Class exercise:  Temperatures and computing correlations (due next Tuesday)

 

 

Temperatures depend on a number of factors.  One of these must certainly be elevation.  From your text, you know that, on average, atmospheric temperature decreases with height.  So, as elevation increases, temperature should decrease but this may not always be true in every case since other factors may be present.

 

We can test the dependence of temperature on elevation using a correlation coefficient.  Standard statistics texts can provide formulations for the correlation coefficient (CC).  One such formula which you will use for this exercise is as follows:

 

CC =                           (nΣxy – ΣxΣy)                  .

Square root{[nΣx2 - (Σx)2][nΣy2 – (Σy)2]}

 

where n is the number of points, x and y are the two series to be correlated.  In this case, x is elevation and y is temperature.  A perfect correlation would be +1 and a perfect anti-correlation would be -1.  The expected correlation between elevation and temperature would be negative because as elevation increases, temperature decreases. If elevation were the only factor, the correlation coefficient would be -1.  If CC is close to 0 or actually equals 0, that means there is little or no relationship between the two variables.

 

A map from early morning (6:43 a.m. Mountain Standard Time) on Jan 7, 2008 is shown at this link.  You also have a paper copy (given out in class). The paper map is analyzed for temperature, i.e., isotherms have been drawn.   Elevations in meters above sea level have also been plotted under each station code.  It was very cold at Big Piney, WY (BPI, -17°F ), but that location is not the highest elevation station.   A map showing the topographical features can be found by clicking this link.

 

Here are the stations:

 

IDA                 RKS                VEL                 CAG                SMY                GXY   

PIH                  EVW               COD                SHR                 GCC                GTC

P60                  LGU                RIW                 BYG                DGW               CYS

JAG                 SLC                 GEY                CPR                 BRX

BPI                  PUC                WRL                RWL                LAR

 

Assignment:

 

1. Using the formula above, calculate the correlation coefficient for the 28 stations whose elevations have been plotted under each station’s 3-letter code.  I did it using Excel but if you don’t know how to program with the spreadsheet, enter them into a calculator.  Write down the summations on your answer sheet.  Then use the formula to put them together.

 

2. Assess the correlation coefficient in terms of the expected relationship.  Was it a strong correlation?  Why or why not?

 

3. The correlation will not be perfect (+1 or -1), therefore something else is acting on the temperature in addition to the elevation.  What other factors would have an effect on temperature?