Greenwood press
Download 1.81 Mb. Pdf ko'rish
|
book-20600
- Bu sahifa navigatsiya:
- Standard Deviation
156
VECTORS Since the concepts of addition, dot product, and angle between vectors scale up to many dimensions, vector mathematics adapts well to statistical computa- tions. Consider the test-score data on five students shown in the table below. The deviation scores form vectors with five components. The science vector is s = (5, 0, 1,-1, -5). The math vector is m = (5,-3, 3,-2,-3). science math science math raw score raw score deviation score deviation score student X Y x y Albert 85 25 5 5 Manuel 80 17 0 -3 Bonnie 81 23 1 3 Sharon 79 18 -1 -2 Elena 75 17 -5 -3 average (mean) 80 20 Test data on five students. Deviation scores are computed by taking the test score minus the mean (e.g., x Bonnie = 81 − 80 = 1;. y Elena = 17 − 20 = -3). Computations with the vectors give the lengths to be | s| = 5 2 + 0 2 + 1 2 + (−1) 2 + (−5) 2 = √ 52 and | m| = √ 56. When each of these lengths is scaled by the reciprocal of the square root of dimensions, 1 √ 5 , the computation produces the standard deviation for each vec- tor. These are about 3.22 for s and 3.34 for m. (See Standard Deviation for uses and formulas.) The cosine of the angle between the two vectors is cos θ = s• m |s || m| = 5(5)+0(−3)+1(3)+(−1)(−2)+(−5)(−3) √ 52 √ 56 ≈ 0.83. This is called the correlation coefficient for the two vectors and is commonly designated with the letter r. We say that for this group of students, science scores correlate 0.83 with math scores. Because it is a cosine, the correlation coefficient r ranges from -1 to +1. Correlations at +1 (angle θ = 0 ◦ ) and -1 (angle θ = 180 ◦ ) indicate that the vectors are collinear. Correlations close to 0 (angle θ = 90 ◦ ) indicate that the vectors are going in different directions. In the first case ( r = 1), the vectors are pulling in the same direction. In the second case ( r = -1), they are opposites. Our correlation coefficient for science and math tests ( r = 0.83) cor- responds to an angle between the vectors of about 33.5°. In a space of five dimensions, these vectors are separate enough that each one is measuring some underlying skills that are different for different students, but they are also meas- uring something that is the same for all students. Generally, students who scored high on science also scored high on math. The square of the cosine provides a measure of overlap. This coefficient of determination is r 2 = 0.83 2 ≈ 0.70. It Download 1.81 Mb. Do'stlaringiz bilan baham: |
Ma'lumotlar bazasi mualliflik huquqi bilan himoyalangan ©fayllar.org 2024
ma'muriyatiga murojaat qiling
ma'muriyatiga murojaat qiling