If one or both of the factor values are ranked, you
must use Spearman *rs*.
If one of the factors is continuous, but not ranked (e.g., scores on an exam),
you will need to transform it to ranks before calculating *rs*.

Example: Physical Education teachers are interested in the relationship between body weight and fitness in 10 year-olds. They have BMI information (Body Mass Index - a ratio of height to weight) for 10 girls. They have them run a race, keeping track of the finish order (e.g., first, second, etc.). Here are the results:

Subject

BMI

Finish orderJuanita

17.2

3

Susan

17.5

1

Chin

17.8

2

Kimberly

18.0

4

Cynthia

19.2

6

Celeste

19.3

5

Audrey

20.0

10

Mee

21.0

8

Fatima

21.4

7

Nicole

25.1

9

The BMI data are normally-distributed. However, the race
outcome is not. The results are ranked data. Also, we can't make any assumptions
about the amount of time between each finish place (e.g., 1st vs. 2nd, 2nd
to 3rd, etc.). In this situation we use the Spearman (*r*s)
rather than the Pearson (*r*) formula.

Before calculating *r*s,
the BMI data must be changed to ranks. In this example we rank the heaviest
child as 1. Our prediction is that the heavier the child, the slower the
speed (an inverse or
negative correlation). Looking at the raw data gives you some idea of the
outcome. Nicole is the heaviest, and she came in ninth in the race. The lightest
runner, Juanita finished third.

Subject

BMI rank

Finish orderJuanita

10

3

Susan

9

1

Chin

8

2

Kimberly

7

4

Cynthia

6

6

Celeste

5

5

Audrey

4

10

Mee

3

8

Fatima

2

7

Nicole

1

9

Here is the formula in case you need it -- no need to memorize it.

Results from the above data:

Hand calculation example

**Tied ranks**

If there are * ties *(e.g., 2 or more cases
of the same rank), add the ranks they would use, and give each of the tied
cases the mean. For example, 2 cases in 3rd place would take up ranks 3 and
4. Assign each a rank of 3.5 = (3+4)/2. The next case is ranked **5** (because
the places for 3 and 4 have been taken). If there are 3 cases tied for 5th
place, they will cover ranks 5, 6, and 7. Each is assigned a rank of 6 (the
mean of 5, 6, and 7). The next rank will be 8. Be sure you end up with neither
more nor fewer ranks than equals the number of paired scores (the last rank
may not be exact if there are cases tied for it).

When you have understood this module along with Pearson *r* take
the self-tests

Self-test #1: Scatterplots

Self test #2: Estimating direction and strength of correlation (this is challenging)

Next section: Effect size