Introduction

For two-variable paired statistical data the closeness of the relationship between the two sets of figures is usually judged by their correlation coefficient.
For data with precise numerical values, Pearson's correlation coefficient can be calculated and the hp 39g+ calculator will perform this calculation by selecting 2VAR from the statistics aplet menu, then the STAT button, and scrolling down to the value of CORR.

Sometimes though exact numerical values may not be available or may be considered too complicated and instead the data is just arranged in order from best to worst or lowest to highest etc. It is then required to judge how similar the two orderings are.

The first step is to assign a sequential number to each of the two orderings, starting at 1 and going up to the number of items, i.e. they are put into rank order. A correlation coefficient between the two sets of ranks is then calculated.

Commonly Spearman's Rank Correlation Coefficient is used and this is found by subtracting the rank numbers in one set individually from the corresponding numbers in the other set to form the difference, d.
Spearman's correlation coefficient is then calculated as:
r_s = 1 - 6*Sd²/n/(n² - 1)
where Sd² is the sum of the squares of the differences between the rankings and n is the number of data pairs.
See later for an example or a statistics textbook for more details.

Spearman's rank correlation coefficient on the hp 39g+ Graphing Calculator

The paired data must first be entered into columns C1 and C2 of the Statistics aplet. Alternatively a copy of the statistics aplet can be made and the data put into that – useful if the data needs to be kept for later review.
Note that only columns C1 and C2 may be used and that there must be at least two pairs of data items, also that C1 and C2 must have the same number of elements so there is no unpaired data.

It does not matter if the original data has not been converted to rank order, e.g. it could be the raw results from a test. The program will sort the data into numerical order on column C1 and then convert both columns to the equivalent rank order from 1 to n.
Notice that this will overwrite the original data so if this needs to be kept it should either be copied to two other columns (e.g. using C1 sto C8 & C2 sto C9 from the home screen), or a copy made of the Statistics aplet containing the data, and this copy started.

Then by running program Spearman the data is checked, sorted and converted to rank order, and the Spearman's correlation coefficient calculated and displayed.

Loading Spearman

Spearman is a single program and should be transferred to the hp 39g+ in the usual way, i.e.:

• Put the files HP39DIR.000, HP39DIR.CUR and SPEARMAN.000 from this zip file into an empty directory on a PC. (The .htm and .gif files can be left in - they will be ignored by the calculator.)
• Connect the calculator to the computer via the USB cable then start the HP39G Connectivity program and point it to the directory holding the files.
• Go to the PROGRAM catalog on the calculator, choose RECV and transfer Spearman from the computer to the hp 39g+'s memory by selecting Spearman as the file to download.

Alternatively it is possible to type the program directly into the calculator from the listing at the end.

Running Spearman

As a simple example, suppose the marks obtained by the same group of six students in French and German tests were:

Student	French Mark	German Mark
A	12	6
B	8	5
C	16	7
D	12	7
E	7	4
F	10	8

We want to know whether there is any significant correlation between the rankings of the marks of these six students in the two tests, for instance is it likely that a student placed near the top in a French test will also be near the top in a German test?
Notice that there are two 'ties' in the test scores. The normal procedure for dealing with equal rank positions is to give the tied data the mean of the rankings they would otherwise have. E.g. if two students were ranked as equal third, they would both be given the rank of 3.5 = (3+4)/2, and the next student would be assigned a rank of 5.

To calculate the rank correlation coefficient manually we would perform the following steps:

1. Sort the first set of data into numerical order, keeping the paired data together. This is not essential but makes it easier to 'see' the level of agreement.
2. Convert the raw numbers into rank order, from 1 to 6 in this example, applying the averaging rule for any tied values.
3. Subtract the rank positions in the second column (German) from those in the first column (French).
4. Square these differences, so that they all become positive.
5. Add up the squared differences.
6. Apply the Spearman's formula to this total, for n = 6 items.

