2b=read_csv("data/ex3_2b.csv", show_col_types = FALSE)

ex4_1a=read_csv("data/ex4_1a.csv", show_col_types = FALSE)
ex4_1b=read_csv("data/ex4_1b.csv", show_col_types = FALSE)
ex4_2a=read_csv("data/ex4_2a.csv", show_col_types = FALSE)
ex4_2b=read_csv("data/ex4_2b.csv", show_col_types = FALSE)

Exercise 1

Plotting the 50% and the 100% of the data from experiment 1 (with a 0 degree angle)

ggplot(data=ex1_1a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex1_1b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex1_2a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex1_2b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

Exercise 2

Calculating the mean and the standard deviation of the above data

df = ex1_1a;
Mean = sum(df$Bin*df$Count)/sum(df$Count)
SD = sqrt(sum((df$Bin-Mean)**2*df$Count)/(sum(df$Count)-1))
cat("Mean=",Mean,"\n","Sample-SD=",SD)

## Mean= 13.52471 
##  Sample-SD= 5.367182

df = ex1_1b;
Mean = sum(df$Bin*df$Count)/sum(df$Count)
SD = sqrt(sum((df$Bin-Mean)**2*df$Count)/(sum(df$Count)-1))
cat("Mean=",Mean,"\n","Sample-SD=",SD)

## Mean= 12.84914 
##  Sample-SD= 5.09997

df = ex1_2a;
Mean = sum(df$Bin*df$Count)/sum(df$Count)
SD = sqrt(sum((df$Bin-Mean)**2*df$Count)/(sum(df$Count)-1))
cat("Mean=",Mean,"\n","Sample-SD=",SD)

## Mean= 12.803 
##  Sample-SD= 4.720793

df = ex1_2b;
Mean = sum(df$Bin*df$Count)/sum(df$Count)
SD = sqrt(sum((df$Bin-Mean)**2*df$Count)/(sum(df$Count)-1))
cat("Mean=",Mean,"\n","Sample-SD=",SD)

## Mean= 13.43869 
##  Sample-SD= 4.409267

Exercise 3

How do means compare between the different cases?

Between the different cases, the mean of the data all agree with each other: between the range of 12.8 and 13.5; which out of a data of nearly 27 bins, seems to indicate that the mean did not change in the first experiment that greatly.

Exercise 4

How does tilting the Galton board vary the results (Try to use the height of the legs, the distance between legs and the height of the pen to make conclusions)?

Experiment 2

ggplot(data=ex2_1a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex2_1b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex2_2a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex2_2b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

#finding the mean the standard deviation with a small anglular tilt.
df2_1a = ex2_1a;
Mean = sum(df2_1a$Bin*df2_1a$Count)/sum(df2_1a$Count)
SD = sqrt(sum((df2_1a$Bin-Mean)**2*df2_1a$Count)/(sum(df2_1a$Count)-1))
cat("Mean 2_1a =",Mean,"\n","Sample-SD 2_1a=",SD,"\n")

## Mean 2_1a = 17.02365 
##  Sample-SD 2_1a= 4.290046

df2_1b = ex2_1b;
Mean = sum(df2_1b$Bin*df2_1b$Count)/sum(df2_1b$Count)
SD = sqrt(sum((df2_1b$Bin-Mean)**2*df2_1b$Count)/(sum(df2_1b$Count)-1))
cat("Mean 2_1b =",Mean,"\n","Sample-SD 2_1b=",SD,"\n")

## Mean 2_1b = 15.79959 
##  Sample-SD 2_1b= 5.033724

df2_2a = ex2_2a;
Mean = sum(df2_2a$Bin*df2_2a$Count)/sum(df2_2a$Count)
SD = sqrt(sum((df2_2a$Bin-Mean)**2*df2_2a$Count)/(sum(df2_2a$Count)-1))
cat("Mean 2_2a =",Mean,"\n","Sample-SD 2_2a=",SD,"\n")

## Mean 2_2a = 15.89836 
##  Sample-SD 2_2a= 5.91973

df2_2b = ex2_2b;
Mean = sum(df2_2b$Bin*df2_2b$Count)/sum(df2_2b$Count)
SD = sqrt(sum((df2_2b$Bin-Mean)**2*df2_2b$Count)/(sum(df2_2b$Count)-1))
cat("Mean 2_2b =",Mean,"\n","Sample-SD 2_2b=",SD,"\n")

