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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 1.1 stored 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 skipremove missing values. To do this, you need to insert the argument na.rm = T or na.rm = TRUE, (not available, remove) into the function.

For arch1, 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.