Body fat percentage (BF%) is easily measured by skinfold calipers. It can be mea
ID: 3128854 • Letter: B
Question
Body fat percentage (BF%) is easily measured by skinfold calipers. It can be measured much more reliably by Dual Energy X-Ray Absorptiometry (DEXA). Body Mass Index (BMI) and waist-to-hip ratio (WH ratio) are two other measures of body composition commonly used in health assessment.
Use the following health data, which was collected from 7 college students, to establish the concurrent validity of WH ratio as a measure of body composition.
What is the correlation coefficient associated with the criterion validity of WH ratio with respect to BF% as measured by DEXA? (Answer to three decimal places)
Subject BF%DEXA BF%
skinfold BMI WH ratio 1 14.4 14.1 24.0 0.96 2 18.7 25.9 26.4 1.05 3 11.7 15.2 19.5 0.74 4 18.2 23.4 28.9 1.01 5 12.4 13.4 19.7 1.03 6 21.3 22.9 31 1.05 7 28.7 35.2 38.2 1.09
Explanation / Answer
Correlation Co-efficient Definition:
A measure of the strength of linear association between two variables. Correlation will always between -1.0 and +1.0. If the correlation is positive, we have a positive relationship. If it is negative, the relationship is negative.
Formula:
Correlation Co-efficient :
Correlation(r) =[ N%u03A3XY - (%u03A3X)(%u03A3Y) / Sqrt([N%u03A3X2 - (%u03A3X)2][N%u03A3Y2 - (%u03A3Y)2])]
where
N = Number of values or elements
X = First Score
Y = Second Score
%u03A3XY = Sum of the product of first and Second Scores
%u03A3X = Sum of First Scores
%u03A3Y = Sum of Second Scores
%u03A3X2 = Sum of square First Scores
%u03A3Y2 = Sum of square Second Scores
Correlation Co-efficient Example: To find the Correlation of
X Values Y Values
60 3.1
61 3.6
62 3.8
63 4
65 4.1
Step 1: Count the number of values.
N = 5
Step 2: Find XY, X2, Y2
See the below table
X Value Y Value X*Y X*X Y*Y
60 3.1 60 * 3.1 = 186 60 * 60 = 3600 3.1 * 3.1 = 9.61
61 3.6 61 * 3.6 = 219.6 61 * 61 = 3721 3.6 * 3.6 = 12.96
62 3.8 62 * 3.8 = 235.6 62 * 62 = 3844 3.8 * 3.8 = 14.44
63 4 63 * 4 = 252 63 * 63 = 3969 4 * 4 = 16
65 4.1 65 * 4.1 = 266.5 65 * 65 = 4225 4.1 * 4.1 = 16.81
Step 3: Find %u03A3X, %u03A3Y, %u03A3XY, %u03A3X2, %u03A3Y2.
%u03A3X = 311
%u03A3Y = 18.6
%u03A3XY = 1159.7
%u03A3X2 = 19359
%u03A3Y2 = 69.82
Step 4: Now, Substitute in the above formula given.
Correlation(r) =[ N%u03A3XY - (%u03A3X)(%u03A3Y) / Sqrt([N%u03A3X2 - (%u03A3X)2][N%u03A3Y2 - (%u03A3Y)2])]
= ((5)*(1159.7)-(311)*(18.6))/sqrt([(5)*(1%u2026
= (5798.5 - 5784.6)/sqrt([96795 - 96721]*[349.1 - 345.96])
= 13.9/sqrt(74*3.14)
= 13.9/sqrt(232.36)
= 13.9/15.24336
= 0.9119
This example will guide you to find the relationship between two variables by calculating the Correlation Co-efficient from the above steps.