NORMAL DISTRIBUTION

1. NORMAL DISTRIBUTION BPT 2423 – STATISTICAL PROCESS CONTROL

2. CHAPTER OUTLINE • Frequency Distribution • Normal Distribution / Probability • Areas Under The Normal Curve • Application of Normal Distribution (N.D.)

3. LESSON OUTCOMES • Understand the importance of the normal curve in quality assurance • To know how to find the area under a curve using the standard normal probability distribution (Z tables) • Able to interpret the information analyzed

4. FREQUENCY DISTRIBUTIONS What is a Frequency Distribution? A frequency distribution is a list or a table … containing the values of a variable (or a set of ranges within which the data falls) ... and the corresponding frequencies with which each value occurs (or frequencies with which data falls within each range)

5. Why Use Frequency Distributions? FREQUENCY DISTRIBUTIONS • A frequency distribution is a way to summarize data • The distribution condenses the raw data into a more useful form and allows for a quick visual interpretation of the data

6. FREQUENCY DISTRIBUTIONS Example: An advertiser asks 200 customers how many days per week they read the daily newspaper.

7. FREQUENCY DISTRIBUTIONS Relative Frequency : What proportion is in each category? 22% of the people in the sample report read the newspaper 0 days per week

8. NORMAL DISTRIBUTION The most important continuous probability distribution in statistics is the normal distribution (also referred to as Gaussian distribution), where its graph is called the normal curve. The continuous random variable X having the bell-shaped distribution is called normal random variable

9. NORMAL DISTRIBUTION • The area under the normal curve can be determined if the mean and the standard deviation are known. • Mean (average) – locates the center of the normal distribution • Standard deviation – defines the spread of the data about the center of the distribution.

10. NORMAL DISTRIBUTION

11. NORMAL DISTRIBUTION • Properties of the normal curve: • A normal curve is symmetrical about µ, the central value. • The mean, mode and median are all equal. • The curve is unimodal and bell-shaped. • Data values concentrate around the mean value of the distribution and decrease infrequency as the values get further away from the mean. • The area under the normal curve equals 1. 100% of the data are found under the normal curve, 50% on the left-hand side and another 50% on the right.

12. NORMAL DISTRIBUTION contains about 68.3% of the values in the population or the sample 68.3%

13. NORMAL DISTRIBUTION contains about 95.5% of the values in the population or the sample contains about 99.7% of the values in the population or the sample 95.5% 99.7%

14. AREAS UNDER THE NORMAL CURVE The area under the curve of any density function bounded by the two ordinates x = x1 and x = x2 equals the probability that the random variable X assumes a value between x = x1 and x = x2.

15. AREAS UNDER THE NORMAL CURVE where: • x = original data value • μ = population mean • σ = population standard deviation • z = standard score (number of standard deviations x is from μ) where: • x = original data value • x = sample mean • s = sample standard deviation • z = standard score (number of standard deviations x is from x bar)

16. AREAS UNDER THE NORMAL CURVE The distribution of a normal random variable with mean 0 and variance 1 is called a standard normal distribution.

17. AREAS UNDER THE NORMAL CURVE Example 1 :

18. AREAS UNDER THE NORMAL CURVE Example 2 :

19. AREAS UNDER THE NORMAL CURVE Example 3 :

20. AREAS UNDER THE NORMAL CURVE Example 4 :

21. AREAS UNDER THE NORMAL CURVE Example 5 :

22. AREAS UNDER THE NORMAL CURVE Solution :

23. APPLICATIONS OF N.D. Exercise 1 : Exercise 2 :

24. APPLICATIONS OF N.D. Answer For Exercise 1:

25. APPLICATIONS OF N.D. Answer For Exercise 2:

26. APPLICATIONS OF N.D. Exercise 3 : Exercise 4 :

27. APPLICATIONS OF N.D. Answer For Exercise 3:

28. APPLICATIONS OF N.D. Answer For Exercise 4: