Univariate Data
3 Univariate Data
There is a distinction between types of data in statistics and
R
knows about some of these differences. In particular, initially, data
can be of three basic types: categorical, discrete numeric and
continuous numeric. Methods for viewing and summarizing the data
depend on the type, and so we need to be aware of how each is handled
and what we can do with it.
Categorical data is data that records categories. Examples could be, a
survey that records whether a person is for or against a proposition.
Or, a police force might
keep track of the race of the individuals they pull over on the
highway. The
U.S. census, which takes
place every 10 years, asks several different questions of a
categorical nature. Again, there was one on race which in the year
2000 included 15 categories with writein space for 3 more for this
variable (you could mark yourself as multiracial). Another example,
might be a doctor's chart which records data on a patient. The gender or the
history of illnesses might be treated as categories.
Continuing the doctor example, the age of a person and their weight
are numeric quantities. The age is a discrete numeric quantity
(typically) and the weight as well (most people don't say they are 4.673
years old). These numbers are usually reported as integers.
If one really needed to know precisely, then they could in theory take
on a continuum of values, and we would consider them to be continuous.
Why the distinction? In data sets, and some tests it is important to
know if the data can have ties (two or more data points with the same value). For discrete data it is true, for
continuous data, it is generally not true that there can be ties.
A simple, intuitive way to keep track of these is to ask what is the mean
(average)? If it doesn't make sense then the data is categorical (such
as the average of a nonsmoker and a smoker), if it
makes sense, but might not be an answer (such as 18.5 for age when you
only record integers integer) then the data is discrete otherwise it
is likely to be continuous.
3.1 Categorical data
We often view categorical data with tables but we may also look at
the data graphically with bar
graphs or pie charts.
3.2 Using tables
The
table command allows us to look at tables.
Its simplest usage looks like
table(x) where
x is a
categorical variable.
Example: Smoking survey
A survey asks people if they smoke or not. The data is
Yes, No, No, Yes, Yes
We can enter this into
R with the
c() command, and
summarize with the
table command as follows
> x=c("Yes","No","No","Yes","Yes")
> table(x)
x
No Yes
2 3
The
table command simply adds up the frequency of each unique
value of the data.
3.3 Factors
Categorical data is often used to classify data into various
levels or factors. For example, the smoking data could be part of a
broader survey on student health issues.
R has a special
class for working with factors which is occasionally important to know
as
R will automatically adapt itself when it knows it has a factor. To make a
factor is easy with the command
factor or
as.factor. Notice the difference
in how
R treats factors with this example
> x=c("Yes","No","No","Yes","Yes")
> x # print out values in x
[1] "Yes" "No" "No" "Yes" "Yes"
> factor(x) # print out value in factor(x)
[1] Yes No No Yes Yes
Levels: No Yes # notice levels are printed.
3.4 Bar charts
A bar chart draws a bar with a a height proportional to the count in
the table. The height could be given by the frequency, or the
proportion. The graph will look the same, but the scales may be
different.
Suppose, a group of 25 people are surveyed as to their beerdrinking
preference. The categories were (1) Domestic can, (2) Domestic
bottle, (3) Microbrew and (4) import. The raw data is
3 4 1 1 3 4 3 3 1 3 2 1 2 1 2 3 2 3 1 1 1 1 4 3 1
Let's make a barplot of both frequencies and proportions. First, we
use the
scan function to read in the data then we plot (figure
1)
> beer = scan()
1: 3 4 1 1 3 4 3 3 1 3 2 1 2 1 2 3 2 3 1 1 1 1 4 3 1
26:
Read 25 items
> barplot(beer) # this isn't correct
> barplot(table(beer)) # Yes, call with summarized data
> barplot(table(beer)/length(beer)) # divide by n for proportion
Figure 1: Sample barplots
Notice a few things:
3.5 Pie charts
The same data can be studied with pie charts using the
pie
function.
^{2}^{3} Here are some simple examples illustrating the usage
(similar to
barplot(), but with some added features.
