Jane M. Horgan - Probability with R

Standard deviation:The standard deviation ( sd) measures how much the data values deviate from their average. It is the square root of the average squared deviations from the mean. A small standard deviation implies most values are near the mean. A large standard deviation indicates that values are widely spread above and below the mean.

In R

sd(downtime)

yields

[1] 14.27164.

Recall that we calculated the mean to be 25.04 minutes. We might loosely describe the downtime as being “25 minutes on average give or take 14 minutes.”

For the data in

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

gives the standard deviation of each examination subject in :

arch1 prog1 arch2 prog2 24.37469 23.24012 21.99061 27.08082

Quantiles:The quantiles divide the data into proportions, usually into quarters called quartiles, tenths called deciles, and percentages called percentiles. The default calculation in R is quartiles.

quantile(downtime)

gives

0% 25% 50% 75% 100% 0.0 16.0 25.0 31.5 51.0

The first quartile (16.0) is the value that breaks the data so that 25% is below this value and 75% is above.

The second quartile (25.0) is the value that breaks the data so that 50% is below and 50% is above (notice that the 2nd quartile is the median).

The third quartile (31.5) is the value that breaks the data so that 75% is below and 25% is above.

We could say that 25% of the computer systems in the laboratory experienced less than 16 minutes of downtime, another 25% of them were down for between 16 and 25 minutes, and so on.

Interquartile range : The difference between the first and third quartiles is called the interquartile range and is sometimes used as a rough estimate of the standard deviation. In downtime it is , not too far away from 14.27, which we calculated to be the standard deviation.

Deciles : Deciles divide the data into tenths. To get the deciles in R , first define the required break points

deciles <- seq(0, 1, 0.1)

The function seqcreates a vector consisting of an equidistant series of numbers. In this case, seqassigns values in [0, 1] in intervals of 0.1 to the vector called deciles. Writing in R

deciles

shows what the vector contains

[1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Adding this extra argument to the quantile function

quantile(downtime, deciles)

yields

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0.0 4.0 12.8 19.8 22.6 25.0 29.2 30.0 34.8 44.8 51.0

Interpreting this output, we could say that 90% of the computer systems in the laboratory experienced less than 45 minutes of downtime.

Similarly, for the percentiles, use

percentiles <- seq(0, 1, 0.01)

as an argument in the quantile function, and write

quantile(downtime, percentiles)

A quicker way of summarizing the data is to use the summaryfunction.

summary(downtime)

returns

Min. 1st Qu. Median Mean 3rd Qu. Max. 0.00 16.00 25.00 25.04 31.50 51.00

which are the minimum the first quartile, the median, the mean, the third quartile, and the maximum, respectively.

For , we might write

summary(arch1)

which gives

Min. 1st Qu. Median Mean 3rd Qu. Max. NA's 3.00 46.75 68.50 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 . First, it is wise to make a declaration about the categorical variable gender.

gender <- factor(gender)

designates the variable gender as a factor, and ensures that it is treated as such in the summaryfunction.

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.

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 . The formula for the mean of a variable with values is given by

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 data points stored in an vector is

We illustrate step by step how this is calculated for .