The working could be set out in a table thus:

Student (ordered by French mark)	French Mark	German Mark	French Rank	German Rank	Difference d in ranks	d²
E	7	4	1	1	0	0
B	8	5	2	2	0	0
F	10	8	3	6	-3	9
A	12	6	4.5	3	1.5	2.25
D	12	7	4.5	4.5	0	0
C	16	7	6	4.5	1.5	2.25
				Total	Sd²	13.5

Then Spearman's rank correlation coefficient is given by
r_s = 1 - 6*13.5/6/(6² - 1) = 0.614

To perform the same calculations on the hp 39g+, START the Statistics aplet, or a copy of it, from the calculator's aplet catalog.

Choose the NUM view and enter the pairs of test marks into columns C1 and C2.
When this is done the display should show:

Next change to the PROGRAM catalog and START program Spearman. This brings up the following warning screen to remind you that the original data will be overwritten:

Press Y or ENTER to proceed, or any other key to cancel the program.

Assuming you pressed Y, the program checks that columns C1 and C2 are the same size and that there are at least two data pairs. If this is not the case an appropriate error message is shown and the program terminates.

Otherwise the data is sorted into numerical order by column C1 and a progress screen is shown:

The percentage complete is only an estimate and may not reach 100%, depending on how disordered the original list is.

Next the two columns are converted to rank order numbers, and again the progress is shown:

For a small number of data points the sorting and ranking processes take just a few seconds, but unfortunately will take several minutes for longer data lists.

Finally the Spearman's Rank Correlation Coefficient is calculated and displayed thus:

Spearman's rank correlation coefficient = 0.614 to 3 figures.
For further use the value of the correlation coefficient is also stored in variable S (for Spearman).

Pressing the NUM key returns you to a view of the data, which you will see is sorted and ranked:

If there are many 'ties' in the rankings then statistics manuals generally recommend that the (more complicated) formula for the Pearson correlation coefficient is used rather than Spearman's formula, even for ranked data. (If there are no ties then both formulas will give the same answer.)

By making sure that STAT2 is selected and pressing the STATS function key, the calculator will compute Pearson's correlation coefficient for the ranked data:

Pearson's correlation coefficient = 0.603

It can be seen that even in the present example, with ties, the two correlation coefficients are very similar. (For interest, calculating the Pearson coefficient on the raw data before converting to rank order gives a value of 0.634)

Of course it is possible to apply the other features of the Statistics aplet to the ranked data, such as drawing a scatter plot after setting suitable axes:

The scatter diagram seems to show some relationship between the rankings but is the Spearman's correlation coefficient value of 0.614 statistically significant?

There are tables of critical values of the correlation coefficient (i.e. the minimum value which is significant) published in statistical handbooks and a graph of these critical values for up to 100 data pairs is shown here:



It can be read off that for six data pairs the minimum value of r_s at 90% confidence is approx 0.65 (the exact value is 0.657) – since our value is a little less than this there is a more than 10% probability that the apparent correlation between the two sets of test marks is just due to chance and hence we cannot be particularly sure that a good score in French will generally mean a good score in German.

If the trend were for a high mark in French to be associated with a low mark in German then the calculated correlation coefficient would of course be negative. As with other definitions of correlation coefficient, Spearman's always lies between -1 and +1.

Hardware Requirements

Spearman runs on a Hewlett-Packard 39g+ calculator. It should also work on the 39g, 39gs and 40gs, and may work on the 38g, 48g, 49g and 50g but has not been tried.
The program takes up 3.7 kilobytes of RAM.

Variables Used

Spearman uses the HOME variables I, N, S, X and Y, and will therefore overwrite any existing information stored in them. It also changes the values in columns C1 and C2 of the current Statistics aplet.

Known Bugs

None known other than that sorting and ranking can be very slow once the number of data pairs exceeds a few tens.

Program Listing

Note: the character Þ represents the 'STO' arrow.
True line breaks in the program are shown by ¿, other line breaks are just to fit the listing into the table more neatly.