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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 ). Basics. 6+7*3/2 #general expression [1] 16.5 x <- 1:4 #integers are assigned to the vector x x #print x [1] 1 2 3 4

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CA200 (based on the book by Prof. Jane M. Horgan )

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  1. 3. Basics of R – cont. Summarising Statistical DataGraphical 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] 1 2 3 4 • x2 <- x**2 #square the element, or x2<-x^2 x2 [1] 1 4 9 16 • X <- 10 #case sensitive! prod1 <- X*x prod1 [1] 10 20 30 40 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] 25.04348 median(downtime) [1] 25 range(downtime) [1] 0 51 sd(downtime) [1] 14.27164 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 99 98 83 94 m NA NA 86 77 m 97 97 92 93 m 99 97 95 96 m 89 92 86 94 m 91 97 91 97 m 100 88 96 85 f 86 82 89 87 and so on CA200

  9. Data Entry – cont. • 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] [1] 89 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] 83.33333 • 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 94.42857 93.00000 89.75000 90.37500

  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. 0.00 16.00 25.00 25.04 31.50 51.00 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.: 79.25 1st Qu.:80.75 1st Qu.:77.75 1st Qu.:77.50 Median : 89.00 Median :82.50 Median :85.50 Median :84.00 Mean : 83.33 Mean :84.25 Mean :81.15 Mean :83.85 3rd Qu.: 96.00 3rd Qu.:90.25 3rd Qu.:91.00 3rd Qu.:92.50 Max. :100.00 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] 83.33333 summary(arch1, na.rm=T) Min. 1st Qu. Median Mean 3rd Qu. Max. NA's 3.00 79.25 89.00 83.33 96.00 100.00 2.00

  15. Programming in R • 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. • 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 • 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", main = " ", ylim = c(0, 35)) 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 | 1134589 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 | 01112223 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 y4 10 8.04 10 9.14 10 7.46 8 6.58 8 6.95 8 8.14 8 6.77 8 5.76 13 7.58 13 8.74 13 12.74 8 7.71 9 8.81 9 8.77 9 7.11 8 8.84 11 8.33 11 9.26 11 7.81 8 8.47 14 9.96 14 8.10 14 8.84 8 7.04 6 7.24 6 6.13 6 6.08 8 5.25 4 4.26 4 3.10 4 5.39 19 12.50 12 10.84 12 9.13 12 8.15 8 5.56 7 4.82 7 7.26 7 6.42 8 7.91 5 5.68 5 4.74 5 5.73 8 6.89 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) x1 y1 9.000000 7.500909 mean(data.frame(x1,x2,x3,x4)) x1 x2 x3 x4 9 9 9 9 mean(data.frame(y1,y2,y3,y4)) y1 y2 y3 y4 7.500909 7.500909 7.500000 7.500909 • Calculate the standard deviation: sd(data.frame(x1,x2,x3,x4)) x1 x2 x3 x4 3.316625 3.316625 3.316625 3.316625 sd(data.frame(y1,y2,y3,y4)) y1 y2 y3 y4 2.031568 2.031657 2.030424 2.030579 Everything seems the same! CA200

  37. 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!

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