Descriptives
Mean
The mean of a random variable is given as
\[ \frac{1}{n}\Sigma_{i=1}^{n}x_{i} \]
data = c(10,22,53,46,54,62,71,38,69,10)
mean(data)## [1] 43.5Median
data = c(10,22,53,46,54,62,71,38,69,10)
median(data)## [1] 49.5Mode
data = c(10,22,53,46,54,62,71,38,69,10)
find_mode <- function(x) {
u <- unique(x)
tab <- tabulate(match(x, u))
u[tab == max(tab)]
}
find_mode(data)## [1] 10Basic Graphs
Bar Chart
starwars %>%
select(eye_color) %>%
ggplot(aes(x = eye_color)) +
geom_bar(fill = '#FA6713') +
labs(y = 'Number of Individuals', x = 'Eye Color', title = 'Number of Individuals by Eye Color') +
theme(axis.text.x = element_text(angle = 90))
Histogram
set.seed(1234)
d = rnorm(200, mean = 0, sd = 1)
hist(d, col = 'maroon', main='Histogram of Data', xlab = 'Data')
Box Plot
starwars %>%
drop_na %>%
select(sex, height) %>%
ggplot(aes(x = sex, y = height, fill = sex)) +
geom_boxplot() +
labs(x = 'Gender', y = 'Height', title = 'Heights of individuals by Gender', fill = 'Gender')
Pie Chart
Prop <- c(3,7,9,1,2)
myPalette <- brewer.pal(5, "Set2")
# You can change the border of each area with the classical parameters:
pie(Prop , labels = c("A","B","C","D","E"), border="white", col=myPalette )
set.seed(2348907)
x <- rnorm(100)
y <- 2*(x+rnorm(100))
m = as.data.frame(cbind(x,y))
model1 = lm(y~x, data = m)
summary(model1)##
## Call:
## lm(formula = y ~ x, data = m)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8412 -1.0767 -0.1808 1.2216 4.1540
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0454 0.1851 0.245 0.807
## x 2.1087 0.1814 11.627 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.848 on 98 degrees of freedom
## Multiple R-squared: 0.5797, Adjusted R-squared: 0.5754
## F-statistic: 135.2 on 1 and 98 DF, p-value: < 2.2e-16
The regression model is given as
\[ Y = 0.0454 + 2.1087X \]
Line of best fit
ggplot()+geom_point(aes(x, y))+
geom_smooth(aes(x,y),method = 'lm', se=F) +
labs(title = 'Relationship between X and Y', x = 'X', y = 'Y')## `geom_smooth()` using formula = 'y ~ x'
Random Sampling
# Getting some random values to use here
set.seed(seed = 14412)
thevalues <- sample(x = 1:100,size = 1000,replace = TRUE)
head(thevalues,20)## [1] 22 10 34 56 31 7 54 77 65 44 92 55 45 93 99 38 23 50 59 68# Obtaining the unique vector of those values
thevalues.unique <- unique(thevalues)
thevalues.unique## [1] 22 10 34 56 31 7 54 77 65 44 92 55 45 93 99 38 23 50
## [19] 59 68 16 96 87 47 19 15 43 60 82 49 88 6 40 98 53 13
## [37] 11 18 72 71 66 76 81 61 42 8 3 39 100 57 79 5 32 17
## [55] 27 94 26 67 1 85 36 58 97 4 70 35 95 12 80 30 75 90
## [73] 84 46 91 83 21 37 14 86 51 9 78 74 69 64 41 25 89 2
## [91] 63 73 20 48 52 24 29 28 62 33# Create a sample without replacement (i.e. take the ball out and don't put it back in)
sample1 <- sample(x = thevalues.unique,size = 10,replace = FALSE)
sample1## [1] 16 5 43 18 46 70 83 56 85 100# Remove the sampled items from the vector of values
thevalues.unique <- thevalues.unique[!(thevalues.unique %in% sample1)]
thevalues.unique## [1] 22 10 34 31 7 54 77 65 44 92 55 45 93 99 38 23 50 59 68 96 87 47 19 15 60
## [26] 82 49 88 6 40 98 53 13 11 72 71 66 76 81 61 42 8 3 39 57 79 32 17 27 94
## [51] 26 67 1 36 58 97 4 35 95 12 80 30 75 90 84 91 21 37 14 86 51 9 78 74 69
## [76] 64 41 25 89 2 63 73 20 48 52 24 29 28 62 33# Another sample, and another removal
sample2 <- sample(x = thevalues.unique,size = 10,replace = FALSE)
thevalues.unique <- thevalues.unique[!(thevalues.unique %in% sample2)]
thevalues.unique## [1] 22 34 31 7 54 77 65 44 92 55 45 99 50 59 68 96 87 47 19 15 60 82 49 88 6
## [26] 40 98 53 13 11 72 71 66 76 81 61 42 3 39 57 79 32 17 26 67 1 36 58 97 4
## [51] 35 95 12 80 30 75 84 91 21 37 14 86 51 9 78 74 69 64 41 25 89 2 63 20 48
## [76] 52 24 29 28 62Coin Toss Sample Space Generation
# toss a coin
coin <- c('heads', 'tails')
sample(coin, size = 1)## [1] "tails"# draw 2 elements without replacement
sample(coin, size = 2)## [1] "tails" "heads"sample(coin, size = 2, replace = FALSE)## [1] "heads" "tails"# draw 4 elements with replacement
sample(coin, size = 4, replace = TRUE)## [1] "tails" "tails" "tails" "heads"# five tosses
sample(coin, size = 5, replace = TRUE)## [1] "heads" "tails" "heads" "heads" "tails"# another five tosses
sample(coin, size = 5, replace = TRUE)## [1] "tails" "heads" "heads" "tails" "tails"# set random seed
# set.seed(1257)
# toss a coin with replacement
sample(coin, size = 4, replace = TRUE)## [1] "tails" "heads" "tails" "heads"# tossing a fair coin
coin <- c("heads", "tails")
sample(coin)## [1] "heads" "tails"# equivalent
sample(coin, prob = c(0.5, 0.5))## [1] "tails" "heads"# tossing a loaded coin (20% heads, 80% tails)
sample(coin, size = 5, replace = TRUE, prob = c(0.2, 0.8))## [1] "tails" "tails" "tails" "tails" "heads"# number of flips
num_flips <- 100
# flips simulation
coin <- c('heads', 'tails')
flips <- sample(coin, size = num_flips, replace = TRUE)
# number of heads and tails
freqs <- table(flips)
freqs## flips
## heads tails
## 56 44# one more 100 flips
flips <- sample(coin, size = num_flips, replace = TRUE, prob=c(0.5, 0.5))
freqs <- table(flips)
freqs## flips
## heads tails
## 54 46heads_freq <- cumsum(flips == 'heads') / 1:num_flips
plot(heads_freq, # vector
type = 'l', # line type
lwd = 2, # width of line
col = 'tomato', # color of line
las = 1, # orientation of tick-mark labels
ylim = c(0, 1), # range of y-axis
xlab = "number of tosses", # x-axis label
ylab = "relative frequency") # y-axis label
abline(h = 0.5, col = 'gray50')