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Normal Distribution
Normal Distribution: Suppose 𝑋 is the set of continuous set of data with mean 𝜇 and variance
𝜎 2 and the graph of p.d.f of this set of data is bell shape then X is said to be normally
distributed. 𝜇 and 𝜎 2 are said to be parameter of 𝑋.
If X is normally distributed then it can be written as 𝑋~(𝜇, 𝜎 2 )
Examples.
𝑋~(3, 0.9)
𝑋~(14, 4)
𝑋~(78, 25)
Curve of normal distribution: The normal distribution curve has the following properties.
 It is bell shaped.
 It is symmetrical about mean.
 It extends from − ∞ 𝑡𝑜 + ∞.
 The total area under the curve is 1.
 Approximately 99.8% of the distribution lies within 3 standard deviation of mean.
Standard normal variable: If a normal has mean 0 and variance 1 then it is called standardized
normal distribution and it is denoted by Z.
𝑍~(0, 1)
Probability density function of standard normal distribution: The p.d.f of standard normal
distribution is denoted by ∅(𝑧) and it gives the area under the curve before the value of 𝑧.
Use of standard normal table:
 To find the value of ∅(𝑧) at given value of z the number must be in three decimal places.
 First decimal place is looked in first column and second decimal place is looked in first
row. Find the crossed value.
 Third decimal place is looked in “Column of ADD”. This value is added into last two digits
of crossed value.
Q.1
Find the value of function ∅(𝑧),
i.
∅(1.234)
iii.
∅(1.6789)
ii.
∅(2.5)
iv.
∅(2.45)
Relation between area under the graph and probability of standardized normal Variable:
Basic Rules:
 P(Z < 𝑎) = ∅(𝑧)
 P(Z > 𝑎) = P(Z < −a)
 ∅(−𝑎) = 𝟏 − ∅(𝑎)
 P(a < Z < b) = ∅(𝑏) − ∅(𝑎)
 P(|Z| < 𝑎) = P(−a < Z < a)
 P(|Z| > 𝑎) = P(Z < −𝑎 𝑜𝑟 Z > 𝑎) = 𝑃(𝑍 < −𝑎) + 𝑃(𝑍 > 𝑎)
Q.2
If 𝑍~(0, 1), find
i.
P(Z < 0.16)
ii.
P(Z < −1.346)
iii.
P(Z > 2.9)
iv.
P(Z < 0.1362)
v.
P(|Z| < 1.12)
vi.
P(|Z| > 1.89)
Use of standard normal table to find ∅−𝟏 (𝑧):
 Look the probability in main body of the table. If is not appear then look the value is less
than it.
 Find in which row and column it is lying.
 Column and row will make first and second decimal places of given value.

Compiled By : Sir Rashid Qureshi
www.levels.org.pk