## Mean 2_2b = 15.71726 
##  Sample-SD 2_2b= 5.164004

Experiment 3

ggplot(data=ex3_1a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex3_1b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex3_2a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex3_2b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

#finding the mean the standard deviation with a large anglular tilt.
df3_1a = ex3_1a;
Mean = sum(df3_1a$Bin*df3_1a$Count)/sum(df3_1a$Count)
SD = sqrt(sum((df3_1a$Bin-Mean)**2*df3_1a$Count)/(sum(df3_1a$Count)-1))
cat("Mean 3_1a =",Mean,"\n","Sample-SD 3_1a=",SD,"\n")

## Mean 3_1a = 16.38649 
##  Sample-SD 3_1a= 5.865318

df3_1b = ex3_1b;
Mean = sum(df3_1b$Bin*df3_1b$Count)/sum(df3_1b$Count)
SD = sqrt(sum((df3_1b$Bin-Mean)**2*df3_1b$Count)/(sum(df3_1b$Count)-1))
cat("Mean 3_1b =",Mean,"\n","Sample-SD 3_1b=",SD,"\n")

## Mean 3_1b = 16.35318 
##  Sample-SD 3_1b= 5.282569

df3_2a = ex3_2a;
Mean = sum(df3_2a$Bin*df3_2a$Count)/sum(df3_2a$Count)
SD = sqrt(sum((df3_2a$Bin-Mean)**2*df3_2a$Count)/(sum(df3_2a$Count)-1))
cat("Mean 3_2a =",Mean,"\n","Sample-SD 3_2a=",SD,"\n")

## Mean 3_2a = 16.10688 
##  Sample-SD 3_2a= 5.706999

df3_2b = ex3_2b;
Mean = sum(df3_2b$Bin*df3_2b$Count)/sum(df3_2b$Count)
SD = sqrt(sum((df3_2b$Bin-Mean)**2*df3_2b$Count)/(sum(df3_2b$Count)-1))
cat("Mean 3_2b =",Mean,"\n","Sample-SD 3_2b=",SD,"\n")

## Mean 3_2b = 16.39186 
##  Sample-SD 3_2b= 5.389525

Experiment 4

ggplot(data=ex4_1a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex4_1b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex4_2a)+geom_bar(aes(x=Bin, y=Count), stat='identity')

ggplot(data=ex4_2b)+geom_bar(aes(x=Bin, y=Count), stat='identity')

#finding the mean the standard deviation with a small anglular tilt in the opposing direction.
df4_1a = ex4_1a;
Mean = sum(df4_1a$Bin*df4_1a$Count)/sum(df4_1a$Count)
SD = sqrt(sum((df4_1a$Bin-Mean)**2*df4_1a$Count)/(sum(df4_1a$Count)-1))
cat("Mean 4_1a =",Mean,"\n","Sample-SD 4_1a=",SD,"\n")

## Mean 4_1a = 11.61041 
##  Sample-SD 4_1a= 5.349967

df4_1b = ex4_1b;
Mean = sum(df4_1b$Bin*df4_1b$Count)/sum(df4_1b$Count)
SD = sqrt(sum((df4_1b$Bin-Mean)**2*df4_1b$Count)/(sum(df4_1b$Count)-1))
cat("Mean 4_1b =",Mean,"\n","Sample-SD 4_1b=",SD,"\n")

## Mean 4_1b = 11.44335 
##  Sample-SD 4_1b= 5.093633

df4_2a = ex4_2a;
Mean = sum(df4_2a$Bin*df4_2a$Count)/sum(df4_2a$Count)
SD = sqrt(sum((df4_2a$Bin-Mean)**2*df4_2a$Count)/(sum(df4_2a$Count)-1))
cat("Mean 4_2a =",Mean,"\n","Sample-SD 4_2a=",SD,"\n")

## Mean 4_2a = 11.48582 
##  Sample-SD 4_2a= 5.326175

df4_2b = ex4_2b;
Mean = sum(df4_2b$Bin*df4_2b$Count)/sum(df4_2b$Count)
SD = sqrt(sum((df4_2b$Bin-Mean)**2*df4_2b$Count)/(sum(df4_2b$Count)-1))
cat("Mean 4_2b =",Mean,"\n","Sample-SD 4_2b=",SD,"\n")

## Mean 4_2b = 11.88596 
##  Sample-SD 4_2b= 5.373549

Answer: Through these above measurements and graphs, one realizes that tilting the graph does not affect the binomial distribution; but it shifts the graph along the x-axis.