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Statistics

Notes
1 15 1 1
(a) P (5 X£ £ 15 ) = ò dX = (15 5- ) =
20 5 20 2

1 1
(b) P (0 X£ £ 10 ) = ´ 10 =
20 2

20 15 1
-
(c) P (15 X£ £ 20 ) = = .
20 4

15.3 Normal Distribution

The normal probability distribution occupies a place of central importance in Modern Statistical
Theory. This distribution was first observed as the normal law of errors by the statisticians of
the eighteenth century. They found that each observation X involves an error term which is
affected by a large number of small but independent chance factors. This implies that an observed
value of X is the sum of its true value and the net effect of a large number of independent errors
which may be positive or negative each with equal probability. The observed distribution of
such a random variable was found to be in close conformity with a continuous curve, which was
termed as the normal curve of errors or simply the normal curve.

Since Gauss used this curve to describe the theory of accidental errors of measurements involved
in the calculation of orbits of heavenly bodies, it is also called as Gaussian curve.

15.3.1 The Conditions of Normality

In order that the distribution of a random variable X is normal, the factors affecting its observations
must satisfy the following conditions :
(i) A large number of chance factors: The factors, affecting the observations of a random
variable, should be numerous and equally probable so that the occurrence or non-
occurrence of any one of them is not predictable.
(ii) Condition of homogeneity: The factors must be similar over the relevant population
although, their incidence may vary from observation to observation.
(iii) Condition of independence: The factors, affecting observations, must act independently of
each other.
(iv) Condition of symmetry: Various factors operate in such a way that the deviations of
observations above and below mean are balanced with regard to their magnitude as well
as their number.
Random variables observed in many phenomena related to economics, business and other
social as well as physical sciences are often found to be distributed normally. For example,
observations relating to the life of an electrical component, weight of packages, height of persons,
income of the inhabitants of certain area, diameter of wire, etc., are affected by a large number
of factors and hence, tend to follow a pattern that is very similar to the normal curve. In addition
to this, when the number of observations become large, a number of probability distributions
like Binomial, Poisson, etc., can also be approximated by this distribution.

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