cqe经典复习资料4

Chapter 4 Statistics 45
CHAPTER 4
BASIC QUALITY CONCEPTS
1.0 Continuous Distributions
2.0 Measures of Central Tendency
3.0 Measures of Spread or Dispersion
4.0 Histograms and Frequency Distributions
5.0 Shapes of Distributions
6.0 The Normal Curve
7.0 Discrete Distributions
8.0 Tolerances
9.0 Determination of Sample Size
10.0 Process Capability Analysis
11.0 Pareto Analysis
“In earlier times they had no statistics. They did it with lies and
we do it with statistics.”
Stephen Leacock
Chapter 4 Statistics 47
STATISTICS
1.0 CONTINUOUS DISTRIBUTIONS
Continuous distributions are formed because everything in the world that can be measured
varies to some degree. Measurements are like snowflakes and fingerprints, no two are
exactly alike. The degree of variation will depend on the precision of the measuring
instrument used. The more precise the instrument, the more variation will be detected. A
distribution, when displayed graphically, shows the variation with respect to a central value.
Everything that can be measured forms some type of distribution that contains the following
characteristics:
Measures of central tendency:
•Arithmetic mean or average
•Median
•Mode
Measures of spread or dispersion from the center:
•Range
•Variance
•Standard deviation
Shapes of distributions:
•Symmetrical - normal
•Symmetrical - not normal
•Skewed right or left
•More than one peak
2.0 MEASURES OF CENTRAL TENDENCY
Measures of central tendency are values that represent the center of the distribution.
2.1 Arithmetic Mean or Average
The arithmetic mean or average of sample data is denoted by x . The mean or average
of an entire population or universe is denoted by . The value of x may always be used
as an estimate of .
x
x
n
x x x x
n
i n


( ... ) 1 2 3
The symbol stands for “sum of.”
QReview 48
Five parts are measured and the following data are obtained:
2.6’’, 2.2’’, 2.4’’, 2.3’’, 2.5’’
x
x
n
i

(2.6 2.2 2.4 .2.3 2.5)
5
= 2.4
2.2 Median
The median is the middle value of the data points.
To find the median, the data must be rank ordered in either ascending or descending
order.
2.2, 2.3, 2.4, 2.5, 2.6
The Median is 2.4
For an even number of data points, the median is the average of the two middle points.
2.3 Mode
The mode is the value that occurs most frequently.
The data 2.6, 2.2, 2.4, 2.3, 2.5 do not contain a mode because no value occurs more
than any other.
The following data are taken from another product:
6, 8, 13, 13, 20
The Mode is 13
3.0 MEASURES OF SPREAD OR DISPERSION FROM THE CENTER
How much can data points vary from a center or central value and still be considered
reasonable variation? The question can be answered by calculating what is considered to be
the natural spread of the data values.
3.1 Range
The calculation of the range provides a simple method of obtaining the spread or
dispersion of a set of data. The range is the difference between the highest and lowest
number in the set and is denoted by the letter r. The range and average are points plotted
on control charts (a subject covered in a subsequent chapter). For the data set 2.6, 2.2,
2.4, 2.3, and 2.5, the high value is 2.6 and the low value is 2.2.
Range = r = (2.6 - 2.2) = .4
Chapter 4 Statistics 49
3.2 Variance
The variance is the mean squared deviation from the average in a set of data. It is used
to determine the standard deviation, which is an indicator of the spread or dispersion of a
data set.
Variance Sigma Squared
x x
n
i

2
( )2
3.3 Standard deviation
The standard deviation is the square root of the variance. It is also known as the rootmean-
square deviation because it is the square root of the mean of the squared
deviations.
S dard Deviation Sigma
x x
n
tan i
( )



2
The average and standard deviation together can provide a great deal of information
about a process or product. These two statistics are very powerful values used to make
inferences about the entire population based on sample data.
When an inference is made about a population from sample data, (n - 1) is used instead
of n in the denominator of the variance formula. The term (n - 1) is defined as degrees of
freedom. When (n - 1) is used, the calculated value is called the unbiased estimator of
the true variance and is usually denoted by s2. When the standard deviation is obtained
from the unbiased estimator of the variance it is denoted by
s or ’.
If a sample is taken and the average and standard deviation are not used to make
inferences about the entire population, then the sample is considered to be the population
and the standard deviation is indicated by . The symbol is used to denote the
population average and x is used to denote the sample average. The value of x may
always be used as an estimate of .
3.4 Variance and Standard Deviation Formulas
The following terminology and formulas will be used for the variance and associated
standard deviation:
•Variance and standard deviation using all data values of a finite population:
Variance
x
N
S dard Deviation Variance
i





