Basic Statistical Methods
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BASIC STATISTICAL METHODS
1
THE STATISTICAL TOOL KIT
the collection, analysis, and interpretation of data or, more broadly, as “the scie
Sec 44 BASIC STATISTICAL METHODS
2
SOURCES AND SUMMARIZATION OF DATA
tion. Investigators using historical data are like blind people probing an elephan
tion, analysis of the data to draw statistical conclusions, and making the transit
tement that can be evaluated by statistical methods.
ive than attributes data (go or no-go data), but the information is much more usef d the hazards of historical data sets. ic consequences of a wrong decision.
Sec 44 BASIC STATISTICAL METHODS
3
SOURCES AND SUMMARIZATION OF DATA
a.
f a parameter, define the precision needed for the estimate. large enough to influence the sample size or the method of data analysis; laborat te the required sample size. dering the desired precision of the result, statistical risk, variability of the d he order of measurements when time is a key parameter. g data in groups defined so as to reflect the different conditions that are to be ny assumptions required. grams that will be needed.
Sec 44 BASIC STATISTICAL METHODS
4
SOURCES AND SUMMARIZATION OF DATA
a.
selected in a random manner. present at the time of each observation. he process shows sufficient stability to make predictions valid for the future.
ed for determining the sample size and for analyzing the data. Take corrective st the original problem. re needed. key sample estimates and other factors in the analysis and noting the effect on f
Sec 44 BASIC STATISTICAL METHODS
5
SOURCES AND SUMMARIZATION OF DATA
Data.
sis to determine if the original technical problem has been evaluated or if it ha
e summary. form by emphasizing results in terms of the original problem rather than the stat where appropriate. Use simple statistical methods in the body of the report, and p ific problem apply to other problems or if the data and calculations could be a u
Sec 44 BASIC STATISTICAL METHODS
6
SOURCES AND SUMMARIZATION OF DATA
ed during the production process, for example. If a satisfactory process goes out
analysis is an indication of the most important variables to include in the desig
Sec 44 BASIC STATISTICAL METHODS
7
SOURCES AND SUMMARIZATION OF DATA
bles that could conceivably have an effect on the output (xk +1 ,…, xm)
reason (such as equipment malfunction), and where the basic model does not change
performing this examination are called data screening methods. Among the most pow
Sec 44 BASIC STATISTICAL METHODS
8
SOURCES AND SUMMARIZATION OF DATA
smaller L is, the more good observations one will wrongly detect as potential ou
of the data points) should be subjected to examination to see if there are proble
Sec 44 BASIC STATISTICAL METHODS
9
SOURCES AND SUMMARIZATION OF DATA
often, two or more methods will be used to attain the clarity of description that
ence of the various values. The following are the steps taken to construct a freq
Sec 44 BASIC STATISTICAL METHODS
10
SOURCES AND SUMMARIZATION OF DATA
without rhyme or reason to it. These characteristics of the data may relate to a
concepts in all statistical analysis.
Sec 44 BASIC STATISTICAL METHODS
11
SOURCES AND SUMMARIZATION OF DATA
ion.
e range is based on only two values, it is most useful when the number of observa
Sec 44 BASIC STATISTICAL METHODS
12
SOURCES AND SUMMARIZATION OF DATA
nts. Construction of computer programs to perform the calculations is a task that
Sec 44 BASIC STATISTICAL METHODS
13
PROBABILITY MODELS FOR EXPERIMENTS
pulation is a large source of measurements from which the sample is taken. (Note t
pulation. Figures on slides (16-18) summarizes some distributions.
