Probability in Statistics Normal Distribution Normal Probability Distribution and its Problems Sample Examples

2. Normal Probability Distribution
And its Problems

3. Group Members
WALEED AHMED 14-ARID-1458
ALI RAZA 14-ARID-1402
NOMAN ALI 14-ARID-1449

5. What is Normal Distribution?
• A probability distribution that plots all of its
values in a symmetrical fashion and most of the
results are situated around the probability's
mean.
• A normal distribution is a continuous,
symmetric, bell-shaped distribution of a variable.

6. History
• In 1733, the French Mathematician , Abraham DeMoivre.
• About 100 years later
• Pierre Laplace in France
• Carl Gauss in Germany (“Gaussian Distribution”)
• In 1924, Karl Pearson found that DeMoivre was Correct

7. The Normal Curve
• Represented by the Bell-shaped Curve.
• Symmetric Curve
• Continuous Curve.

8. The Normal Equation
The Mathematical Equation for Normal Distribution is
Where
x = Normal Random Variable
μ = Mean “mu”
Standard Deviation “sigma”
 
 2
2
2
1
2
x
f x e


 
 
 
 
x   

10. Effect of Mean
• Mean Effects On The Position Of The Curve.

11. Effect of Standard Deviation
• Standard Deviation Effects On the Disperse of the
Curve.

12. The Standard Normal
Distribution
• The Standard Normal Distribution is a Normal Distribution
With a Mean of 0 and a Standard Deviation Of 1.
• Formula Becomes:
 
2
2
2
z
e
f x




13. Representation of Standard curve
• Total area under the curve is 1.

14. Properties of a Normal Curve
1. The mean, median, and mode are equal and are located at
the center of the distribution (Highest Point = μ).
2. A normal distribution curve is continuous, unimodal and
symmetric about the mean.
3. The total area under every normal curve is 1.
4. It is completely determined by its Mean and S.D.

15. Properties
• The Area that Lies Within 1 Standard Deviation Of the
Mean is Approximately 0.68, Or 68%
• Within 2 S.Ds, About 0.95, Or 95%
• And Within 3 S.Ds, About 0.997, Or 99.7%.

16. Properties

17. Standardizing the Variables
• All Normally Distributed variables can be transformed into
Standard Normally distributed variables by the given
formula.
x
z



value - mean
standard deviation
z 

19. Questions
Find Probability for each.
a) P(0< z <2.32)
b) P(z < 1.65)
c) P(z > 1.91)

20. Questions
Sol.a) P(0 < z < 2.32)
1. Look up the Area Corresponding to 2.32. It is 0.9898.
2. Then lookup the Area Corresponding to 0 in z-table. It is 0.500
3. Subtract the Two Areas: 0.9898 - 0.5000 = 0.4898.
4. Hence the probability is 0.4898, or 48.98%.

21. Questions
Sol.b) P(z < 1.65)
1. Look up the area corresponding to z < 1.65 in Table Z. It is
0.9505.
2. Hence, P(z < 1.65) = 0.9505, or 95.05%.

22. Questions
Sol.c) P(z > 1.91)
1. Look up the area that corresponds to z > 1.91. It is 0.9719.
2. Then subtract this area from 1.0000.
3. P(z > 1.91) = 1.0000 - 0.9719 =0.0281, or 2.81%.