2
( )2
tan
QReview 50
•Variance and standard deviation using a subset (sample) of an infinite (very large)
population:
Variance s
x x
n
S dard Deviation Variance s or
i



2 2
2
1
( )
( )
tan


This is called the unbiased estimator of the population variance 2.
•Variance and standard deviation using a subset (sample) of a finite population:
Variance s x x
n
N
N
S dard Deviation Variance s or
i 

•

2 2
2
1
( ) ( ) ( 1)
tan


This is also called the unbiased estimate of the population variance 2.
•The standard deviation for a distribution of averages is called the standard error.
S dard Error
x x
N
or
s
n x
i
s
tan
( )


s
2
Ns is the number of samples and n is the sample size.
Example 1
Compute the variance and standard deviation for the data: 2.6'' 2.2'', 2.4'', 2.3'', 2.5''.
Assume that the data is the entire population.


2 
2
2 4


( )
.
x
N
i where
(2.6 - 2.4)2 = ( .2)2 = .04
(2.2 - 2.4)2 = ( - .2)2 = .04
(2.4 - 2.4)2 = ( 0)2 = 0
(2.3 - 2.4)2 = ( - .1)2 = .01
(2.5 - 2.4)2 = ( .1)2 = .01
Total = .10
Therefore 2 = .10/5 = .02
Chapter 4 Statistics 51
The standard deviation is the square root of the variance. For this example, the standard
deviation is





( )
. .
x
N
i
2
02 1414
Many scientific hand calculators have a function to compute the mean, variance and
standard deviation. The calculator is the preferred method of obtaining the values. The
example is to ensure that you know what your calculator is doing when performing the
calculations.
Another formula known as the working formula may also be used to calculate the
variance and standard deviation. When the calculation for the variance and standard
deviation must be done manually, the working formula may be easier than the formula
given above. The answer will be the same using either formula. The working formula for
the variance is
2 
2
2


(x )
N
i
(xi)2
(2.6)2 = 6.76
(2.2)2 = 4.84
(2.4)2 = 5.76
(2.3)2 = 5.29
(2.5)2 = 6.25
Total = 28.90
Variance
x
N
S dard Deviation
i 



