Jane M. Horgan - Probability with R

Download RStudio and familiarize yourself with its workings, by using it to do Exercise 1. Decide, at this point, whether you prefer using RStudio for your data analysis rather than using R directly. Follow your preference throughout the rest of the book. It is up to you!

1 Venables, W.N., Smith, D.M. and the R Core Team (2018), An Introduction to R, Notes on R: A Programming Environment for Data Analysis and Graphics, Version 3.6.1 (2019-07-05).

In this chapter, we explore some of the procedures available in R to summarize statistical data, and we give some examples of writing programs.

Measures of central tendency are typical or central points in the data. The most commonly used are the mean and the median.

Mean:The mean is the sum of all values divided by the number of cases, excluding the missing values.

To obtain the mean of the data in Example 1.1stored in write

mean(downtime)[1] 25.04348

So the average downtime of all the computers in the laboratory is just over 25 minutes.

Going back to the original data in Exercise 1.1 stored in marks , to obtain the mean, write

mean(marks)

which gives

[1] 57.44

To obtain the mean marks for females, write

mean(marks[1:23]) [1] 65.86957

For males,

mean(marks[24:50]) [1] 50.25926

illustrating that the female average is substantially higher than the male average.

To obtain the mean of the corrected data in Exercise 1.1, recall that the mark of 86 for the 34th student on the list was an error, and that it should have been 46. We changed it with

marks[34] <- 46

The new overall average is

mean(marks) 56.64

and the new male average is

mean(marks[24:50]) [1] 48.77778

increasing the gap between the male and female averages even further.

If we perform a similar operation for the variables in the examination data given in Example 1.2, we run into trouble. Suppose we want the mean mark for Architecture in Semester 1. In R

mean(arch1)

gives

[1] NA

Recall that, in the results file, we recorded the missing marks with the special value to indicate that these marks were “not available”. R will not perform arithmetic operations on objects containing NA , unless specifically mandated to skip remove missing values. To do this, you need to insert the argument na.rm = Tor na.rm = TRUE,(not available, remove) into the function.

For arch 1, writing

mean(arch1, na.rm = TRUE)

yields

[1] 63.56897

To obtain the mean of all the variables in results file, we use the R function sapply.

sapply(results, mean, na.rm = T)

yields

gender arch1 prog1 arch2 prog2 NA 63.56897 59.01709 51.97391 53.78378

Notice that a message is returned for gender. The reason for this is that the gender variable is nonnumeric, and R cannot calculate its mean. We could, instead specify the columns that we want to work on.

sapply(results[2:5], mean, na.rm = TRUE)

gives

arch1 prog1 arch2 prog2 63.56897 59.01709 51.97391 53.78378

Median:The median is the middle value of the data set; 50% of the observations is less and 50% is more than this value.

In R

median(downtime)

yields

[1] 25

which means that 50% of the computers experienced less than 25 minutes of downtime, while 50% experienced more than 25 minutes of downtime.

Also,

median(marks) [1] 55.5

In both of these examples ( and ), you will observe that the medians are not too far away from their respective means.

The median is particularly useful when there are extreme values in the data. Let us look at another example.

Examining the nine apps with greatest usage on your smartphone, you may find the usage statistics (in MB) are

To enter the data, write

usage <- c(39.72, 35.27, 5.73, 5.6, 3.3, 3.22, 2.52, 1.71, 1.55)

The mean is

mean(usage) [1] 10.95778

while the median is

median(usage) [1] 3.3

Unlike the previous examples, where the mean and median were similar, here the mean is more than three times the median. Looking at the data again, you will notice that the usage of the first two apps, Facebook and Chrome, is much larger than the usages of the other apps in the data set. These values are the cause of the mean being so high. Such values are often designated as outliers and are analyzed separately. Omitting them and calculating the mean and median once more, we get

mean(usage[3:9]) [1] 3.375714 median(usage[3:9]) [1] 3.22

Now, we see that there is not much difference between the mean and median.

When there are extremely high values in the data, using the mean as a measure of central tendency gives the wrong impression. A classic example of this is wage statistics where there may be a few instances of very high salaries, which will grossly inflate the average, giving the impression that salaries are higher than they actually are.

Measures of dispersion, as the name suggests, estimate the spread or variation in a data set. There are many ways of measuring spread, and we consider some of the most common.

Range:The simplest measure of spread of data is the range, which is the difference between the maximum and the minimum values.

rangedown <- max(downtime) - min(downtime) rangedown [1] 51

tells us that the range in the downtime data is 51 minutes.

rangearch1 <- max(arch1, na.rm = T) - min(arch1, na.rm = T) rangearch1 [1] 97

gives the range of the marks awarded in Architecture in Semester 1.

The R function rangemay also be used.

range(arch1, na.rm = TRUE) [1] 3 100

which gives the minimum (3) and the maximum (100) of the marks obtained in Architecture in Semester 1.

Note that, since arch1 contains missing values, the declaration of na.rm = Tor equivalently na.rm = TRUEneeds to be used.

To get the range for all the examination subjects in results, we use the function sapply.

sapply(results[2:5], range, na.rm = TRUE)

gives the minimum and maximum of each subject.

arch1 prog1 arch2 prog2 [1,] 3 12 6 5 [2,] 100 98 98 97