1. CA200 (based on the book by Prof. Jane M. Horgan)
3. Basics of R – cont.
Summarising Statistical Data Graphical Displays 4. Basic distributions with R
CA200
(based on the book by Prof. Jane M. Horgan)

2. Basics 6+7*3/2 #general expression [1] 16.5
x <- 1:4 #integers are assigned to the vector x
x #print x
[1]
x2 <- x**2 #square the element, or x2<-x^2 x2
[1]
X < #case sensitive!
prod1 <- X*x
prod1
[1]
CA200

3. Getting Help click the Help button on the toolbar help() help.start()
demo()
?read.table
help.search ("data.entry")
apropos (“boxplot”) - "boxplot", "boxplot.default", "boxplot.stat”
CA200

4. Statistics: Measures of Central Tendency
Typical or central points:
Mean: Sum of all values divided by the number of cases
Median: Middle value. 50% of data below and 50% above
Mode: Most commonly occurring value, value with the highest frequency
CA200

5. Statistics: Measures of Dispersion
Spread or variation in the data
Standard Deviation (σ): The square root of the average squared deviations from the mean
- measures how the data values differ from the mean
- a small standard deviation implies most values are near the average
- a large standard deviation indicates that values are widely spread above and below the average.
CA200

6. Statistics: Measures of Dispersion
Spread or variation in the data
Range: Lowest and highest value
Quartiles: Divides data into quarters. 2nd quartile is median
Interquartile Range: 1st and 3rd quartiles, middle 50% of the data.
CA200

7. Data Entry Entering data from the screen to a vector Example: 1.1
downtime <-c(0, 1, 2, 12, 12, 14, 18, 21, 21, 23, 24, 25, 28, 29, 30,30,30,33,36,44,45,47,51)
mean(downtime)
[1]
median(downtime)
[1] 25
range(downtime)
[1] 0 51
sd(downtime)
[1]
CA200

8. Data Entry – cont. Entering data from a file to a data frame
Example 1.2: Examination results: results.txt
gender arch1 prog1 arch2 prog2 m
m NA NA m m m m m f
and so on
CA200

9. Data Entry – cont. results$arch1[5] NA indicates missing value.
No mark for arch1 and prog1 in second record.
results <- read.table ("C:\\results.txt", header = T)
# download the file to desired location
results$arch1[5]
[1] 89
Alternatively
attach(results)
names(results)
allows you to access without prefix results.
arch1[5]
CA200

10. Data Entry – Missing values
mean(arch1)
[1] NA #no result because some marks are missing
na.rm = T (not available, remove) or
na.rm = TRUE
mean(arch1, na.rm = T)
[1]
mean(prog1, na.rm = T)
[1] 84.25
mean(arch2, na.rm = T)
mean(prog2, na.rm = T)
mean(results, na.rm = T)
gender arch1 prog1 arch2 prog2 NA

11. Data Entry – cont.
Use “read.table” if data in text file are separated by spaces
Use “read.csv” when data are separated by commas
Use “read.csv2” when data are separated by semicolon
CA200

12. Data Entry – cont. Entering a data into a spreadsheet:
newdata <- data.frame()
#brings up a new spreadsheet called newdata
fix(newdata)
#allows to subsequently add data to this data frame
CA200

13. Summary Statistics
Example 1.1: Downtime: summary(downtime) Min. 1st Qu. Median Mean 3rd Qu. Max Example 1.2: Examination Results: summary(results) Gender arch1 prog1 arch2 prog2 f: 4 Min. : 3.00 Min. :65.00 Min. :56.00 Min. :63.00 m:22 1st Qu.: st Qu.: st Qu.: st Qu.:77.50 Median : Median :82.50 Median :85.50 Median :84.00 Mean : Mean :84.25 Mean :81.15 Mean : rd Qu.: rd Qu.: rd Qu.: rd Qu.:92.50 Max. : Max. :98.00 Max. :96.00 Max. :97.00 NA's : 2.00 NA's : 2.00

14. Summary Statistics - cont.
Example 1.2: Examination Results: For a separate analysis use: mean(results$arch1, na.rm=T) # hint: use attach(results) [1] summary(arch1, na.rm=T) Min. 1st Qu. Median Mean 3rd Qu. Max. NA's

15. Programming in R x <- sum(downtime) # sum of elements in downtime
Example 1.3: Write a program to calculate the mean of downtime
Formula for the mean:
x <- sum(downtime) # sum of elements in downtime
n <- length(downtime) #number of elements in the vector
mean_downtime <- x/n or
mean_downtime <- sum(downtime) / length(downtime)

16. Programming in R – cont. #hint - use sqrt function
Example 1.4: Write a program to calculate the standard deviation of downtime
#hint - use sqrt function
CA200

17. Graphical displays - Boxplots
Boxplot – a graphical summary based on the median, quartile and extreme values
boxplot(downtime)
box represents the interquartile
range which contains 50% of cases
whiskers are lines that extend
from max and min value
line across the box represents median
extreme values are cases on more than
1.5box length from max/min value
CA200

18. Graphical displays – Boxplots – cont.
To improve graphical display use labels:
boxplot(downtime, xlab = "downtime", ylab = "minutes")

19. Graphical displays – Multiple Boxplots
Multiple boxplots at the same axis - by adding extra arguments to boxplot function:
boxplot(results$arch1, results$arch2,
xlab = " Architecture, Semesters 1 and 2" )
Conclusions:
marks are lower in sem2
Range of marks in narrower in sem2
Note outliers in sem1! 1.5 box length
from max/min value. Atypical values.