2
2
2 28 90 2
5
2 4 5 78 5 76 02
02 1414
( ) .
( . ) . . .
tan . .
4.0 HISTOGRAMS AND FREQUENCY DISTRIBUTIONS
4.1 Histograms
A histogram is a simple frequency distribution. It is a plot of the actual data showing the
data values versus the number of occurrences for each value. The plot will give a general
indication of the shape of the distribution. It is a picture of a number of observations. The
more data values that are plotted, the more informative it will be. As more observations
are plotted, the histogram will approach the distribution of the population from which the
data were obtained.
QReview 52
Histogram
0
1
2
3
4
Number of occurrences
Data Values (x)
4.2 Frequency Distributions
A frequency distribution is a model that indicates how the entire population is distributed
based on sample data. Since the entire population is rarely considered, sample data and
frequency distributions are used to estimate the shape of the actual distribution. This
estimate allows inferences to be made about the population from which the sample data
were obtained. It is a representation of how data points are distributed. It shows whether
the data are located in a central location, scattered randomly or located uniformly over
the whole range. The graph of the frequency distribution will display the general variability
and the symmetry of the data. The frequency distribution may be represented in the form
of an equation and as a graph.
Data Value (xi)
Frequency of Occurrence
When using a frequency distribution, the interest is rarely in the particular set of data
being investigated. In virtually all cases, the data are samples from a larger set or
population. The population may be a specified number of items already produced or an
infinite set of items that are continually made by some process. Sometimes, it is
wrongfully assumed that data follow the pattern of a known distribution such as the
normal. The data should be tested to determine if this is true. Goodness of Fit tests are
used to compare sample data with known distributions. This topic will be covered in a
subsequent chapter. The inferences made from a frequency distribution apply to the
entire population.
Chapter 4 Statistics 53
Quality engineers and statisticians deal with distributions formed from individual
measurements as well as distributions formed by sets of averages. Control charts,
which are covered in a subsequent chapter, are applications of a distribution of averages.
If the data are taken from the same population, there is a relationship between the
distribution of individual measurements and the distribution of averages. The means will
be equal ( x x ). If the standard deviation for individual measurements is s, then the
standard error for the distribution of averages is s n . If a sample of 100 parts is divided
into 20 subsets of 5 parts each, then n is 100 when calculating the variance and standard
deviation of individual measurements and n is 5 when calculating the standard error
using s n .
Distribution of averages Distribution of
individual measurements (x)
(x)
Comparison of x and x distributions
Some distributions have more than one point of concentration and are called multimodal.
When multimodal distributions occur, it is likely that portions of the output were produced
under different conditions. A distribution with a single point of concentration is called
unimodal.
A distribution is symmetrical if the mean, median and mode are at the same location.
The symmetry of variation is indicated by skewness. If a distribution is asymmetrical it is
considered to be skewed. The tail of a distribution indicates the type of skewness. If the
tail goes to the right, the distribution is skewed to the right and is positively skewed. If the
tail goes to the left, the distribution is skewed to the left and is negatively skewed. A
symmetrical distribution has no skewness.
Kurtosis is defined as the state or quality of flatness or peakedness of a distribution. If a
distribution has a relatively high concentration of data in the middle and out on the tails,
but little in between, it has large kurtosis. If it is relatively flat in the middle and has thin
tails, it has little kurtosis.
If the frequencies of occurrence of a frequency distribution are cumulated from the lower
end to the higher end of a scale, a cumulative frequency distribution is formed.
QReview 54
5.0 SHAPES OF DISTRIBUTIONS
Unimodal Bimodal
Small Variability Large Variability
Positively Skewed Negatively Skewed
Symmetrical and possibly Normal
Large Kurtosis Little Kurtosis
Chapter 4 Statistics 55
6.0 THE NORMAL CURVE
The normal curve is one of the most frequently occurring distributions in statistics. The
pattern that most distributions form tend to approach the normal curve. It is sometimes
referred to as the Gaussian curve named after Karl Friedrich Gauss (1777-1855) a German
mathematician and astronomer. The normal curve is symmetrical about the average, but not
all symmetrical curves are normal. For a distribution or curve to be normal, a certain
proportion of the entire area must occur between specific values of the standard deviation.
There are two ways that the normal curve may be represented: The actual normal curve and
the standard normal curve.
6.1 Actual Normal
The curve represents the distribution of actual data. The actual data points (xi) are
represented on the abscissa (x-scale) and the number of occurrences are indicated on
the ordinate (y-scale).
6.2 Standard Normal
The sample average and standard deviation are transformed to standard values with a
mean of zero and a standard deviation of one. The area under the curve represents the
probability of being between various values of the standard deviation. By transforming the
actual measurements to standard values, one table is used for all measurement scales.
A Standard Normal Curve table is included in appendix A and various iterations of the
table can be found in most probability and statistics textbooks.
The abscissa on the actual normal curve is denoted by x and the abscissa on the
standard normal curve is denoted by Z.
The relationship between x and Z:
Z
x x
s
i
( )
This is known as the transformation formula. It transforms the x value to its
corresponding Z value. A distribution of averages may also be represented with the
normal curve. The abscissa on the actual normal curve for a distribution of averages is
denoted by x . The center is denoted by x , the average of averages.
The relationship between x and Z:
Z
x x
s
n
i
( )
The statistic s
n
is the standard error or the standard deviation for a set of averages.
The statistic x is an estimate of the parameter , the population average.
The standard normal curve areas are used to make certain forecasts and predictions
about the population from which the data were taken. The standard normal curve areas
are probability numbers. The area indicates the probability of being between two values
QReview 56
on the Z scale.
Chapter 4 Statistics 57
6.3 Areas Under the Standard Normal Curve
34.1% 34.1%
2.1% 13.6% 13.6% 2.1%
-3 -2 -1 0 +1 +2 +3
68.3%
95.5%
99.73%

