Using R

Perhaps the simplest way to arrange data is in a table. The table() function of R creates tables. Notice that the next example also shows how to read categorical data in using the scan() function (by specifying that the data to be read will be character data rather than the default—numeric data).

To plot the same table as a bar chart, pass the output of the table to the barplot() function:

and as a pie chart

Use stem() to generate a stem-leaf plot, and hist() to generate a histogram.

A word of caution is in order. The stem() command in R will attempt to make plots that are of a "nice" size, and it may create the stems differently than you would. If you don’t read the plot carefully, you can make a mistake. Consider for example:

Notice, in the plot above, that the value 92 is lumped in with the 80s, coming after 80+8 as 80+12. The 55 is also very unintuitively lumped into the 40s and the 31 is in the 20s. You can adjust the scale of the plot to give it an appearance more sensible to us. Is this case, if we ask for a plot 2 times as long as normal, the appearance is better:

Use boxplot() to draw single box plots, or multiple boxplots together.

Use qqnorm() to generate a normal probability plot, and qqline() to add a line to an existing plot. If you prefer your normal reference line to be computed the same way as SPSS, use the abline() function to add the line by hand.

The plot() function generates scatter plots. The abline() function adds a line to an existing scatter plot and is particularly useful when fed via the linear model function lm() (to add a regression line).

For the data below, the correlation coefficient is returned by cor() and a line of best fit is calculated via lm().

We can get a picture of the normal distribution function. This is the curve we usually draw; the areas under it represent probabilities.

Other normal distributions (that require arguments) can be plotted with the help of "anonymous functions", like the following distribution which has a standard deviation of 2.

R can perform 1 variable and 2 variable t tests. Alternative hypotheses can be "less", "greater", and "two.sided". For example, we can solve this exercise from the book (Section 10.3 #23): The mean waiting time at a restaurant is 84.3 seconds. A manager devises a new system and randomly measures some wait times, 108.5, 67.4, 58.0, 75.9, 65.1, 80.4, 95.5, 86.3, 70.9, 72. Is there evidence that the new method works? Use alpha=0.01.

With a p-value much greater than the stated alpha=0.01 we fail to reject.

We can get a picture of the t distribution function. This is the curve we usually draw; the areas under it represent probabilities. For example, this is the t distribution with 3 degrees of freedom.

To see that the t distribution becomes more normal as the number of degrees of freedom increases, we could plot several together.

R can perform chi square tests. For example, we can solve this exercise from the book (section 12.1 #11): A bag of plain M&Ms should contain 13% brown, 14% yellow, 13% red, 24% blue, 20% orange, and 16% green candies. A randomly selected bag contained 61 brown, 64 yellow, 54 red, 61 blue, 96 orange, 64 green. Test whether the bag follows the stated distribution.

If our significance level was alpha=0.05, we can see to reject.

R can do hypothesis tests for proportions, although it uses the chi-squared distribution for the calculation. Alternative hypotheses can be "less", "greater", or "two.sided". For example, we can work this exercise from the book (Section 12.1 #23): According to the Census Bureau, 7.1% of babies with non-smoking mothers have low birth weight. To see if mothers from 35 to 39 had had more low weight babies, a random sample measured 15 low weight babies out of 160 births. Test using alpha = 0.05.

Since the p-value is larger than 0.05, we fail to reject.

In some situations, you may wish to have R embed its fonts into PDF files you create (particularly if you substitute different fonts for the standard font set). You can add them using the embedFonts() function on a file you’ve already created to run ghostscript and process it.

Uniformly distributed random numbers (i.e. the kind of random numbers you get from rolling a fair die) are generated with the runif() function. By default, the random numbers come from the interval (0,1) but that may be adjusted via arguments. One argument is always required: a count of random numbers to produce.

For simple "die rolling," i.e. random integer numbers, this function may be convenient:

And here are some examples:

The distributions that R knows about can be used to generate example data (or samples). Say we want to conduct a binomial experiment where we roll 5 dice and count the number of ones. We’d like to get an idea of the shape of this distribution, so we’ll repeat the experiment 100 times and see what the outcomes are. We use rbinom(n,k,p), which generates random numbers according to the binomial distribution.

This is decidedly non-normal, but we can verify that things are different if the number of dice is different. Verify that the number of ones appearing on a roll of 72 dice is approximately normally distributed with a mean of 12 and a standard deviation of 3.162278.

We can generate random numbers based on the normal distribution just as easily. For example, we’d like a random sample of 20 items from a normally distributed population that has mean 50 and standard deviation about 8.

Perhaps we would prefer integers for our example data set.

I have an entire page devoted to modeling the central limit theorem using R’s built-in random number generators. Check it out at:

ModelCLTmean.html Central limit theorem for the sample mean.

ModelCLTprop.html Central limit theorem for the sample proportion.

From the book, we learn that human pregnancies last an average of 266 days and are approximately normally distributed with a standard deviation of 16. We can simulate a data set of 10,000 pregnancies using random numbers.

Now that we have a simulated population, we can use the sample() function to take random samples from it. For example, if we would like a random sample of size 4:

Or we might compute the mean of a sample of size 4:

The central limit theorem tells us that the sampling distribution of the mean will have the same center as the parent population, but a smaller standard deviation. We can verify the sampling distribution of the mean (for samples of size 4) looks normal also with standard deviation of 16/sqrt(4). Here we use replicate() to find the means of 100 different samples of size 4; then we compute the mean and standard deviation. Note how close the results are to 266 and 8.