The right figure shows (a) truncated and (b) ordinary notmal distribution of many evens.
(1) The frequency distribution of x.
(2) The cumulative distribution of x. The ordinate is initially the number of events, but may be understood as probability.
(3) Operating cum^-1 on y gives x', where x' vs x is a line.
(4) Operate a linear transformation g on x' so that a cumulative distribution can be obtained, where g(x'(x_min))=0% and g(x'(x_max))=100%.

<Possible to be ranged 0%〜100%?>

Yes for Model (a). But no for Model (b) since x' extends to infinity and since no unique g can be determined .

His has adopted Model (b), where the only restriction is that z=1/2 if x is the average. Then y-z relation is given by

<Model (b) continuous cumulative distribution>

Let assume that x=[2, 8]. The ordinary distribution is x vs y, where y ranges between 0% and 100%. x vs z is the transformed distribution, where ○ is 5% of y and ● 5% of z.
It is possible to make ε smaller, say 0.001, but then x-z extends further, losing the practicability.

Take the data for Fig.2.2 of Dr. Berendsen (p.7). Thin line is the ordinary cumulative distribution. Thick line is the transformed distribution with ε = 0.01 (1%). The shaded regions should be excluded since z<ε, z>1-ε there. This figure seems consistent with Fig.2.2.