Example 2
The following data represent ten measurements (timing in seconds) from an electronic
device. This is a sample taken from a production run.
10, 11, 11, 12, 12, 12, 12, 13, 13, 14
A histogram is drawn to get a general idea of the shape of the distribution.
0
1
2
3
4
10 11 12 13 14
Measurement
The mean and standard deviation are calculated: x
x
n
i ( ) 120
10
12
Number of Occurrences
QReview 58
The standard deviation from the unbiased estimator of the variance using the working
formula: (Using the calculator is much easier.)
s
x
n
x
n
n
i 

è

÷


è

÷

è

÷

è

÷

è

÷
( )
. . .
2
2
1
1452
10
144
10
9
12
10
9
1333 115
The normal curve areas are used to make predictions about the process.
8.55 x 9.7 10.85 12.0 13.15 14.3 15.45
To use the standard normal tables the x values must be converted to their equivalent Z
values.
Using Z
x x
s
i
( )
, the x value 10.85 converts to Z = -1.0, the x value 12 converts to
Z = 0, the x value 13.15 converts to Z = +1.0, the x value 14.3 converts to the
Z = +2.0, etc.
-3.0 -2.0 -1.0 0 +1.0 +2.0 +3.0 Z
Area from - to + = 1.0
Area from - to 0 = .5
Area from 0 to + = .5
Chapter 4 Statistics 59
Example 3
Use the standard normal curve table to find the area between Z = +1.0 and Z = +2.0.
Area from 0 to +2.0 = .4772
Area from 0 to +1.0 = .3413
Area between +1.0 and +2.0 = .4772 - .3413 = .1359
Example 4
For x = 12.0 and s = 1.15, find the probability that a measurement will be greater than
12.0. This is written as P(x > 12). P(x > 12) = .50 which is the same as the probability
that Z > 0 since the mean value on the x scale corresponds to 0 on the Z scale.
Example 5
What is the probability that a part will have a measurement greater than 13.5?
The first step is to draw a diagram indicating the area that represents the probability of a
measurement greater than 13.5. This is a very important step because the areas under
the normal curve are difficult to visualize and a diagram makes it easy.
The next step is to convert the x value into a Z value. Z
x x
s
i




( ) ( . . )
.
.
13 5 120
115
130
This is the area from Z = 0 to Z = +1.30, therefore P(x > 13.5) = P(Z > + 1.30) =
QReview 60
(.5000 - .4032) = .0968.
Example 6
What percentage of the population will have measurements between 9.0 and 10.0?
Z1 = (9.0 - 12.0)/1.15 = -3.0/1.15 = -2.61
Z2 = (10.0 - 12.0)/1.15 = -2.0/1.15 = -1.74
The standard normal curve table gives the following results:
Area from Z1 to 0 = area from 9.0 to 12.0 = .4955
Area from Z2 to 0 = area from 10.0 to 12.0 = .4591
Area from Z1 to Z2 = area from 9.0 to 10.0 = .4955 - .4591 = .0364
Therefore, 3.64% of the population will have measurements between 9.0 and 10.0.
7.0 DISCRETE DISTRIBUTIONS
There are many applications where the areas under the normal curve are used to
approximate probabilities associated with discrete distributions. The mean and standard
deviation are calculated using the formulas shown below. The procedures are the same as
previously described for continuous distributions.
7.1 Hypergeometric Distribution
Mean and standard deviation for the hypergeometric distribution:
In terms of np: 


np npq
N n
N
,
( )
( 1)
In terms of p: 


