Normal Distribution
Normal distribution is an approximation of binomial distribution under certain conditions.
1. n, the number of trails is indefinitely large, i.e. n  
2. neither p nor q is very small.
Definition: A continuous random variable X is said to have a normal distribution with parameters 
and  2 if its density function is given by the probability law
 1  x   2 
exp  
 
2 
 2    
Where    X  
    ,  0
e  2.7183 ,   3.1416
( 2  2.5066 )
f ( x /  , 2)  N (  ,  2 ) 
1
Here  and  2 are the mean and variance of the normal distribution respectively.
Note: A random variable X with mean  and variance  2 and following the normal law can be
expressed by X ~ N (  ,  2 )
Properties of normal distribution:
The following points are important properties of normal distribution.
Normal curve
1. The normal curve is symmetrical and bell shaped. The range of
the distribution is   to 
2. The value of mean, median, mode will coincide as the
distribution is symmetrical.
i.e. mean = median = mode


3. The parameters  and  2 represent the mean and variance of
the distribution. For different values of the parameters we get different normal distributions.
4. It has only one mode i.e. the distribution is unimodal and it occurs at x   .
5. The skew ness of the distribution is 1  0 and kurtosis is  2  3 .
6. The odd ordered moments about mean vanishes i.e.  2r 1  0
7. The even ordered moments about mean  2 r  (2r  1) 2  2 r 2 .
8. The mean deviation from mean is 
2
4
 .
 5
9. The total area bounded by the curve and horizontal axis is equal to 1.
10. The maximum ordinate occurs at x   and its value is
1
2 
.
Q3  Q1
 0.6745
2
12. The first quartile Q1 =  -0.6745  and third quartile Q3 =  +0.6745 
11. The quartile deviation is
Q3  Q1

 0.6745 .
Q3  Q1

14. A linear function a1 X 1  a 2 X 2  a3 X 3  .........  a n X n of n independent normal variables
13. The co-efficient of quartile deviation
X 1 , X 2 , X 3 ,............, X n with means 1 ,  2 ,  3 ,........,  n and variances  1 ,  2 ,  3 ,..........,  n is also
a normal variable with mean a1 1  a 2  2  a3  3  ........  a n  n and
2
2
2
2
variance a1  1  a2  2  a3  3  ..........  an  n .
2
2
2
2
2
2
2
2
Proof:
Let Z= a1 X 1  a 2 X 2  a3 X 3  .........  a n X n
Now, E (Z ) = Ea1 X 1  a2 X 2  a3 X 3  .........  an X n 
= a1 E[ X 1 ]  a2 E[ X 2 ]  a3 E[ X 3 ]  .........  an E[ X n ]
= a1 1  a 2  2  a3  3  ........  a n  n
And
V (Z )  Va1 X 1  a2 X 2  a3 X 3  .........  an X n 
= a1 V ( X 1 )  a2 V ( X 2 )  a3 V ( X 3 )  ..........  an V ( X n )
2
2
2
2
= a1  1  a2  2  a3  3  ..........  an  n
2
2
2
2
2
2
2
2
15. If X is a normal variate with mean  and standard deviation  , then the distribution of
Z
X 

is also normal with mean 0 and variance 1. Here Z is called standard normal
variable.
Symbolically if X ~ N (  ,  2 ) then Z 
X 

~ N (0,1)
Proof:
1
X  1
E (Z )  E 
 E ( X )          0


   
1
1
X 
V (Z )  V 
 2 V ( X )  2  2  1 .


   
 
Note: The probability density function of standard normal variable Z is
f ( z / 0,1)  N (0,1) 
1
2 
e
1
 z2
2
,  Z  
Area under normal curve:
As the normal variable is a continuous random variable, the probability that the random variable X
assumes a value x  x1 and x  x2 is represented by the area under the probability curve bounded by
the values x1 and x2 can be defined as
x2
Pr ob( x1  x  x2 ) = 
x1
1
2
e
1 x   


2  
2
dx
Since the normal curve depends on two parameters  and  2 , the area represented by
Pr ob( x1  x  x2 ) is also dependent on  and  2 . Though theoretically this probability can be
calculated by using the method of integral calculus, normal integral tables are available for the use
of practicing statisticians. It is very voluminous work to compile tables for all possible values of 
and  2 . In fact such tables would be infinitely many because       ,   0 .
To facilitate the preparation of tables, the normal variable is standardized or is transformed to
a new variable which is also normal, but having mean 0 and variance 1. Thus if X is normal variable
X 
with mean  and variance  2 , then Z 

is a standardized normal variable having mean 0 and
variance 1
 x1  
And thus Pr ob( x1  x  x2 ) = Pr ob



x


x2   

 
= Pr ob( z1  z  z 2 )
i.e.
x2
1
Pr ob( x1  x  x2 ) = 
2
x1
z2
1
=
2
z1
e
e
1 x   


2  
1 2
z 
2
2
dx
dz = Pr ob( z1  z  z 2 )
Cumulative distribution functions of Z:
The cumulative distribution function of Z is defined as
 t   P(Z  t ) =
t


1
2
1
e2
 z 2
dz
And
P( Z  t ) =1- P ( Z  t )
Note: since the normal curve is symmetrical
P( Z  t ) = P( Z  t )
Some of the areas of standardized normal curve:
Distance from the
mean ordinate
±0.6745
±1
±1.96
±2
±2.58
±3
Area under curve
50%
68.27%
95%
95.45%
99%
99.73%