The Normal Distribution
Properties of The Normal
Distribution

The curve is bell-shaped with the
highest point over the mean, .
Properties of The Normal
Distribution

The curve is symmetrical about a
vertical line through .
Properties of The Normal
Distribution

The curve approaches the horizontal
axis but never touches or crosses it.
Properties of The Normal
Distribution
–


The transition points between cupping
upward and downward occur
above  +  and  –  .
The Normal Density Function
( x   )2
P( x) 
2 2
e
 2
This formula generates the density curve which
gives the shape of the normal distribution.
The Empirical Rule
Approximately 68% of the data values lie is
within one standard deviation of the mean.
68%

One standard deviation from the mean.
The Empirical Rule
Approximately 95% of the data values lie within
two standard deviations of the mean.
95%
x
Two standard deviations from the mean.
The Empirical Rule
Almost all (approximately 99.7%) of the data
values will be within three standard deviations of
the mean.
99.7%
x
Three standard deviations from the mean.
Application of the Empirical
Rule
The life of a particular type of light bulb is
normally distributed with a mean of 1100
hours and a standard deviation of 100
hours.
What is the probability that a light bulb of
this type will last between 1000 and 1200
hours?
Approximately 68%
Control Chart
a statistical tool to track data over
a period of equally spaced time
intervals or in some sequential
order
Statistical Control
A random variable is in statistical
control if it can be described by
the same probability distribution
when it is observed at successive
points in time.
To Construct a Control Chart
• Draw a center horizontal line at .
• Draw dashed lines (control limits) at
    and   .
• The values of  and  may be target
values or may be computed from past data
when the process was in control.
• Plot the variable being measured using
time on the horizontal axis.
Control Chart





1
2
3
4
5
6
7
Control Chart





1
2
3
4
5
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7
Out-Of-Control Warning Signals
I
One point beyond the 3 level
II
A run of nine consecutive points on
one side of the center line at
target 
IIIAt least two of three consecutive
points beyond the 2 level on the same
side of the center line.
Probability of a False Alarm
Warning Signal
Probability of false
alarm
I Point beyond 3
0.003
II Nine conscecutive
points on same side of

III At least 2/3 points
beyond 2
0.004
0.004
Is the Process in Control?





1
2
3
4
5
6
7
Is the Process in Control?





1 2 3 4 5 6 7 8 9 10 11 12 13
Is the Process in Control?





1
2
3
4
5
6
7
Is the Process in Control?





1
2
3
4
5
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7