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HP 12c Platinum Financial Calculator - Statistics - Correlation Coefficient

The correlation coefficient
Statistics can be understood as a set of tools involving the study of methods and procedures for collecting, classifying, and analyzing data. Statistical tools also offer the means for making scientific inferences from summarized data, like linear regression and forecasting. Both linear regression and forecasting are computed with the parameters that define a curve (or line) that best represents the sample, given the plotted data of a two-variable sample is close enough to a straight line that it can be represented this way. The correlation coefficient r is in the range (-1 ≤ r ≤ 1) and measures how close the sample is to a straight line. The closer the sample is to a straight line, the closer |r| will be to 1 and the closer the forecasted points will be to the sample behavior.
HP 12c Platinum correlation coefficient
On the HP 12c Platinum, statistics data are stored as a set of summations resulting from the originally collected data. The collected data set must be typed in prior to using any statistics features available in the HP 12c Platinum. The HP 12c Platinum memory organization allows the study of statistic data organized as one- or two-variable sample. As a general procedure, data is always collected as a pair of n numbers, or (xn,yn) values, and the HP 12c Platinum computes the following summations:
Figure : Calculating the values for x and y
With these values updated and stored in memory, the HP 12c Platinum computes the correlation coefficient with the following expression:
Figure : Method for calculating the correlation coefficient
There are also functions to forecast values for each of the two variables. These functions are associated to (forecast x given y) and (forecast y given x) keys. Each time a new forecast is calculated, the correlation coefficient is computed too, and pressing right after or shows its value. The HP 12c Platinum uses the following expressions to compute the forecasted values:
Figure : Method for calculating the forecasted values
Practice finding the correlation coefficient
Example 1
A land researcher wants to compute the relationship between the constructed area and the land area of a community in order to suggest the construction area for a new home with a land area of 12500 sq yards and the suggested land area for a construction with 3520 sq yards based on forecasted values. He also wants to know how good this relationship fits in a straight line to know if his suggestions are valid. The table below summarizes his measurements.
Land Area (sq yards)
Construction Area (sq yards)
Land Area (sq yards)
Construction Area (sq yards)
12000
3120
9000
2080
10000
2560
10000
2700
11000
2920
13000
3280
14000
3300
12000
3080
Solution
Be sure to clear the statistics/summation memories before starting the problem.
Keystroke
Display
Figure : Clearing the statistics/summation memory
Then enter the first data point.
Keystroke
Display
Figure : Entering the first data
The first entered value (construction area) is used as the y variable and the second value (land area) is used as the x variable. The display shows the number of entries each time is pressed. Make sure that all data is entered:
Keystroke
Display
Figure : Entering all the data
Since the y-values are the construction area, the forecast construction area (y) for a 12500 sq yards land area is computed by pressing:
Keystroke
Display
Figure : Calculating the forecast construction area
The correlation coefficient is automatically computed each time a forecasting is performed. Press:
Keystroke
Display
Figure : Calculating the correlation coefficient
The land area is stored as x-values, so for a 3520 sq yards construction area, the forecasted land area is:
Keystroke
Display
Figure : Calculating the forecasted land area
Although the correlation coefficient has the same value for the same sample, it's easy to check for it:
Keystroke
Display
Figure : Calculating the correlation coefficient
Answer
For a construction area of 3,520 sq yards, the estimated land area needed is approximately 14,140 sq yards. For a land area of 12,500 sq yards, a construction area of approximately 3,140 sq yards is recommended. The sample shows a correlation coefficient of 0.95, so the forecast is close to the actual data.
Example 2
A stockholder observes the foreign stock market for some time in order to compose a curve that relates the amount of investment with amount of earnings in a particular brand. He decides to analyze the data and use the correlation coefficient to measure the margin of error when predicting the amount of earnings given the amount of investment. If the correlation coefficient is lower than 0.80, then he will not use the data to make future predictions. The collected data so far is as follows:
Investment amount
Amount of earnings
Investment amount
Amount of earnings
$1,200,000
$91,000
$1,450,000
$112,000
$1,000,000
$98,000
$1,300,000
$109,000
$900,000
$85,000
$1,150,000
$99,000
Solution
Be sure to clear the statistics/summation memories before starting the problem.
Keystroke
Display
Figure : Clearing the statistics/summation memory
Consider that each pair must be entered prior to add it to the statistics summations.
Keystroke
Display
Figure : Entering the first data
Remember that the display shows the number of entries each time is pressed. Make sure that all data is entered:
To compute the correlation coefficient, press:
Keystroke
Display
  note:
Any value will do in place of the 1 shown here.
Figure : Calculating the correlation coefficient
Answer
The correlation coefficient for the collected data is 0.84, and it is higher than the stockholder defined. Therefore, the available data will be used to predict future investments.

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