.. Copyright (C)  Google, Runestone Interactive LLC
    This work is licensed under the Creative Commons Attribution-ShareAlike 4.0
    International License. To view a copy of this license, visit
    http://creativecommons.org/licenses/by-sa/4.0/.

Example: Correlation and College Data
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Correlation is also used to find which variable might be useful in
predicting another variable. For example, consider a dataset that
contains the following variables.

-  shoe size
-  age
-  salary
-  height

Using this dataset, you want to be able to predict the height of a
person, using only one other variable: shoe size, age, or salary. To get
the best prediction, you would choose whichever variable has the
strongest relationship with height. Correlation is the most standard
metric for quantifying how strong a relationship is between two
variables.

Now, you will apply the concepts you’ve learned so far to help identify
relationships in `a dataset about colleges and
universities
<https://docs.google.com/spreadsheets/d/1tqE_wB1rXJc19RgHy9sXqQGhT_ke5e-uq6xNx_oOtTo/edit?usp=sharing>`__
across the USA. For each school, the dataset has information about the
following.

-  The state where the school is located.
-  The type of institution: public, private non-profit, or private
   for-profit.
-  The median SAT scores for reading, math, and overall.
-  The average net tuition, i.e. the mean of the amount paid by the
   student after applying scholarships and other forms of financial aid.
-  The percentage of enrolled students receiving Pell Grants.
-  The percentage of enrolled students receiving federal loans.
-  The median earnings for employed students 10 years after first
   enrolling.
-  The median debt of graduates of the school.
-  The completion rate, i.e. the percentage of full-time, first-time
   students who graduate within 6 years.

Suppose you want to know which variable has the strongest relationship
with completion rate. In this dataset, completion is defined as the
percentage of full time students who graduate within 6 years without
transferring to another institution. In other words, “completion” is
used as a proxy for graduation rate. Using the *CORREL* function, you
can calculate the correlation between completion rate and the other
variables.

You can watch this `video
<https://www.youtube.com/watch?v=omIT5V7naqM>`__ to learn more about 
how to use the *CORREL* function.

Out of all the explanatory variables, it is “SAT average” that has the
closest correlation to -1 or 1 with completion rate, at r = 0.84.
Looking at the scatter plot, SAT average and completion rate have a
strong, positive, linear relationship.

.. image:: figures/college_data.png
   :align: center
   :alt: Scatter plot with average SAT score as the x-axis and the 
         completion rate as the y-axis with a positive trend.

.. shortanswer:: completion_rate_1200

   Use the scatter plot to estimate the completion rate at a
   school with an average SAT score of 1200.

.. shortanswer:: completion_rate_1400

   Use the scatter plot to estimate the completion rate at a
   school with an average SAT score of 1400.

.. shortanswer:: completion_rate_accuracy
   
   Which estimate do you think was more accurate? Why?

.. shortanswer:: completion_rate_median_debt
    
   Which variable has the strongest relationship with median
   debt?

.. shortanswer:: completion_rate_correlation_coefficient 
   
   What two variables have a correlation coefficient closest to -1?