Answers to correlation & regression question
1) Small frame
Height (cm)
158
160
163
165
168
170
173
175
178
180
183
186
188
191
193
Weight (kg)
58
59
60
61
62
63
64
64
65
66
68
69
70
72
73
Medium frame
Height (cm)
158
160
163
165
168
170
173
175
178
180
183
186
188
191
193
Weight (kg)
59
60
61
62
63
64
66
67
68
70
71
73
74
76
78
Large frame
Height (cm)
158
160
163
165
168
170
173
Weight (kg)
63
64
64
65
66
68
69
175
178
180
183
186
188
191
193
70
72
73
74
76
78
80
82
Small frame
90
193, 82
80
70
Weight (kg)
60
50
40
Weight (kg)
30
20
10
0
0
50
100
150
Height (cm)
2)
200
250
Medium frame
90
80
193, 78
70
Weight (kg)
60
50
40
Weight (kg)
30
20
10
0
0
50
100
150
200
250
Height (cm)
Large frame
90
193, 82
80
70
Weight (kg)
60
50
40
Weight (kg)
30
20
10
0
0
50
100
150
Height (cm)
3) Key formula to know:
r=
𝑆𝑥𝑦
√𝑆𝑥𝑥𝑆𝑦𝑦
200
250
Using the Calculator trick I showed on the article titled ‘How to find the sum variables
needed for calculations of Sxx, Sxy, & Syy [S1, Degree level which requires statistics]’ you
find the following;
Small frame:
Σx= 2631
Σx2=463279
Σy=974
Σy2=63550
Σxy=171576
Then you must find out Sxx, Sxy & Syy using the following 3 formulae;
Σ𝑥Σ𝑦
𝑛
Sxy=Σxy-
(Σ𝑥)2
𝑛
Sxx=Σx2-
(Σ𝑦)2
𝑛
Syy=Σy2-
You find out that;
Sxy=736.4
Sxx=1801.6
Syy=4574/15
Then sub it into the correlation coefficient formula
r=0.994 (3.s.f)
Medium frame:
Σx= 2631
Σx2=463279
Σy=1012
Σy2=68786
Σxy=178459
Sxy=954.2
Sxx=1801.6
Syy=7646/15
r=0.996 (3.s.f)
Large Frame:
Σx= 2631
Σx2=463279
Σy=1064
Σy2=76000
Σxy=187590
Sxy=964.4
Sxx=1801.6
Syy=7904/15
r=0.990 (to 3.s.f)
4) All groups have a significant correlation as they are all very close to 1.
5) All groups show a strong positive relationship between height & weight.
6Σ𝑑 2
6) Spearmann’s rank formula is rs=1-𝑛(𝑛2 −1)
d stands for the difference between the ranks.
Small frame:
Height (cm)
158
160
163
165
168
170
173
175
178
180
183
186
188
191
193
R1
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
The sum of d2=0.5
6(0.5)
15(152 −1)
rs=1-
rs=0.999 (3.s.f)
Medium frame: rs=1
Large frame: rs=0.999 (3.s.f)
Weight (kg)
58
59
60
61
62
63
64
64
65
66
68
69
70
72
73
R2
15
14
13
12
11
10
8.5
8.5
7
6
5
4
3
2
1
R1-R2 (d)
0
0
0
0
0
0
0.5
0.5
0
0
0
0
0
0
0
D2
0
0
0
0
0
0
0.25
0.25
0
0
0
0
0
0
0
7) We will be needing the regression equation; y=a+bx
𝑆𝑥𝑦
Where a=(ӯ-b𝑥̅ ) & b=𝑆𝑥𝑥
Find the mean of y & x values in each group.
Small Frame: y=-2.56+0.41x
Sub in 200
y=-2.56+0.41(200)
y=79.2kg (3.s.f)
Medium frame: y=-25.4+0.53x
y=-25.4+0.53(200)
y=80.5kg
Large frame: y=-23.0+0.59x
y=-23.0+0.59(200)
y=84.1kg
8) The medium frame has the most confidence as it has the highest possible spearmann’s rank.