Z-Score Calculation and Analysis of Age and Height Data

This assignment involves a statistical analysis of age and height data using Z-scores to normalize the data. The analysis includes calculating Z-scores for age and height, determining the mean and standard deviation for each variable, and establishing acceptance regions based on 2.5% and 5% cut-off points. The student identifies individuals whose Z-scores fall outside the acceptance regions, indicating potential outliers. The document presents the calculated Z-scores, the distances from the cut-off points, and the identification of individuals whose values are the minimum distance from the cut-off points, providing a comprehensive understanding of the data's distribution and identifying extreme values. The analysis also includes the sex and year in college of the participants.

Normalization of Age and Height

Table 1: Standard Normalized Age and Height of participants
ID# age age_Z_score height height_Z_score sex year in college
1 19 -2.0904 65 -1.3305 F 1
2 19 -2.0904 70 0.9057 M 1
3 20 -1.6572 63 -2.2250 F 1
4 20 -1.6572 72 1.8002 M 1
5 20 -1.6572 65 -1.3305 F 1
6 19 -2.0904 70 0.9057 M 1
7 22 -0.7907 66 -0.8833 F 1
8 22 -0.7907 70 0.9057 M 1
9 22 -0.7907 65 -1.3305 F 1
10 22 -0.7907 66 -0.8833 M 1
11 23 -0.3574 66 -0.8833 F 2
12 23 -0.3574 65 -1.3305 F 2
13 25 0.5091 68 0.0112 F 2
14 24 0.0758 68 0.0112 M 2
15 24 0.0758 67 -0.4361 F 2
16 25 0.5091 68 0.0112 M 2
17 23 -0.3574 69 0.4584 F 2
18 23 -0.3574 69 0.4584 M 2
19 24 0.0758 70 0.9057 M 2
20 23 -0.3574 72 1.8002 M 2
21 24 0.0758 67 -0.4361 F 3
22 24 0.0758 67 -0.4361 F 3
23 25 0.5091 68 0.0112 F 3
24 25 0.5091 68 0.0112 F 3
25 25 0.5091 68 0.0112 F 3
26 26 0.9423 70 0.9057 M 3
27 26 0.9423 70 0.9057 M 3
28 27 1.3756 71 1.3529 M 3
29 25 0.5091 71 1.3529 M 3
30 25 0.5091 72 1.8002 M 3
31 26 0.9423 69 0.4584 M 4
32 26 0.9423 68 0.0112 M 4
33 27 1.3756 69 0.4584 M 4
34 27 1.3756 69 0.4584 M 4
35 26 0.9423 68 0.0112 M 4
36 25 0.5091 65 -1.3305 F 4
37 26 0.9423 65 -1.3305 F 4
38 26 0.9423 66 -0.8833 F 4
39 25 0.5091 67 -0.4361 F 4
40 25 0.5091 67 -0.4361 F 4

Mean Age = 23.825 years, Standard deviation of age = 2.308 years
Mean Height = 67.975 inches, Standard deviation = 2.236 inches
Z-scores for Age and Height have been calculated using the transformation Z = X
¿
−μ
σ where μ is the
mean and σ is the standard deviation of the population (Colan, 2013).
At 5% cut-off, the acceptance region is ( −1. 645 , 1. 645 )
Hence, at 5% cut-off, lower critical limit = -1.645.
Hence, at 5% cut-off, upper critical limit = 1.645
At 2.5% cut-off, the acceptance region is ( −1. 96 ,1 . 96 )
Hence, at 2.5% cut-off, lower critical limit = -1.96.
Hence, at 2.5% cut-off, upper critical limit = 1.96
For AGE:
From Table 2, minimum distance from 2.5% upper cut-off = 0.5844
ID numbers are: 28, 33, and 34.
From Table 2, minimum distance from 2.5% lower cut-off = 0.1304
ID numbers are: 1, 2, and 6.
From Table 2, minimum distance from 5% upper cut-off = 0.2694
ID numbers are: 28, 33, and 34.
From Table 2, minimum distance from 2.5% lower cut-off = 0.0122
ID numbers are: 3, 4, and 5.
For HEIGHT:
From Table 3, minimum distance from 2.5% upper cut-off = 0.1598
ID numbers are: 4, 20, and 30.
From Table 2, minimum distance from 2.5% lower cut-off = 0.2650
ID number is: 3.
From Table 2, minimum distance from 5% upper cut-off = 0.1552
ID numbers are: 4, 20, and 30.
From Table 2, minimum distance from 2.5% lower cut-off = 0.3145
ID numbers are: 1, 5, 9, 12, 36, and 37.

