63.57 83.25 100.00 3.00
An entire data frame may be summarized by using the summary command. Let us do this in the data frame
gender <- factor(gender)
designates the variable gender as a factor, and ensures that it is treated as such in the summary function.
summary(results) gender arch1 prog1 arch2 prog2 f: 19 Min. : 3.00 Min. :12.00 Min. : 6.00 Min. : 5.00 m:100 1st Qu.: 46.75 1st Qu.:40.00 1st Qu.:40.00 1st Qu.:30.00 Median : 68.50 Median :64.00 Median :48.00 Median :57.00 Mean : 63.57 Mean :59.02 Mean :51.97 Mean :53.78 3rd Qu.: 83.25 3rd Qu.:78.00 3rd Qu.:61.00 3rd Qu.:76.50 Max. :100.00 Max. :98.00 Max. :98.00 Max. :97.00 NA's : 3.00 NA's : 2.00 NA's : 4.00 NA's : 8.00
Notice how the display for gender is different than that for the other variables; we are simply given the frequency for each gender.
2.4 Programming in R
One of the great benefits of R is that it is possible to write your own programs and use them as functions in your analysis. Programming is extremely simple in R because of the way it handles vectors and data frames. To illustrate, let us write a program to calculate the mean of
In standard programming languages, implementing this formula would necessitate initialization and loops, but with R, statistical calculations such as these are much easier to implement. For example,
sum(downtime)
gives
576
which is the sum of the elements in
length(downtime)
gives
23
gives the number of elements in
To calculate the mean, write
meandown <- sum(downtime)/length(downtime) meandown [1] 25.04348
Let us also look at how to calculate the standard deviation of the data in
The formula for the standard deviation of
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,