> beer.counts = table(beer) # store the table result
> pie(beer.counts) # first pie  kind of dull
> names(beer.counts) = c("domestic\n can","Domestic\n bottle",
"Microbrew","Import") # give names
> pie(beer.counts) # prints out names
> pie(beer.counts,col=c("purple","green2","cyan","white"))
# now with colors
The first one was kind of boring so we added names. This is done with
the
names which allows us to specify names to the
categories. The resulting piechart shows how the names are
used. Finally, we added color to the piechart. This is done by setting
the piechart attribute
col. We set this equal to a vector of
color names that was the same length as our
beer.counts. The
help command (
?pie) gives some examples for automatically
getting different colors, notably using
rainbow and
gray.
Notice we used additional
arguments to the function
pie The syntax for these is
name=value. The ability
to pass in named values to a function, makes it easy to have fewer
functions as each one can have more functionality.
3.6 Numerical data
There are many options for viewing numerical data. First, we
consider the common numerical summaries of center and spread.
3.7 Numeric measures of center and spread
To describe a distribution we often want to know where is it
centered and what is the spread. These are typically measured with
mean and variance (or standard deviation), or the median and more generally the
fivenumber summary. The
R commands for these are
mean,
var,
sd,
median,
fivenum and
summary.
Example: CEO salaries
Suppose, CEO yearly compensations are sampled and the following are
found (in millions). (This is before being indicted for cooking
the books.)
12 .4 5 2 50 8 3 1 4 0.25
> sals = scan() # read in with scan
1: 12 .4 5 2 50 8 3 1 4 0.25
11:
Read 10 items
> mean(sals) # the average
[1] 8.565
> var(sals) # the variance
[1] 225.5145
> sd(sals) # the standard deviation
[1] 15.01714
> median(sals) # the median
[1] 3.5
> fivenum(sals) # min, lower hinge, Median, upper hinge, max
[1] 0.25 1.00 3.50 8.00 50.00
> summary(sals)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.250 1.250 3.500 8.565 7.250 50.000
Notice the
summary command. For a numeric variable it prints
out the five number summary and the median. For other variables, it adapts
itself in an intelligent manner.
Some Extra Insight: The difference between fivenum and the quantiles.
You may have noticed the slight difference between the
fivenum and the
summary command. In particular, one gives 1.00 for the lower
hinge and the other 1.250 for the first quantile. What is the
difference? The story is below.
The median is the point in the data that splits it into half. That is,
half the data is above the data and half is below. For example, if our
data in sorted order is
10, 17, 18, 25, 28
then the midway number is clearly 18 as 2 values are less and 2 are
more. Whereas, if the data had an additional point:
10, 17, 18, 25, 28, 28
Then the midway point is somewhere between 18 and 25 as 3 are larger
and 3 are smaller. For concreteness, we average the two values giving
21.5 for the median. Notice, the
point where the data is split in half depends on the number of data
points. If there are an odd number, then this point is the
(
n+1)/2 largest data point. If there is an even number of data
points, then again we use the (
n+1)/2 data point, but since this is
a fractional number, we average the actual data to the left and the
right.
The idea of a quantile generalizes this median. The
p quantile,
(also known as the 100p%percentile) is the point in the data where
100p% is less, and 100(1p)% is larger. If there are
n data points,
then the
p quantile occurs at the position 1+(
n1)
p with weighted
averaging if this is between integers. For example the .25 quantile of
the numbers 10,17,18,25,28,28 occurs at the position 1+(61)(.25) =
2.25. That is 1/4 of the way between the second and third number which
in this example is 17.25.
The .25 and .75 quantiles are denoted the
quartiles. The
first quartile is called
Q_{1}, and the third quartile is called
Q_{3}. (You'd think the second quartile would be called
Q_{2}, but use
``the median'' instead.) These values are in the
R function
RCodesummary. More generally, there is a
quantile
function which will compute any quantile between 0 and 1. To find the
quantiles mentioned above we can do
> data=c(10, 17, 18, 25, 28, 28)
> summary(data)
Min. 1st Qu. Median Mean 3rd Qu. Max.
