Download NORMAL DISTRIBUTION

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)

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
44
 .22
200
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.
μ  1σ
contains about 68.3% of the values
in the population or the sample
68.3%
μ
μ  1σ
μ  2σ contains about 95.5% of the values in the
population or the sample
μ  3σ contains about 99.7% of the values in the
population or the sample
95.5%
99.7%
μ  2σ
μ  3σ
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.
x μ
z
σ
where:
 x = original data value
 μ = population mean
 σ = population standard deviation
 z = standard score (number of
standard deviations x is from μ)
xx
z
s
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: