Types of Correlation

Date: 06/25/2002 at 15:18:25
From: Katie
Subject: Correlation

Hello! I am really stuck on a question that deals with weak, strong,
positve, and negative correlation in scatter plots. You have a post
about this topic but I really don't know what correlation is so it
didn't help. 

In the problem it shows you a scatter plot and it asks you to tell
what kinda of correlation is shown. If you could just tell me what all
this correlation stuff is about I think I can get it from there.

Thanks!

Date: 06/25/2002 at 22:54:42
From: Doctor Achilles
Subject: Re: Correlation

Hi Katie,

Thanks for writing to Dr. Math.

A correlation is when two variables are related.  That definition is 
probably not terribly helpful, so let me give you a few examples.

First, there is a positive correlation between the amount of 
education a person has and the amount of money that person makes (at 
least I hope there is, since I'm a college student).  That is really 
just a fancy way of saying: "On average, if someone has more 
education, then that person will make more money."

You might graph that using a scatter plot by taking a random sample 
of people, finding out how much money they make and asking how much 
education they have.  Then for each person you put a point down.  
The farther to the right the point is, the more education that person 
has; the higher up the point is the more money that person makes.  
I've made up some data that look like this:

     |       .
     |            . ..
  m  |           ...
  o  |         .   .
  n  |  .    .
  e  |          .
  y  | ...
     | .  .
     |   .
     -----------------

        education

Notice that the highest paid person actually only has about an 
average education.  But still, on average more education means more 
money.  So there is a POSITIVE correlation between education and 
money.

For another example, let's look at the relationship between the 
number of cars on the freeway and the average speed.  Here's my 
(made-up) scatter plot:

     |   .   .
     |.     
  s  |   . .     
  p  |      .   
  e  |      .
  e  |          .
  d  |   .     .
     |           ..
     |            ..
     -----------------

        number of cars

Again, there are a few exceptions, but for the most part, the more 
cars there are, the slower the average speed. (Think of rush hour, 
when there are a lot of cars but everyone goes really slowly.)  This 
means that there is a NEGATIVE correlation between the number of cars 
and the average speed.

Finally, let's imagine that we go to a mall and take a survey.  What 
we do is we ask random people to tell us their height.  Then we look 
to see if there is any correlation between the answer they give and 
the time of day it is.  Here's my made up scatter plot:

     |       .
     |    .        
  h  |    .  ..
  e  |         .   .
  i  |  .    .
  g  |          .
  h  |      .     .
  t  | .  .
     |   .     ..
     -----------------

       time of day

This time the points are all over.  There doesn't seem to be any
relationship at all.  So we answer that there is NO correlation 
here.

As far as strong vs. weak correlations, that just has to do with how 
many exceptions there are to the general rule.  The examples of car 
speed and money are moderate or weak correlations.  Here's a very 
strong (positive) correlation:

     |                .
     |              .
     |            .
     |          .
  A  |        .
     |      .  
     |    .
     |  .  
     |.
     -------------------

            B

Hope this helps.  If you have other questions about this or you're 
still stuck, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/26/2002 at 16:15:36
From: Katie
Subject: Thank you (Correlation)

I just wanted to say thanks again for your time. Your 
examples really helped!

-Katie