BPT 2423 STATISTICAL PROCESS CONTROL Frequency Distribution Normal

BPT 2423 – STATISTICAL PROCESS CONTROL

§ Frequency Distribution § Normal Distribution / Probability § Areas Under The Normal Curve § Application of Normal Distribution (N. D. )

§ 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

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)

WHY USE 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

Example: An advertiser asks 200 customers how many days per week they read the daily newspaper. Number of Days Read Frequency 0 44 1 24 2 18 3 16 4 20 5 22 6 26 7 30 Total 200

Relative Frequency : What proportion is in each category? Number of Days Read Frequency Relative Frequency 0 44 . 22 1 24 . 12 2 18 . 09 3 16 . 08 4 20 . 10 5 22 . 11 6 26 . 13 7 30 . 15 Total 200 1. 00 22% of the people in the sample report read the newspaper 0 days per week

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

�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.

§ Properties of the normal curve: 1. A normal curve is symmetrical about µ, the central value. 2. The mean, mode and median are all equal. 3. The curve is unimodal and bell-shaped. 4. Data values concentrate around the mean value of the distribution and decrease infrequency as the values get further away from the mean. 5. 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.

contains about 68. 3% of the values in the population or the sample 68. 3%

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%

The area under the curve of any density function bounded by the two ordinates x = x 1 and x = x 2 equals the probability that the random variable X assumes a value between x = x 1 and x = x 2.

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)

The distribution of a normal random variable with mean 0 and variance 1 is called a standard normal distribution.

Example 1 :

Example 2 :

Example 3 :

Example 4 :

Example 5 :

Solution :

Exercise 1 : Exercise 2 :

Answer For Exercise 1:

Answer For Exercise 2:

Exercise 3 : Exercise 4 :

Answer For Exercise 3:

Answer For Exercise 4: