Transcript Chapter 19

TMAT 103
Chapter 19
Statistics for Process Control
TMAT 103
§19.1
Graphic Presentation of Data
§19.1 – Graphic Presentation of Data
• Collected data is presented graphically to:
– Understand distribution of data
– Identify trends
• Current
• Future
– Draw conclusions
§19.1 – Graphic Presentation of Data
• An automobile manufacturer is analyzing data
gathered in regards to cars coming off an assembly
line and not starting on the first try. In the month
of April, the following data was collected:
42
25
12
44
17
18
27
18
48
30
36
9
33
46
29
14
26
28
31
39
35
31
32
21
29
20
16
45
23
35
§19.1 – Graphic Presentation of Data
• A frequency table can make the data more
readable:
First-Try Nonstarts
Tallies
Frequency
9 – 16
17 – 24
25 – 32
33 – 40
41 – 48
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4
6
10
5
5
§19.1 – Graphic Presentation of Data
• A histogram can also be used:
– Rectangles must be of equal width
§19.1 – Graphic Presentation of Data
• A frequency polygon can also be used:
– X coordinate is center of interval, y coordinate is frequency
§19.1 – Graphic Presentation of Data
• Example:
– Using 10 intervals containing 6 numbers each, construct a
frequency table, histogram, and frequency polygon for the
following situation:
A local restaurant counted the number of hamburgers served
on 25 consecutive weekends – the data is given below.
268
252
222
279
234
260
261
220
246
268
253
250
228
273
243
268
245
272
254
225
230
240
250
279
231
TMAT 103
§19.2
Measures of Central Tendency
§19.2 – Measures of Central Tendency
• Central Tendency
– Finding a number which describes a set of data
– 3 general methods
• Mean
• Median
• Mode
§19.2 – Measures of Central Tendency
• The Mean
– The Mean of a set of n numbers, a1, a2, …, an is given by:
a1  a2  ...  an
x
n
§19.2 – Measures of Central Tendency
• Examples:
– 13 students took an exam, with the following scores. Find the mean score.
98, 92, 90, 85, 85, 82, 77, 76, 75, 74, 74, 68, 52
– 10 people were surveyed for their salary. The following data was
collected. Find the mean salary.
35,000,000
99,500
88,300
67,200
60,000
59,500
55,300
30,200
25,400
22,000
§19.2 – Measures of Central Tendency
• The Median
– The Median of an ordered set of n numbers, a1, a2, …, an is
the middle number if n is odd, and the mean of the two middle
numbers if n is even.
§19.2 – Measures of Central Tendency
• Examples:
– 13 students took an exam, with the following scores. Find the median
score.
98, 92, 90, 85, 85, 82, 77, 76, 75, 74, 74, 68, 52
– 10 people were surveyed for their salary. The following data was
collected. Find the median salary.
35,000,000
99,500
88,300
67,200
60,000
59,500
55,300
30,200
25,400
22,000
§19.2 – Measures of Central Tendency
• The Mode
– The Mode of an set of n numbers, a1, a2, …, an is the number
which occurs most often. There may be more than one mode.
§19.2 – Measures of Central Tendency
• Examples:
– 13 students took an exam, with the following scores. Find the mode.
98, 92, 90, 85, 85, 82, 77, 76, 75, 74, 74, 68, 52
– 10 people were surveyed for their salary. The following data was
collected. Find the mode.
35,000,000
99,500
88,300
67,200
60,000
59,500
55,300
30,200
25,400
22,000
TMAT 103
§19.3
Measures of Dispersion
§19.3 – Measures of Dispersion
• Terminology
– Range
• Difference between largest and smallest values
– Population
• Collection of all items being considered
– Sample
• Items selected to be in calculation
– Random
• When each item has an equal chance to be selected
– Sample Standard Deviation
• One way to measure dispersion
§19.3 – Measures of Dispersion
• Sample Standard Deviation
– The sample standard deviation of a set of data x1, x2, …, xn is
given by:
( x1  x ) 2  ( x2  x ) 2  ...  ( xn  x ) 2
sx 
n 1
§19.3 – Measures of Dispersion
• Example:
– A furniture company manufactures 28-in. table legs.
Acceptable lengths are between 27.9375 and 28.0625 in. A
random sample of 30 legs were measured each day for a week.
The number of acceptable legs produced each day were:
41, 41, 43, 44, 46, 46, 48.
Find the range and sample standard deviation for this set.
TMAT 103
§19.4
The Normal Distribution
§19.4 – The Normal Distribution
• Normal distribution
– Histogram of sample means with smooth curve drawn through
centers of rectangles
§19.4 – The Normal Distribution
• Important features of normal distribution
–
–
–
–
Bell shaped
Symmetric about a vertical line passing through the mean
Smaller sx implies more data is closer to the mean
Distribution of data is predictable
§19.4 – The Normal Distribution
• Smaller sx implies more data is closer to the mean
§19.4 – The Normal Distribution
• Distribution of data is predictable
– 68.2% within 1 standard deviation of mean
– 95.4% within 2 standard deviations of mean
– 99.7% within 3 standard deviations of means
§19.4 – The Normal Distribution
• Examples:
– Does the following set of numbers meet the criteria for
a normal distribution in terms of the percent of values
with one standard deviation (allow a 2% margin of
error)?
35, 36, 38, 43, 47, 48, 48, 55, 67
– The scores from a test resulted in a mean of 72 and
standard deviation of 8.5. Mark scored 89. Assuming
the scores were normally distributed, what percent of
students can he estimate scored below him?
TMAT 103
§19.5
Fitting Curves to Data Sets
§19.5 – Fitting Curves to Data Sets
• Regression Analysis
– Finding an equation which relates to a data set
as closely as possible
– Allows for analysis and prediction
– Advanced regression analysis uses matrix
theory and calculus
TMAT 103
§19.6
Statistical Process Control
§19.6 – Statistical Process Control
• Using statistics for quality control
– Specifications
• Does it meet or exceed established specifications
– Durability
• Does the item perform as long as expected
– Reliability
• How often are repairs needed
– Service
• Is item easy to repair? Are shipping/billing errors rare?
– Customer needs
• Does item meet expectations and needs of customer?
§19.6 – Statistical Process Control
• Terminology
– Common cause variation
• Variations always occur within a product
– Process in control
• Produced items consistently fall within common cause tolerance limits,
and measurements fit a normal curve
– Limits
•
•
•
•
UCL – upper control limit
LCL – lower control limit
UTL – upper tolerance limit
LTL – Lower tolerance limit
– Capable process
• Control limits within tolerance limits (i.e. specifications)
§19.6 – Statistical Process Control
• Process in control
§19.6 – Statistical Process Control
• Capable Process
– A capable process is ALWAYS in control
§19.6 – Statistical Process Control
• Process not in control, but within tolerance limits
– Unpredictable, and undesirable
§19.6 – Statistical Process Control
• Determine which of the processes are
capable
– Both processes are in control, and all
measurements are in centimeters
Process
Control Limits
Order Specifications
1
44.9992 to 45.0008
45  0.001
2
3.9995 to 4.0005
4  0.0001