Page 305 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 15.3 Properties of Normal Probability Curve

A normal probability curve or normal curve has the following properties:
1. It is a bell shaped symmetrical curve about the ordinate at X = . The ordinate is maximum
at X = .
2. It is unimodal curve and its tails extend infinitely in both directions, i.e., the curve is
asymptotic to X axis in both directions.

3. All the three measures of central tendency coincide, i.e.,
mean = median = mode
4. The total area under the curve gives the total probability of the random variable taking
values between – to . Mathematically, it can be shown that
2
1 X
1 2
P X p X dX e dX 1.
2
5. Since median = , the ordinate at X = divides the area under the normal curve into two
equal parts, i.e.,

p X dX p X dX 0.5

6. The value of p(X) is always non-negative for all values of X, i.e., the whole curve lies above
X axis.
7. The points of inflexion (the point at which curvature changes) of the curve are at
X = ± .

8. The quartiles are equidistant from median, i.e., M – Q = Q – M , by virtue of symmetry.
d 1 3 d
Also Q = – 0.6745 , Q = + 0.6745 , quartile deviation = 0.6745 and mean deviation
1 3
= 0.8s, approximately.
9. Since the distribution is symmetrical, all odd ordered central moments are zero.
10. The successive even ordered central moments are related according to the following
recurrence formula
= (2n - 1) 2 for n = 1, 2, 3, ......
2n 2n - 2
11. The value of moment coefficient of skewness is zero.
1
3 4
4
12. The coefficient of kurtosis 2 2 4 3.
2
Note that the above expression makes use of property 10.
13. Additive or reproductive property

If X , X , ...... X are n independent normal variates with , ,  means and variances
1 2 n 1 2 n
2 2 , 2
1 , 2 , respectively, then their linear combination a X + a X + ...... + a X is
1
1
2
n
2
n
n n
2 2
also a normal variate with mean a i i and variance a i i .
i 1 i 1
300 LOVELY PROFESSIONAL UNIVERSITY