10.00 17.25 21.50 21.00 27.25 28.00
> quantile(data,.25)
25%
17.25
> quantile(data,c(.25,.75)) # two values of p at once
25% 75%
17.25 27.25
There is a historically popular set of alternatives to the quartiles,
called the hinges that are somewhat easier to compute by hand. The
median is defined as above. The lower hinge is then the median of all
the data to the left of the median, not counting this particular data
point (if it is one.) The upper hinge is similarly defined. For
example, if your data is again 10, 17, 18, 25, 28, 28, then the median is 21.5, and the
lower hinge is the median of 10, 17, 18 (which is 17) and the upper hinge is
the median of 25,28,28 which is 28. These are available in the function
fivenum(), and later appear in the boxplot function.
Here is an illustration with the
sals data, which has
n=10. From above we should have the median at (10+1)/2=5.5, the
lower hinge at the 3rd value and the upper hinge at the 8th largest
value. Whereas, the value of
Q_{1} should be at the 1+(101)(1/4) =
3.25 value. We can check that this is the case by sorting the data
> sort(sals)
[1] 0.25 0.40 1.00 2.00 3.00 4.00 5.00 8.00 12.00 50.00
> fivenum(sals) # note 1 is the 3rd value, 8 the 8th.
[1] 0.25 1.00 3.50 8.00 50.00
> summary(sals) # note 3.25 value is 1/4 way between 1 and 2
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.250 1.250 3.500 8.565 7.250 50.000
3.8 Resistant measures of center and spread
The most used measures of center and spread are the mean and standard deviation
due to their relationship with the normal distribution, but they
suffer when the data has long tails, or many outliers. Various
measures of center and spread have been developed to handle this.
The median is just such a resistant measure. It is oblivious
to a few arbitrarily large values. That is, is you make a measurement
mistake and get 1,000,000 for the largest value instead of 10 the
median will be indifferent.
Other resistant measures are available. A common one for the center
is the trimmed mean
. This is useful if the data has many
outliers (like the CEO compensation, although better if the data is symmetric). We trim off a certain
percentage of the data from the top and the bottom and then take the
average. To do this in
R we need to tell the
mean() how
much to trim.
> mean(sals,trim=1/10) # trim 1/10 off top and bottom
[1] 4.425
> mean(sals,trim=2/10)
[1] 3.833333
Notice as we trim more and more, the value of the mean gets closer to
the median which is when
trim=1/2. Again notice how we used a named argument to the
mean function.
The variance and standard deviation are also sensitive to outliers.
Resistant measures of spread include the
IQR and the
mad.
The IQR or interquartile range is the difference of the 3rd
and 1st quartile. The function
IQR calculates it
for us
> IQR(sals)
[1] 6
The median average deviation (MAD) is also a useful, resistant measure of
spread. It finds the median of the absolute differences from the
median and then multiplies by a constant. (Huh?) Here is a formula
median  X_{i}  median(X)  (1.4826)
That is, find the median, then find all the differences from the
median. Take the absolute value and then find the median of this new
set of data. Finally, multiply by the constant. It is easier to do with
R than to describe.
> mad(sals)
[1] 4.15128
And to see that we could do this ourself, we would do
> median(abs(sals  median(sals))) # without normalizing constant
[1] 2.8
> median(abs(sals  median(sals))) * 1.4826
[1] 4.15128
(The choice of 1.4826 makes the value comparable with the standard
deviation for the normal distribution.)
3.9 Stemandleaf Charts
There are a range of graphical summaries of data. If the data set is
relatively small, the stemandleaf diagram is very useful for
seeing the shape of the distribution and the values. It takes a
little getting used to. The number on the left of the bar is the
stem, the number on the right the digit. You put them together to
find the observation.
Suppose you have the box score of a basketball game and find the
following points per game for players on both teams
2 3 16 23 14 12 4 13 2 0 0 0 6 28 31 14 4 8 2 5
To create a stem and leaf chart is simple
> scores = scan()
1: 2 3 16 23 14 12 4 13 2 0 0 0 6 28 31 14 4 8 2 5
21:
Read 20 items
> apropos("stem") # What exactly is the name?
