How to normalize data to have mean zero?

[chiaoyi]

I have a variable called "Post-Test Score Yi^obs". The book said "normalized the rank of the variable to have mean zero". I think the rank is from 1 (the lowest value) to 16 (the highest value).

54.6
60.6
56.5
55.5
75.2
84.8
75.6
101.9
55.3
70.6
59.3
78.4
87
84.2
73.7
108.6

The normalized rank post-test score Ri is:

-7.5
2.5
-4.5
5.5
0.5
4.5
1.5
7.5
-6.5
-1.5
-3.5
2.5
5.5
3.5
-0.5
7.5

I don't know how to get this column. I tried first to get the average, and then the standard deviation. Then I use the formula (x-xbar)/std.dev. But I got the different column. The original rank column should be (I did):

1
6
4
3
9
13
10
15
2
7
5
11
14
12
8
16

[HanOostdijk]

Starting with your first column of numbers I get different result.
But I have to admit that I do not know why I would do this calculation.

x1 = c(54.6, 60.6, 56.5, 55.5, 75.2, 84.8, 75.6, 101.9,
55.3, 70.6, 59.3, 78.4, 87, 84.2, 73.7, 108.6)
x2 = order(x1)
x2
#>  [1]  1  9  4  3 11  2 10 15  5  7 12 14  6 13  8 16
x3 = x2-mean(x2)
x3
#>  [1] -7.5  0.5 -4.5 -5.5  2.5 -6.5  1.5  6.5 -3.5 -1.5  3.5  5.5 -2.5  4.5 -0.5
#> [16]  7.5

^{Created on 2020-06-15 by the reprex package (v0.3.0)}