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Measures of central tendency, variability, and spread summarize a single variable by providing important information about its distribution. Often,more than one variable is collected on each individual. For example, in large health studies of populations it is common to obtain variables suchas age, sex, height, weight, blood pressure, and total cholesterol on each individual. Economic studies may be interested in, among other things,personal income and years of education. As a third example, most university admissions committees ask for an applicant's high school grade point averageand standardized admission test scores (e.g., SAT). In this chapter we consider bivariate data, which for now consists of two quantitative variables for each individual. Our first interest is in summarizing such data in a way that is analogous to summarizing univariate (single variable) data.

By way of illustration, let's consider something with which we are all familiar: age. It helps to discuss something familiar since knowing thesubject matter goes a long way in making judgments about statistical results. Let's begin by asking if people tend to marry other people of about the sameage. Our experience tells us "yes," but how good is the correspondence? One way to address the question is to look at pairs of ages for a sample ofmarried couples. Table 1 below shows the ages of 10 married couples. Going across the columns we see that, yes, husbands and wives tend to be of aboutthe same age, with men having a tendency to be slightly older than their wives. This is no big surprise, but at least the data bear out our experiences,which is not always the case.

Sample of spousal ages of 10 white american couples.
Husband 36 72 37 36 51 50 47 50 37 41
Wife 35 67 33 35 50 46 47 42 36 41

The pairs of ages in are from a dataset consisting of 282 pairs of spousal ages, too many to make sense of from a table. What we need is a way to summarize the282 pairs of ages. We know that each variable can be summarized by a histogram (see ) and by a mean and standard deviation (See ).

Histograms of spousal ages.
Means and standard deviations of spousal ages.
Mean Standard Deviation
Husband 49 11
Wife 47 11

Each distribution is fairly skewed with a long right tail. From we see that not all husbands are older than their wives and it isimportant to see that this fact is lost when we separate the variables. That is, even though we provide summary statisticson each variable, the pairing within couple is lost by separating the variables. We cannot say, for example, based on the meansalone what percentage of couples have younger husbands than wives. We have to count across pairs to find this out. Only by maintainingthe pairing can meaningful answers be found about couples per se. Another example of information not available from the separatedescriptions of husbands and wives' ages is the mean age of husbands with wives of a certain age. For instance, what is the averageage of husbands with 45-year-old wives? Finally, we do not know the relationship between the husband's age and the wife's age.

We can learn much more by displaying the bivariate data in a graphical form that maintains the pairing. shows a scatter plot of the paired ages. The x-axis represents the age of the husband and the y-axis the age of the wife.

Scatterplot showing wife age as a function of husband age.

There are two important characteristics of the data revealed by . First, it is clear that there is a strong relationship between the husband's age and the wife's age: the older the husband,the older the wife. When one variable ( y ) increases with the second variable ( v ), we say that x and y have a positive association . Conversely, when y decreases as x increases, we say that they have a negative association .

Second, the points cluster along a straight line. When this occurs, the relationship is called a linear relationship .

shows a scatterplot of Arm Strength and Grip Strength from 149 individuals working in physically demanding jobs including electricians,construction and maintenance workers, and auto mechanics. Not surprisingly, the stronger someone's grip, the stronger their arm tends to be. There is therefore a positiveassociation between these variables. Although the points cluster along a line, they are not clustered quite as closely as they are for the scatter plot of spousal age.

Scatter plot of Grip Strength and Arm Strength.

Not all scatter plots show linear relationships. shows the results of an experiment conducted by Galileo on projectile motion.In the experiment, Galileo rolled balls down incline and measured how far they traveled as a function of the release height. It is clear from that the relationship between "Release Height" and "Distance Traveled" is not described well by astraight line: If you drew a line connecting the lowest point and the highest point, all of the remaining points would be above the line. The data arebetter fit by a parabola.

Galileo's data showing a non-linear relationship.

Scatter plots that show linear relationships between variables can differ in several ways including the slope of the line about which they clusterand how tightly the points cluster about the line. A statistical measure of the strength of the relationship between variables that takes thesefactors into account is the subject of the next section.

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
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