[1] "stem" "system" "system.file" "system.time"
> stem(scores)
The decimal point is 1 digit(s) to the right of the 
0  000222344568
1  23446
2  38
3  1
R Basics: help, ? and apropos
Notice we use
apropos() to help find the name for the
function. It is
stem() and not
stemleaf(). The
apropos() command is convenient when you think you know the
function's name but aren't sure. The
help command will help us
find help on the given function or dataset once we know the name. For
example
help(stem) or the abbreviated
?stem will
display the documentation on the
stem function.
Suppose we wanted to break up the
categories into groups of 5. We can do so by setting the ``scale''
> stem(scores,scale=2)
The decimal point is 1 digit(s) to the right of the 
0  000222344
0  568
1  2344
1  6
2  3
2  8
3  1
Example: Making numeric data categorical
Categorical variables can come from numeric variables by aggregating
values. For example. The salaries could be placed into broad
categories of 01 million, 15 million and over 5 million. To do
this using
R one uses the
cut() function and the
table() function.
Suppose the salaries are again
12 .4 5 2 50 8 3 1 4 .25
And we want to break that data into the intervals
[0,1],(1,5],(5,50]
To use the cut command, we need to specify the cut points. In this
case 0,1,5 and 50 (=
max(sals)). Here is the syntax
> sals = c(12, .4, 5, 2, 50, 8, 3, 1, 4, .25) # enter data
> cats = cut(sals,breaks=c(0,1,5,max(sals))) # specify the breaks
> cats # view the values
[1] (5,50] (0,1] (1,5] (1,5] (5,50] (5,50] (1,5] (0,1] (1,5] (0,1]
Levels: (0,1] (1,5] (5,50]
> table(cats) # organize
cats
(0,1] (1,5] (5,50]
3 4 3
> levels(cats) = c("poor","rich","rolling in it") # change labels
> table(cats)
cats
poor rich rolling in it
3 4 3
Notice,
cut() answers the question ``which interval is the
number in?''. The output is the interval (as a
factor). This
is why the
table command is used to summarize the result of
cut. Additionally, the names of the levels where changed as
an illustration of how to manipulate these.
3.10 Histograms
If there is too much data, or your audience doesn't know how to read
the stemandleaf, you might try other summaries. The most common is
similar to the bar plot and is a histogram. The histogram defines a
sequence of breaks and then counts the number of observation in the
bins formed by the breaks. (This is identical to the features of the
cut() function.) It plots these with a bar similar to the
bar chart, but the bars are touching. The height can be the
frequencies, or the proportions. In the latter case the areas sum to
1  a property that will be sound familiar when you study
probability distributions. In either case the area is proportional
to probability.
Let's begin with a simple example. Suppose the top 25 ranked movies
made the following gross receipts for a week
^{4}
29.6 28.2 19.6 13.7 13.0 7.8 3.4 2.0 1.9 1.0 0.7 0.4 0.4 0.3
0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1
Let's visualize it
(figure
3). First we scan it
in then make some histograms
> x=scan()
1: 29.6 28.2 19.6 13.7 13.0 7.8 3.4 2.0 1.9 1.0 0.7 0.4 0.4 0.3 0.3
16: 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1
27:
Read 26 items
> hist(x) # frequencies
> hist(x,probability=TRUE) # proportions (or probabilities)
> rug(jitter(x)) # add tick marks
Figure 3: Histograms using frequencies and proportions
Two graphs are shown. The first is the default graph which makes a
histogram of frequencies (total counts). The second does a histogram
of proportions which makes the total area add to 1. This is preferred
as it relates better to the concept of a probability density. Note
the only difference is the scale on the
y axis.
A nice addition to the histogram is to plot the points using the
rug command. It was used above in the second graph to give the
tick marks just above the
xaxis. If your data is discrete and has ties,
then the
rug(jitter(x)) command will give a little jitter to
the
x values to eliminate ties.