p
pq
n
N n
N
,
( )
( 1)
Chapter 4 Statistics 61
The parameter p is the fraction defective and q = (1 - p) represents the fraction of good
parts. To use the hypergeometric distribution formula the actual number of defective and
goods parts in the lot must be known, not just the fraction defective.
7.2 Binomial Distribution
Mean and standard deviation for the binomial distribution:
In terms of np: np, npq
In terms of p: p 
pq
n
,
The parameter p is the fraction defective and q = (1 - p) represents the fraction of good
parts. The parameter p is also defined as the probability of a single success and must
always be a value between zero and one.
7.3 Poisson Distribution
Mean and standard deviation for the Poisson distribution:
In terms of np: np, np
In terms of p: p 
p
n
,
The parameter p is either defects per unit or fraction defective. If p represents a fraction
defective, it must be a value between zero and one. If p represents defects per unit, it is a
value between zero and infinity. In terms of np, the mean is equal to the variance for the
Poisson distribution.
8.0 TOLERANCES
Tolerances are usually specified in design drawings for interacting dimensions that mate or
merge with other dimensions to obtain a final result.
A simple assembly is shown below:
A B C
2.0 ±0.001 4.0 ±0.003
0.3 ±0.0004
Assembly Length
QReview 62
8.1 Conventional Method of Computing Tolerances
Adding each individual tolerance in an assembly to form a final result is called the
conventional method of computing tolerances.
Nominal value = nominal valueA + nominal valueB + nominal valueC
Nominal value of the example assembly = 2.0 + 0.3 + 4.0 = 6.3
Addition of individual tolerances = TA + TB + TC
Tolerance of the example assembly = 0.001 + 0.0004 + 0.003 = 0.0044
The final value for the example assembly is 6.3 ±0.0044.
Although this method is mathematically correct, the resulting tolerance may in some
cases be quite large. Most mathematicians, statisticians, design engineers and quality
engineers reject this method in favor of the statistical method shown below.
8.2 Statistical Method of Computing Tolerances
The nominal or center value is computed by adding the individual nominal values.
This is the same computation for both the conventional and statistical methods.
Nominal value = nominal valueA + nominal valueB + nominal valueC
Nominal value of the example assembly = 2.0 + 0.3 + 4.0 = 6.3
Statistical method for computing the tolerance = T T T A B C
2 2 2 
Tolerance of the example assembly = (0.001)2 (0.0004)2 (0.003)2 = 0.003187
The final value is 6.3 ±0.003187. Most of the assemblies will fall within this range.
9.0 DETERMINATION OF SAMPLE SIZE
9.1 Sample Size Determination for Variables Data
n
Zs
E

è

÷
2
Z is the Z value corresponding to the level of confidence from the standard normal curve
table. The symbol s is the standard deviation and E is the error factor. On the normal
curve, E is the distance from the center () to Z standard errors.
E Z
s
n
Z
s
n


è

÷

If the standard deviation is unknown, take thirty parts and calculate it using the standard
deviation formula. Use this estimate for s in the above formula, and then recalculate s
from the new sample size.
Chapter 4 Statistics 63
Example 7
What sample size is required so that there is a 90% chance that the sample mean will be
within ±0.2 inch of the true mean? The standard deviation is two.
From the standard normal curve table, Z is ±1.645 for a 90% confidence level.
(E = ±0.2)
n
Zs
E

è

÷

è

÷

2 2 1645 2
0 2
271
( . )( )
.
9.2 Sample Size Determination for Discrete Data - Binomial
n pq
Z
E

è

÷
2
The formula requires a value of p. When p is unknown, the worst case of p = .5 is used.
This gives the largest value of pq (pq = .5 x .5 = .25).
Example 8
In conducting a public opinion poll, what sample size is required so that the poll takers
are 95% confident that the poll is accurate to the nearest one percent?
From the standard normal curve table, Z is ±1.96 for a 95% confidence level.
(E = ±0.01)
p 
è

÷
(. )(. ) 
.
.
5 5
196
01
9604
2
9.3 Sample Size Determination for Discrete Data - Poisson
n p
Z
E

è

÷
2
When used in the above formula, p represents defects per unit. If p is in terms of
defective units, use the sample size formula for the binomial.
Example 9
In checking a characteristic on an assembly, what sample size is required so that there
is a 99% confidence level that the average defects per unit recorded from the sample is
within ±0.1 of the true defects per unit in the population? Data from a random sample of
one hundred parts yielded 0.5 defects per unit.
From the standard normal curve table, Z is ±2.575 for a 99% confidence level.
(E = ±0.1)
n 
è

÷
. 
.
.
5
2 575
1
332
2
QReview 64
10.0 PROCESS CAPABILITY ANALYSIS
The term process capability refers to the normal behavior of product characteristic
measurements when the process is in statistical control. It is the measured range of inherent
variation of product characteristics turned out by the process. Process capability may be
expressed by variables or attributes data. Process capability may also be defined as the
range of values where 99.73% of the data values will fall. If a product characteristic yields an
x of 2.1" and an s of .01", the process capability is the range 2.07" to 2.13". A process
capability study is a scientific procedure for determining the capability of a process to obtain
the desired results.
The standard deviation calculated from the sample data (s) is used as an estimate of the
population standard deviation ().
(n 1)
(x x)
ProcessCapability 6 Sigma 6 , where s
2
i



