Mean, Median, Variance, Standard Deviation

Once data is entered into R, the measures of center and variation are easy!

my_data <- c(number1, number 2, etc)

mean(my_data)   # compute the mean of my_data
median(my_data) # compute the median of my_data

var(my_data)    # compute the variance of my_data
sd(my_data)     # compute the standard deviation of my_data

Notice it only shows one mumber in the output. This is a bit annoying but we will need to deal with this qwirk. One way to handle this is to comment out three of the line so it only displays one at a time.

Histograms

To create a histogram, first enter your data and then use the hist() function.

# Histogram (let R handle everything)
hist(data)

R let you customize almost everything. Here's a simple example:

hist( data,                             # data
      main = "A lovely Title",          # add a title
      xlab = "x values are..."          # specify labels on x-axis
      xlim = c(min,max),                # specify desired range
      breaks = c(seq(list of values)),  # classes / bins
      col = c("blue", "red", etc),      # colors of bars
      freq = FALSE,                     # TRUE by default, when false gives relative freq
    )

Note that R automatically decides on the width of the bins/classes.

An example with a few direct specifications:

Dot Plots

To create a dot plot>, first you need to load a special package called plotrix. A package is a bit of code someone else wrote so we can use it!

Next, enter your data and then use the function dotplot.mtb().

# Dot Plot
library(plotrix) # load a special package

dotplot.mtb( data ) # dot plot

Here's a simple example:

Stem-and-Leaf Plots

To create a stem-and-leaf pot>, use the stem() function.

# Stem-Leaf Plot

stem( data ) # stem-leaf plot

Suppose that we want to a make a stem-and-leaf plot of Babe Ruth's homeruns:

Pie Charts

To create a pie chart>, use the pie() function.

# Pie Charts

slices <- c(list of values)  # enter data of the 'slices'
lbls <- c("category1", "category2", etc) # enter label names for categories matching slice data
pie( slices, 
     label = lbls, 
     main = "Add Title"
     )

We can create pie charts:

One more example: Time Series

We can plot the time series for the 1918-1919 influenze epidemic. The plot for the number of cases is in red, and the plot for 10 times the number of deaths is in blue.

Five Number Summary

To create a five number summary, first enter your data and then use the summary() function.

# Five Number Summary
summary(data)

Box-Whisker Plots

To create a box-whisker plot, use the boxplot() function.

# Box Plot

boxplot(data)

Simple box plot example:

Note that it displays the boxplot vertically by default. Also, it identifies outliers by default.

Here is another example with more options specified:

Binomial Distributions compute the probability of exactly \(x\) successess in \(n\) trials of a binomial probability distribution with probability of success \(p\).

Syntax in R: base distribution: binom().

There's 4 prefixes: d (exactly), p (cumulative), q (quartile aka "inverse"), r (random sample).

dbinom(x = #, size = #, prob = #) 
    # computes the probability of exactly x successess 
        # x = number of successess, 
        # size = number of trials, 
        # prob = probability of success of single trial

pbinom(#, size = #, prob = #) 
    # computes the probability of at most x = # successess (cumulative)
    # note: need to remove 'x=' from first argument for cumulative

qbinom(p = #, size = #, prob = #)
    # inverse binomial: finds z such that P(x <= z) = p (right-tail)

rbinom(n = #, size = #, prob = #) 
    # generates required number of random values of given probability from a given sample
    # n = number of observations     

Examples:

Continuous Distributions compute the probability of a range of values from a given distribution.

Normal Distrbutions

Let's start with the Normal Distributions.

Syntax in R: base distribution: norm().

There's 4 prefixes: d (exactly), p (cumulative), q (quartile aka "inverse"), r (random sample).

In R, to get the exact value of $f(x)$, use dnorm.

dnorm(x = #, mean = #, sd = #) 
    # computes the probability of exactly x successess 
        # x = exact value of x (any real number), 
        # mean = mean of normal distribution
        # sd = standard deviation of normal distribution

We don't really need to know or use dnorm unless we want to plot the curve.

To compute probabilities of intervals of a normal distribution, we use the cumulative distribution. In math notation, if \(X \sim N(\mu,\sigma) \) then to compute $$ P(X \le b) \qquad \text{(i.e. less than b--left tail)} $$ we use:

   
pnorm(b, mean = #, sd = #) 
    # computes the cumulative probability P(X <= b)
        # mean = mean of normal distribution
        # sd = standard deviation of normal distribution
        # lower value = -oo 
        # higher value = b

Often, we need other calculations using the normal curve. For example: $$ P(a \le X \le b) \qquad \text{(i.e. between a and b)} $$ or $$ P(X \ge a) \qquad \text{(i.e. greater than a--right tail)} $$

Thanfully, with a bit of basic geometry (drawing pictures helps!), we can use the cumulative from above to accomplish the other two probabilities.

# P(a <= X <= b): 
pnorm(b, mean = #, sd = #) - dnorm(a, mean = #, sd = #) 
    # computes the probability under a normal distribution between the values of a and b
    # lower value = a 
    # higher value = b

# P(X >= a):   
pnorm(a, mean = #, sd = #, lower.tail = FALSE) 
    # computes the probability under a normal distribution for the values greater than a
    # lower value = a 
    # higher value = +oo 

Examples:

Inverse Normal

qnorm(p = #, mean = #, sd = #, lower.tail = FALSE)
    # inverse normal: finds z such that P(x <= z) = p (right-tail)

Examples:

Additional Continuous Distributions

In addition to the normal distribution, R has other distributions programmed. In the code below, I'll reference a few of the distributions we use in our Intro Statistics course.