Notice these commands opened up a graph window. The graph window in
R has few options available using the mouse, but many using command
line options. The
GGobi package
has more but requires an extra software
installation.
The basic histogram has a predefined set of break points for the
bins. If you want, you can specify the number of breaks or your own
break points (figure
4).
> hist(x,breaks=10) # 10 breaks, or just hist(x,10)
> hist(x,breaks=c(0,1,2,3,4,5,10,20,max(x))) # specify break points
Figure 4: Histograms with breakpoints specified
From the histogram, you can easily make guesses as to the values of
the mean, the median, and the IQR. To do so, you need
to know that the median divides the histogram into two
equal area pieces, the mean would be the point where the histogram
would balance if you tried to, and the IQR captures exactly the middle
half of the data.
3.11 Boxplots
Figure 5: A typical boxplot
The boxplot (eg. figure
5) is used to summarize data
succinctly, quickly displaying if the data is symmetric or has
suspected outliers. It is based on the 5number summary. In its
simplest usage, the boxplot has a box with lines at the lower hinge
(basically
Q_{1}), the Median, the upper hinge (basically
Q_{3}) and
whiskers which extend to the min and max. To showcase possible
outliers, a convention is adopted to shorten the whiskers to a length
of 1.5 times the box length. Any points beyond that are plotted with
points. These may further be marked differently if the data is more
than 3 box lengths away. Thus the boxplots allows us to check
quickly for symmetry (the shape looks unbalanced) and outliers (lots
of data points beyond the whiskers). In figure
5 we see a
skewed distribution with a long tail.
Example: Movie sales, reading in a dataset
In this example, we look at data on movie revenues for the 25
biggest movies of a given week. Along the way, we also introduce
how to ``readin'' a builtin data set. The data set here is from
the data sets accompanying these notes.
^{5}
> library("Simple") # read in library for these notes
> data(movies) # read in data set for gross.
> names(movies)
[1] "title" "current" "previous" "gross"
> attach(movies) # to access the names above
> boxplot(current,main="current receipts",horizontal=TRUE)
> boxplot(gross,main="gross receipts",horizontal=TRUE)
> detach(movies) # tidy up
We plot both the current sales and the gross sales in a boxplot (figure
6).
Figure 6: Current and gross movie sales
Notice, both distributions are skewed, but the gross sales are less so.
This shows why Hollywood is so interested in the ``big hit'', as a
real big hit can generate a lot more revenue than quite a few
medium sized hits.
R Basics: Reading in datasets with library and data
In the above example we read in a builtin dataset. Doing so is
easy. Let's see how to read in a dataset from the package
ts (time series functions). First we need to load the
package, and then ask to load the data. Here is how
> library("ts") # load the library
> data("lynx") # load the data
> summary(lynx) # Just what is lynx?
Min. 1st Qu. Median Mean 3rd Qu. Max.
39.0 348.3 771.0 1538.0 2567.0 6991.0
The
library and
data command can be used in
several different ways

To list all available packages
 Use the command library().
 To list all available datasets
 Use the command
data() without any arguments
 To list all data sets in a given package
 Use
data(package='package name') for example
data(package=ts).
 To read in a dataset
 Use data('dataset name'). As
in the example data(lynx). You first need to load the
package to access its datasets as in the command library(ts).
 To find out information about a dataset

You can use the help command to see if
there is documentation on the data set. For example
help("lynx") or equivalently ?lynx.
Example: Seeing both the histogram and boxplot
The function
simple.hist.and.boxplot will
plot both a histogram and a boxplot to show the relationship
between the two graphs for the same dataset.
The figure shows some
examples on some randomly generated data. The data would be
described as bell shaped (normal), short tailed, skewed and long
tailed (figure
7).
Figure 7: Random distributions with both a histogram and the boxplot
3.12 Frequency Polygons
Some times you will see the histogram information presented in a different way.
