# 8.2a z-Score of a Normal Distribution

8.2a z–Score of a Normal Distribution

Example:
(a)  A normal distribution has a mean, µ = 50 and a standard deviation σ = 10. Calculate the standard score of the value X = 35.
(b)  The masses of Form 5 students of a school are normally distributed with a mean of 60kg and a standard deviation of 15kg.
Find
(i) the standard score of the mass of 65kg,
(ii) the mass of a student that corresponds to the standard score of – ½.

Solution:
(a)
X ~ N (µσ2).
X ~ N (50, 102)
$Z=\frac{X-\mu }{\sigma }=\frac{35-50}{10}=-1.5$

(b)(i)
X – Mass of a Form 5 student
X ~ N (µσ2).
X ~ N (60, 152)
$Z=\frac{X-\mu }{\sigma }=\frac{65-60}{15}=\frac{1}{3}$
Hence, the standard score of the mass of 65kg is .

(b)(ii)
Z = – ½,
$\begin{array}{l}Z=\frac{X-\mu }{\sigma }\\ -\frac{1}{2}=\frac{X-60}{15}\\ X-60=-\frac{1}{2}\left(15\right)\\ X=52.5\end{array}$
Hence, the mass of a Form 5 student that corresponds to the standard score of – ½ is 52.5kg.