10.1 Process Capability Index = Cp
This is the ratio of the specification spread to the measured process variability or sample
distribution (6). The sample distribution is an estimate of the population distribution
because s2 is the unbiased estimator of . The Cp does not indicate the location of the
sample distribution relative to the specification. It is a comparison of the sample distribution
width to the specification width. If the Cp is exactly 1.0, the 6spread is the same width as
the distance between the specification limits. A Cp of 2.0 means that the 6spread is half of
the specification range. A process with a Cp of 1 or greater may be within or totally outside
of the specification limits. A Cp of less than 1 means that the sample distribution is wider
than the specification range. (USL = upper specification limit and
LSL = lower specification limit).
C
USL LSL
p 
6
Chapter 4 Statistics 65
10.2 Process Performance Index = Cpk
This index reflects the location of the sample distribution in relation to the specification
midpoint. The maximum value of Cpk is equal to Cp and occurs when the sample
distribution is centered on the specification midpoint or target. If the Cpk is 1.0 or less,
there is no room for the process average to vary from the nominal dimension of the
engineering specifications. A Cpk that is greater than one indicates that the 6spread is
inside of the specification limits. A Cpk that is less than one indicates that some part of the
distribution is outside of the specification limits. When the process average is located at
one of the specification limits, Cpk is zero and 50% of the measurements will be outside
of the limits. If the process average is outside of the specification limits, Cpk is a negative
value. A Cpk of 1.3 to 2.0 is a respectable process performance index. To compute the
Cpk, enter x , LSL, USL and s into the formulas below. The lesser of the two values is the
Cpk.
Cpk = ú
ù
ê
é
3s
USL x
or
3s
x LSL
minimum
Cpk = 1.0
Cp = 1.0
Cpk = .67
Cp = 1.0
Cpk = 2.0
Cp = 2.0
Cpk = 1.33
Cp = 2.0
Cpk = 0
Cp = 2.0
USL
LSL
Nominal =
Example 10
The specifications for a certain product characteristic are .005" ±.0002". The control
chart data (n = 5) indicate an x of .0051" and an average range of .0001. Calculate the
Cp and Cpk for this characteristic. Is the process capability acceptable? What is the
percent defective?
s
R
d

