CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 12. Non-Parametric Statistics

TABLE12.8:

Math SAT Score

(X)

Final Exam

Score(Y)

X Rank Y Rank d d^2

595 68 4 1 3 9

520 55 8 7 1 1

715 65 1 2 − 1

405 42 14 12 2 4

680 64 2 3 − 1 1

490 45 11 10 1 1

565 56 6. 5 5. 5 1 1

580 59 5 4 1 1

615 56 3 5. 5 − 2. 5 6. 25

435 42 13 12 1 1

440 38 12 14 − 2 4

515 50 9 9 0 0

380 37 15 15 0 0

510 42 10 12 − 2 4

565 53 6. 5 8 − 1. 5 2. 25

Sum 0 36. 50

Using the formula for the Spearman correlation coefficient, we find that:

ρ= 1 − (^6) ∑
d^2
n(n^2 − 1 )

= 1 −

6 ( 36. 50 )

15 ( 225 − 1 )

= 1 − 0. 07 = 0. 93

We interpret this rank correlation coefficient in the same way as we interpret the linear correlation coefficient. This
coefficient states that there is a strong, positive correlation between the two variables.

Lesson Summary

1. We use the rank sum test (also known as the Mann-Whitneyυtest) to assess whether two samples come from the
   same distribution. This test is sensitive to both the median and the distribution of the samples.


2. When performing the rank sum test there are several different conditions that need to be met including that the
   population not be normally distributed, we have continuously distributed observations, there be an independence of
   samples, the samples are greater than 5 observations, and that the observations be on a numeric or ordinal scale.


3. When performing the rank sum test, we need to calculate a figure known as theUstatistic. This statistic takes
   both the median and the total distribution of both samples into account.


4. To calculate the test statistic for the rank sum test, we first must calculate something known as theUstatistic
   which is derived from the ranks of the observations in both samples. When performing our hypotheses tests, we
   calculate the standard score which is defined as

z=

U−μU

σU

5. We use the Spearman rank correlation coefficient (also known as simply the ’rank correlation’ coefficient) to mea-
   sure the strength, magnitude and direction of the relationship between two variables from non-normal distributions.