Rather than draw a rectangle for each bin, put a point at the top of
the rectangle and then connect these points with straight lines. This is
called the
frequency polygon. To generate it, we need to know the
bins, and the heights. Here is a way to do so with
R getting the
necessary values from the
hist command. Suppose
the data is batting averages for the New York Yankees
^{6}
> x = c(.314,.289,.282,.279,.275,.267,.266,.265,.256,.250,.249,.211,.161)
> tmp = hist(x) # store the results
> lines(c(min(tmp$breaks),tmp$mids,max(tmp$breaks)),c(0,tmp$counts,0),type="l")
Figure 8: Histogram with frequency polygon
Ughh, this is just too much to type, so there is a function to do this
for us
simple.freqpoly.R. Notice though that the basic
information was available to us with the values labeled
breaks and
counts.
3.13 Densities
The point of doing the frequency polygon is to tie the histogram in
with the probability density of the parent population. More
sophisticated densities functions are available, and are much less
work to use if you are just using a builtin function.The builtin
data set
faithful (
help faithful) tracks the time
between eruptions of the oldfaithful geyser.
The
R command
density can be used to give more
sophisticated attempts to view the data with a curve (as the
frequency polygon does). The
density() function has means
to do automatic selection of bandwidth. See the help page for the
full description. If we use the default choice it is easy to add a
density plot to a histogram. We just call the
lines function
with the result from density (or
plot if it is the first
graph). For example
> data(faithful)
> attach(faithful) # make eruptions visible
> hist(eruptions,15,prob=T) # proportions, not frequencies
> lines(density(eruptions)) # lines makes a curve, default bandwidth
> lines(density(eruptions,bw="SJ"),col='red') # Use SJ bandwidth, in red
The basic idea is for each point to take some kind of
average for the points nearby and based on this give an estimate for
the density. The details of the averaging can be quite complicated,
but the main control for them is something called the bandwidth which
you can control if desired. For the last graph the ``SJ'' bandwidth
was selected. You can also set this to be a fixed number if
desired. In
figure
9 are 3 examples
with the bandwidth chosen to be 0.01, 1 and then 0.1. Notice, if the
bandwidth is too small, the result is too jagged, too big and the
result is too smooth.
Figure 9: Histogram and density estimates. Notice choice of bandwidth
is very important.
3.14 Problems

3.1
 Enter in the data
60 85 72 59 37 75 93 7 98 63 41 90 5 17 97
Make a stem and leaf plot.
 3.2
 Read this stem and leaf plot, enter in the data and make a
histogram:
The decimal point is 1 digit(s) to the right of the 
8  028
9  115578
10  1669
11  01
 3.3
 One can generate random data with the ``r''commands. For
example
> x = rnorm(100)
produces 100 random numbers with a normal distribution. Create two
different histograms for two different times of defining x as
above. Do you get the same histogram?
 3.4
 Make a histogram and boxplot of these data sets from these Simple
data sets: south, crime and aid. Which
of these data sets is skewed? Which has outliers, which is
symmetric.
 3.5
 For the Simple data sets bumpers,
firstchi, math make a histogram. Try to
predict the mean, median and standard deviation. Check your guesses
with the appropriate R commands.
 3.6
 The number of Oring failures for the first 23 flights of the US
space shuttle Challenger were
0 1 0 NA 0 0 0 0 0 1 1 1 0 0 3 0 0 0 0 0 2 0 1
(NA means not available  the equipment was lost). Make a table of
the possible categories.
Try to find the mean. (You might need to try
mean(x,na.rm=TRUE) to avoid the value NA, or look at x[!is.na(x)].)
 3.7
 The Simple dataset
pi2000
contains the first 2000 digits of
p. Make a histogram. Is it surprising? Next, find the proportion
of 1's, 2's and 3's. Can you do it for all 10 digits 09?
 3.8
 Fit a density estimate to the Simple dataset
pi2000
.
 3.9
 Find a graphic in the newspaper or from the web. Try to use
R to produce a similar figure.
Copyright © John Verzani, 20012. All rights reserved.