2
0001
2 33
0000429
.
.
.
C
USL LSL
p 
6
0052 0048
6 0000429
0004
0002574
155

. .
(. )
.
.
.
2.33
.0001287
.0003
3 (.0000429)
.0051 .0048
3
x LCL
C (1) pk 





QReview 66
.77
.0001287
.0001
3 (.0000429)
.0052 .0051
3
USL x
C (2) pk 





Cpk (2) is less than Cpk (1), therefore Cpk = Cpk (2) = .77
Since the Cpk is less than one, a portion of the sample distribution will be outside of the
specification limits. As shown below, the process will yield approximately one percent
defective parts. One percent of the parts will be above the upper specification limit. This
may or may not be an acceptable process capability. If the parts are expensive, the
process capability may be unacceptable because of the high dollar value of one percent
of the parts. If the parts are relatively cheap, the process capability may be acceptable.
.0099 or .99% or 1%
of the parts will be
defective
Z
x x
s
i 
. .
.
.
0052 0051
0000429
233
USL x
.0052
LSL
.0048
x
.0051
-6.99 0 +2.33 Z
11.0 PARETO ANALYSIS
Vilfredo Pareto (1848 - 1923) was an Italian economist and sociologist whose theories
influenced the development of Italian fascism. He was initially credited with the theory of
maldistribution of wealth. This theory simply states that in any country a small percentage of the
people own a large percentage of the money. The theory may really belong to M. O. Lorenz
rather than Pareto. Since J. M. Juran identified the maldistribution of wealth and its similarities to
defects in a manufacturing environment as the Pareto Principle in the first edition of his Quality
Control Handbook, the term Pareto Principle been used.
As in the maldistribution of wealth, it is also a fact that quality losses are maldistributed. A small
percentage of the quality characteristics will account for a high percentage of the quality losses.
The Pareto Principle is a simple yet powerful concept that provides a tool (Pareto diagram) for
the analysis of data as well as information for action. Like all statistical tools, it does not provide
the action itself.
A Pareto diagram indicates which problems should be worked on first in eliminating defects and
improving the operation. The Pareto diagram is a way of portraying those problems that have
Chapter 4 Statistics 67
the greatest impact on the process or product, and once solved will yield the greatest return. A
Pareto diagram is simply a bar chart arranged in order of importance.
Example 11
Defects recorded from a circuit board manufacturing operation
0
2
4
6
8
10
Number of defects
0
10
20
30
40
50
60
70
80
90
100
Cumulative % of defects
Insecure
Solder
Connections
Defective
Resistors
Defective
Capacitors
Defective
ICs
Misaligned
Components
Open
path
From this analysis, the first problem that may be pursued is the problem of insecure solder
connections. This may not be obvious unless the frequencies of the various defects are
plotted in some way. In most cases it is easier to see which defects are most important with
a bar graph than by using a table of numbers. The diagram has two distinct parts: the “vital
few” and the “trivial many.” Of course in an actual analysis a great many more defect types
could occur.
Example 12 Simple analysis of defects
Defect Code Number of
Occurrences
Percent of Total
A 34 47.2
B 27 37.5
C 7 9.7
D 2 2.8
E 2 2.8
72 100.0
QReview 68
A B C D E
Defect Type
0
5
10
15
20
25
30
35
40
Number of defects
0
10
20
30
40
50
60
70
80
90
100
Cumulative % defects
Defect A has the highest number of occurrences, but it may not have the greatest impact on
the total operations. The key is to consider costs when making a Pareto analysis. Costs
should always be taken into consideration. A separate study may have to be conducted to
determine the costs of various defects.
Example 13 Pareto analysis considering costs
Defect Code Number of
Occurrences
Repair
Costs*
Other
Costs*
Total
Costs
Percent of
Total Costs
A 34 $1.00 $1.50 $85.00 24.5
B 27 $1.25 $1.60 $76.95 22.2
C 7 $12.75 $8.50 $148.75 42.9
D 2 $10.00 $2.00 $24.00 6.9
E 2 $3.25 $2.75 $12.00 3.5
$346.7 100.0
*Incurred costs for each defect occurrence
C A B D E
Defect Type
0
20
40
60
80
100
120
140
160
Cost
0
10
20
30
40
50
60
70
80
90
100
Cumulative Cost
From this diagram, it is evident that the root cause of defect C should be investigated first.
Chapter 4 Statistics 69
The elimination of this defect would reduce costs by 42.9%.
Pareto diagrams may be used to first identify major problems and then to display the impact of
the improvement activity. The order of the bars will change if significant improvements to the
process are made. The Pareto analysis itself will not actually solve the problem in question. A
plan of attack must be devised after the problem is identified. The objective is to eliminate the
root cause of the problem. Pareto charts and Pareto analyses are techniques to display data in
a form that aids in the identification of the vital “few” and the “trivial many.”
When used alone, the Pareto analysis and associated diagram have several limitations. They
should be used with good judgment and with knowledge of the process. If the samples are
small, the diagram may not show much difference between the various classes of defects. It
does not show variation over time for occurrences of a particular defect. A defect that occurred
several times last month may not occur this month although no corrective action was taken.
The Pareto diagram does not provide the trend of individual defects over time. In some rare
cases, the diagram may show a new defect in the number one position each week although no
corrective action was taken on the last number one defect. This is where knowledge of the
process is important.
One way to make Pareto diagrams more effective is to use them together with trend charts for
each specific defect class. The combination of Pareto diagrams and trend charts have many
benefits. A particular defect class could be considered a significant problem if the Pareto
diagram were used alone. A trend chart, however, may show that the high rate of occurrence of
a particular defect last month was a one-time event. Trend charts show the effect of corrective
actions.
Combining Pareto diagrams and trend charts provides a powerful analysis tool. More
information is available than if they are used separately. This combination allows for the
identification of critical problems and provides a method for determining the effectiveness of
corrective actions.