20. Graphical displays – Multiple Boxplots – cont.
Displays values per gender:
boxplot(arch1~gender,
xlab = "gender", ylab = "Marks(%)",
main = "Architecture Semester 1")
Note the effect of using:
main = "Architecture Semester 1”

21. Par Display plots using par function
par (mfrow = c(2,2)) #outputs are displayed in 2x2 array
boxplot (arch1~gender,
main = "Architecture Semester 1")
boxplot(arch2~gender,
main = "Architecture Semester 2")
boxplot(prog1~gender,
main = "Programming Semester 1")
boxplot(prog2~gender,
main = "Programming Semester 2")
To undo matrix type:
par(mfrow = c(1,1)) #restores graphics to the full screen

22. Par – cont. Conclusions:
- female students are doing less well in programming for sem1
- median for female students for prog. sem1 is lower than for male students

23. Histograms hist(arch1, breaks = 5, xlab ="Marks(%)",
A histogram is a graphical display of frequencies in the categories of a variable
hist(arch1, breaks = 5,
xlab ="Marks(%)",
ylab = "Number of students",
main = "Architecture Semester 1“ )
Note: A histogram with five breaks
equal width
- count observations that fill
within categories or “bins”

24. Histograms hist(arch2, xlab ="Marks(%)", ylab = "Number of students",
main = “Architecture Semester 2“ )
Note: A histogram with default breaks
CA200

25. Using par with histograms
The par can be used to represent all the subjects in the diagram
par (mfrow = c(2,2))
hist(arch1, xlab = "Architecture",
main = " Semester 1", ylim = c(0, 35))
hist(arch2, xlab = "Architecture",
main = " Semester 2", ylim = c(0, 35))
hist(prog1, xlab = "Programming",
main = " ", ylim = c(0, 35))
hist(prog2, xlab = "Programming",
Note: ylim = c(0, 35) ensures that the y-axis is the same scale for all four objects!
CA200

26. CA200

27. Stem and leaf
Stem and leaf – more modern way of displaying data! Like histograms: diagrams gives frequencies of categories but gives the actual values in each category
Stem usually depicts the 10s and the leaves depict units.
stem (downtime, scale = 2)
The decimal point is 1 digit(s) to the right of the |
0 | 012
1 | 2248
2 |
3 | 00036
4 | 457
5 | 1
CA200

28. Stem and leaf – cont. stem(prog1, scale = 2)
The decimal point is 1 digit(s) to the right of the |
6 | 5
7 | 12
7 | 66
8 |
8 | 5788
9 | 012
9 | 7778
Note: e.g. there are many students with mark 80%-85%
CA200

29. Scatter Plots To investigate relationship between variables:
plot(prog1, prog2,
xlab = "Programming, Semester 1",
ylab = "Programming, Semester 2")
Note:
one variable increases with other!
students doing well in prog1 will do well
in prog2!
CA200

30. Pairs If more than two variables are involved:
courses <- results[2:5]
pairs(courses) #scatter plots for all possible pairs or
pairs(results[2:5])
CA200

31. Pairs – cont.
CA200

32. Graphical display vs. Summary Statistics
Importance of graphical display to provide insight into the data!
Anscombe(1973), four data sets
Each data set consist of two variables on which there are 11 observations
CA200

33. Graphical display vs. Summary Statistics
Data Set 1 Data Set 2 Data Set 3 Data Set 4 x1 y1 x2 y2 x3 y3 x4 y
CA200

34. First read the data into separate vectors:
x1<-c(10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5)
y1<-c(8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68)
x2 <- c(10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5)
y2 <-c(9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74)
x3<- c(10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5)
y3 <- c(7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73)
x4<- c(8, 8, 8, 8, 8, 8, 8, 19, 8, 8, 8)
y4 <- c(6.58, 5.76, 7.71, 8.84, 8.47, 7.04, 5.25, 12.50, 5.56, 7.91, 6.89)
CA200

35. For convenience, group the data into frames:
dataset1 <- data.frame(x1,y1)
dataset2 <- data.frame(x2,y2)
dataset3 <- data.frame(x3,y3)
dataset4 <- data.frame(x4,y4)
CA200

36. It is usual to obtain summary statistics: Calculate the mean:
mean(dataset1)
x y1
mean(data.frame(x1,x2,x3,x4))
x1 x2 x3 x4
mean(data.frame(y1,y2,y3,y4))
y y y y4
Calculate the standard deviation:
sd(data.frame(x1,x2,x3,x4))
x x x x4
sd(data.frame(y1,y2,y3,y4))
Everything seems the same!
CA200

37. plot(x1,y1, xlim=c(0, 20), ylim =c(0, 13))
But when we plot:
par(mfrow = c(2, 2))
plot(x1,y1, xlim=c(0, 20), ylim =c(0, 13))
plot(x2,y2, xlim=c(0, 20), ylim =c(0, 13))
plot(x3,y3, xlim=c(0, 20), ylim =c(0, 13))
plot(x4,y4, xlim=c(0, 20), ylim =c(0, 13))
CA200

38. Note:
Data set 1 in linear with some scatter
Data set 2 is quadratic
Data set 3 has an outlier. Without them the data would be linear
Data set 4 contains x values which are equal expect one outlier. If removed, the data would be vertical.
Everything seems different!
Graphical displays are the core of getting insight/feel for the data!