Probability with R. Jane M. Horgan

      We illustrate step by step how this is calculated for

First, subtract the mean from each data point.

       downtime - meandown [1] -25.04347826 -24.04347826 -23.04347826 -13.04347826 -13.04347826 [6] -11.04347826 -7.04347826 -4.04347826 -4.04347826 -2.04347826 [11] -1.04347826 -0.04347826 3.95652174 2.95652174 4.95652174 [16] 4.95652174 4.95652174 7.95652174 10.95652174 18.95652174 [21] 19.95652174 21.95652174 25.95652174

      Then, obtain the squares of these differences.

      (downtime - meandown)^2 [1] 6.271758e+02 5.780888e+02 5.310019e+02 1.701323e+02 1.701323e+02 [6] 1.219584e+02 4.961059e+01 1.634972e+01 1.634972e+01 4.175803e+00 [11] 1.088847e+00 1.890359e-03 1.565406e+01 8.741021e+00 2.456711e+01 [16] 2.456711e+01 2.456711e+01 6.330624e+01 1.200454e+02 3.593497e+02 [21] 3.982628e+02 4.820888e+02 6.737410e+02

      Sum the squared differences.

      sum((downtime - meandown)^2) [1] 4480.957

      Finally, divide this sum by length(downtime)‐1 and take the square root.

      sqrt(sum((downtime -meandown)^2)/(length(downtime)-1)) [1] 14.27164

      You will recall that R has built‐in functions to calculate the most commonly used statistical measures. You will also recall that the mean and the standard deviation can be obtained directly with

      mean(downtime) [1] 25.04348 sd(downtime) [1] 14.27164

      We took you through the calculations to illustrate how easy it is to program in R.

      2.4.1 Creating Functions

Occasionally, you might require some statistical functions that are not available in R. You will need to create your own function. Let us take, as an example, the skewness coefficient, which measures how much the data differ from symmetry.

      The skewness coefficient is defined as

(2.1)

      A perfectly symmetrical set of data will have a skewness of 0; when the skewness coefficient is substantially greater than 0, the data are positively asymmetric with a long tail to the right, and a negative skewness coefficient means that data are negatively asymmetric with a long tail to the left. As a rule of thumb, if the skewness is outside the interval

      Example 2.2 A program to calculate skewness

      The following syntax calculates the skewness coefficient of a set of data and assigns it to a function called

      skew <- function(x) { xbar <- mean(x) sum2 <- sum((x-xbar)^2, na.rm = T) sum3 <- sum((x-xbar)^3, na.rm = T) skew <- (sqrt(length(x))* sum3)/(sum2^(1.5)) skew}

      You will agree that the conventions of vector calculations make it very easy to calculate statistical functions.

      When skew has been defined, you can calculate the skewness on any data set. For example,

      skew(downtime)

      gives

      [1] -0.04818095

      which indicates that the

Looking again at the data given Example 2.1, let us calculate the skewness coefficient

      skew(usage) [1] 1.322147

which illustrates that the data is highly skewed. Recall that the first two values are outliers in the sense that they are very much larger than the other values in the data set. If we calculate the skewness with those values removed, we get

      skew(usage[3:9]) [1] 0.4651059

      which is very much smaller than that obtained with the full set.

      2.4.2 Scripts

      There are various ways of developing programs in R.

      The most useful way of writing programs is by means of R 's own built‐in editor called

      When you want to execute a line or group of lines, highlight them and press Ctrl/R, that is, Ctrl and the letter R simultaneously. The commands are then transferred to the control window and executed.

      Alternatively, if the program is short, it may be developed interactively while working at your computer.

      Programs may also be developed in a text editor, like Notepad, saved with the .R extension and retrieved using the source