Page 316 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 15: Normal Probability Distribution

It is a bell shaped symmetrical curve about the ordinate at X. Notes
The ordinate is maximum at X.

It is unimodal curve and its tails extend infinitely in both directions.

The curve is asymptotic to X-axis in both directions.
The value of p(X) is always non-negative for all values of X, i.e., the whole curve lies
above X-axis
Since the distribution is symmetrical, all odd ordered central moments are zero.

The area between the ordinates at and + is 0.6826. This implies that for a normal
distribution about 68% of the observations will lie between and + .
The area between the ordinates at 2 and + 2 is 0.9544. This implies that for a
normal distribution about 95% of the observations will lie between 2 and + 2 .

The area between the ordinates at 3 and + 3 is 0.9974. This implies that for a
normal distribution about 99% of the observations will lie between 3 and + 3 .
This result shows that, practically, the range of the distribution is 6 although,
theoretically, the range is from to .
Normal distribution can be used as an approximation to binomial distribution when n is
large and neither p nor q is very small.

15.8 Keywords

Condition of homogeneity: The factors must be similar over the relevant population although,
their incidence may vary from observation to observation.

Condition of independence: The factors, affecting observations, must act independently of each
other.
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.
Fitting a Normal Curve: A normal curve is fitted to the observed data with the objectives (1) To
provide a visual device to judge whether it is a good fit or not. (2) Use to estimate the
characteristics of the population.

Method of Areas: Under this method, the probabilities or the areas of the random variable lying
in various intervals are determined. These probabilities are then multiplied by N to get the
expected frequencies.

Method of Ordinates: In this method, the ordinate f(X) of the normal curve, for various values of
the random variate X are obtained by using the table of ordinates for a standard normal variate.
Normal Approximation to Poisson Distribution: Normal distribution can also be used to
approximate a Poisson distribution when its parameter m 10 .
Normal Probability 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.

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