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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 can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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