Table 2: Z-Scores for age and absolute difference from the cut-off points
ID# age_Z_score D1= age_Z_score-Upper (2.5%) D1= age_Z_score-Lower (2.5%) D1= age_Z_score-Upper (5%) D1= age_Z_score-Lower (5%)
1 -2.0904 4.0504 0.1304 3.7354 0.4454
2 -2.0904 4.0504 0.1304 3.7354 0.4454
3 -1.6572 3.6172 0.3028 3.3022 0.0122
4 -1.6572 3.6172 0.3028 3.3022 0.0122
5 -1.6572 3.6172 0.3028 3.3022 0.0122
6 -2.0904 4.0504 0.1304 3.7354 0.4454
7 -0.7907 2.7507 1.1693 2.4357 0.8543
8 -0.7907 2.7507 1.1693 2.4357 0.8543
9 -0.7907 2.7507 1.1693 2.4357 0.8543
10 -0.7907 2.7507 1.1693 2.4357 0.8543
11 -0.3574 2.3174 1.6026 2.0024 1.2876
12 -0.3574 2.3174 1.6026 2.0024 1.2876
13 0.5091 1.4509 2.4691 1.1359 2.1541
14 0.0758 1.8842 2.0358 1.5692 1.7208
15 0.0758 1.8842 2.0358 1.5692 1.7208
16 0.5091 1.4509 2.4691 1.1359 2.1541
17 -0.3574 2.3174 1.6026 2.0024 1.2876
18 -0.3574 2.3174 1.6026 2.0024 1.2876
19 0.0758 1.8842 2.0358 1.5692 1.7208
20 -0.3574 2.3174 1.6026 2.0024 1.2876
21 0.0758 1.8842 2.0358 1.5692 1.7208
22 0.0758 1.8842 2.0358 1.5692 1.7208
23 0.5091 1.4509 2.4691 1.1359 2.1541
24 0.5091 1.4509 2.4691 1.1359 2.1541
25 0.5091 1.4509 2.4691 1.1359 2.1541
26 0.9423 1.0177 2.9023 0.7027 2.5873
27 0.9423 1.0177 2.9023 0.7027 2.5873
28 1.3756 0.5844 3.3356 0.2694 3.0206
29 0.5091 1.4509 2.4691 1.1359 2.1541
30 0.5091 1.4509 2.4691 1.1359 2.1541
31 0.9423 1.0177 2.9023 0.7027 2.5873
32 0.9423 1.0177 2.9023 0.7027 2.5873
33 1.3756 0.5844 3.3356 0.2694 3.0206
34 1.3756 0.5844 3.3356 0.2694 3.0206
35 0.9423 1.0177 2.9023 0.7027 2.5873
36 0.5091 1.4509 2.4691 1.1359 2.1541
37 0.9423 1.0177 2.9023 0.7027 2.5873
38 0.9423 1.0177 2.9023 0.7027 2.5873
39 0.5091 1.4509 2.4691 1.1359 2.1541
40 0.5091 1.4509 2.4691 1.1359 2.1541
MIN DISTANCE 0.5844 0.1304 0.2694 0.0122

Table 3: Z-Scores for height and absolute difference from the cut-off points
ID# height_Z_score D1= height_Z_score-Upper (2.5%) D1= height_Z_score-Lower (2.5%) D1= height_Z_score-Upper (5%) D1= height_Z_score-Lower (5%)
1 -1.3305 3.2905 0.6295 2.9755 0.3145
2 0.9057 1.0543 2.8657 0.7393 2.5507
3 -2.2250 4.1850 0.2650 3.8700 0.5800
4 1.8002 0.1598 3.7602 0.1552 3.4452
5 -1.3305 3.2905 0.6295 2.9755 0.3145
6 0.9057 1.0543 2.8657 0.7393 2.5507
7 -0.8833 2.8433 1.0767 2.5283 0.7617
8 0.9057 1.0543 2.8657 0.7393 2.5507
9 -1.3305 3.2905 0.6295 2.9755 0.3145
10 -0.8833 2.8433 1.0767 2.5283 0.7617
11 -0.8833 2.8433 1.0767 2.5283 0.7617
12 -1.3305 3.2905 0.6295 2.9755 0.3145
13 0.0112 1.9488 1.9712 1.6338 1.6562
14 0.0112 1.9488 1.9712 1.6338 1.6562
15 -0.4361 2.3961 1.5239 2.0811 1.2089
16 0.0112 1.9488 1.9712 1.6338 1.6562
17 0.4584 1.5016 2.4184 1.1866 2.1034
18 0.4584 1.5016 2.4184 1.1866 2.1034
19 0.9057 1.0543 2.8657 0.7393 2.5507
20 1.8002 0.1598 3.7602 0.1552 3.4452
21 -0.4361 2.3961 1.5239 2.0811 1.2089
22 -0.4361 2.3961 1.5239 2.0811 1.2089
23 0.0112 1.9488 1.9712 1.6338 1.6562
24 0.0112 1.9488 1.9712 1.6338 1.6562
25 0.0112 1.9488 1.9712 1.6338 1.6562
26 0.9057 1.0543 2.8657 0.7393 2.5507
27 0.9057 1.0543 2.8657 0.7393 2.5507
28 1.3529 0.6071 3.3129 0.2921 2.9979
29 1.3529 0.6071 3.3129 0.2921 2.9979
30 1.8002 0.1598 3.7602 0.1552 3.4452
31 0.4584 1.5016 2.4184 1.1866 2.1034
32 0.0112 1.9488 1.9712 1.6338 1.6562
33 0.4584 1.5016 2.4184 1.1866 2.1034
34 0.4584 1.5016 2.4184 1.1866 2.1034
35 0.0112 1.9488 1.9712 1.6338 1.6562
36 -1.3305 3.2905 0.6295 2.9755 0.3145
37 -1.3305 3.2905 0.6295 2.9755 0.3145
38 -0.8833 2.8433 1.0767 2.5283 0.7617
39 -0.4361 2.3961 1.5239 2.0811 1.2089
40 -0.4361 2.3961 1.5239 2.0811 1.2089
MIN DISTANCE 0.1598 0.2650 0.1552 0.3145
References
Colan, S. D. (2013). The why and how of Z scores. Journal of the American Society of
Echocardiography, 26(1), 38-40.