Sec 44 BASIC STATISTICAL METHODS
14
PROBABILITY MODELS FOR EXPERIMENTS
tinuous probability distribution. Experience has shown that most continuous chara
ete probability distribution. The common discrete distributions are the Poisson, b
Sec 44 BASIC STATISTICAL METHODS
15
PROBABILITY MODELS FOR EXPERIMENTS
Sec 44 BASIC STATISTICAL METHODS
16
PROBABILITY MODELS FOR EXPERIMENTS
Sec 44 BASIC STATISTICAL METHODS
17
PROBABILITY MODELS FOR EXPERIMENTS
Sec 44 BASIC STATISTICAL METHODS
18
PROBABILITY MODELS FOR EXPERIMENTS
le outcomes of the experiment of interest to us, that set is called the sample space o
or simplicity let us denote these outcomes, respectively, by e1, e2, e3, e4, e5, e6, e7, e
or example, the probability of HHH in our experiment of tossing three coins is usually
large number of experiments, we also must have probabilities that sum to 1 when all ou
Sec 44 BASIC STATISTICAL METHODS
19
PROBABILITY MODELS FOR EXPERIMENTS
) occurs in our example of tossing three coins. The frequency with which we find “
Sec 44 BASIC STATISTICAL METHODS
20
PROBABILITY MODELS FOR EXPERIMENTS
0.0 (impossibility of occurrence), and the most intuitive definition of probability
Sec 44 BASIC STATISTICAL METHODS
21
PROBABILITY MODELS FOR EXPERIMENTS
has occurred, then on those trials of the experiment where A2 has occurred, how of P(A1|A2)
Sec 44 BASIC STATISTICAL METHODS
22
DISCRETE PROBABILITY DISTRIBUTIONS
ure or success or 0, 1, 2, 3,…as a number of occurrences of some event of interest
set of values x1,…, xn. In this case the probability of xi is 1/n. Since the probabi
the same number of times. (This makes values equally likely to occur in the sample
through 1499. Then the chance that an item selected at random from the lot will h
Sec 44 BASIC STATISTICAL METHODS
23
DISCRETE PROBABILITY DISTRIBUTIONS
en the probability of r occurrences in n trials is
robability p of occurrence of an event of interest (commonly termed a success), th
Sec 44 BASIC STATISTICAL METHODS
24
DISCRETE PROBABILITY DISTRIBUTIONS
e probability of exactly r occurrences in n trials from a lot of N items having d
rrence of the event of interest changes from trial to trial because of depletion
Sec 44 BASIC STATISTICAL METHODS
25
DISCRETE PROBABILITY DISTRIBUTIONS
10 times the sample size, and the probability of occurrence p on each trial is le
me, or in space, or in location, for example) with a probability of occurrence roug
hits will follow a Poisson distribution, and the number of shells fired may be se
Sec 44 BASIC STATISTICAL METHODS
26
DISCRETE PROBABILITY DISTRIBUTIONS
ributions in situations where the sample size is not set in advance but rather is
of occurrence of an event is constant from trial to trial and we make trials unt
Sec 44 BASIC STATISTICAL METHODS
27
DISCRETE PROBABILITY DISTRIBUTIONS
is, where the outcome of interest relates to one variable’s value (such as the numb here are any number of categories into which the items may be classified.
ectively p1,…, pk (with p1+ … pk = 1 so that one of them must occur), then the pr
Sec 44 BASIC STATISTICAL METHODS
28
DISCRETE PROBABILITY DISTRIBUTIONS
bility. In either case, a test of the model selected is desirable to check its val
practical situation. For example, if one draws 50 items at random from a large lo
cell totals with those predicted by the model using the chi square test discussed
n be fitted to the data using the relative frequencies observed in the past.
Sec 44 BASIC STATISTICAL METHODS
29
ONTINUOUS PROBABILITY DISTRIBUTIONS
all values greater than zero for the failure time of a motor that is run continuo
are proportional to its length, then the uniform distribution is appropriate. The
value between a and b. For example, if a valve on a water line is spun at random
omputer simulation models and are of great importance in simulation studies of qu
Sec 44 BASIC STATISTICAL METHODS
30
ONTINUOUS PROBABILITY DISTRIBUTIONS
ent below it. In an exponential population, 36.8 percent are above the mean and 63.
Sec 44 BASIC STATISTICAL METHODS
31
ONTINUOUS PROBABILITY DISTRIBUTIONS
gives a good discussion of this and other assumptions in reliability calculation
Sec 44 BASIC STATISTICAL METHODS
32
ONTINUOUS PROBABILITY DISTRIBUTIONS
ely approximates the normal distribution. In practice, βvaries from about 1/3 to
Sec 44 BASIC STATISTICAL METHODS
33
ONTINUOUS PROBABILITY DISTRIBUTIONS
istribution . Many engineering characteristics can be approximated by the
18,
π =3.141, μ=population average,
norma
σ=population standard deviation.
Sec 44 BASIC STATISTICAL METHODS
34
ONTINUOUS PROBABILITY DISTRIBUTIONS
s 1 standard deviation of the population, 95.46 percent of the population will fal
om a normally distributed population plots approximately as a straight line on no
Sec 44 BASIC STATISTICAL METHODS
35
ONTINUOUS PROBABILITY DISTRIBUTIONS
re Z has a normal distribution, Y is said to have a lognormal distribution (since
mixture distribution if Y results from source i a percentage 100pi of the time (i
Sec 44 BASIC STATISTICAL METHODS
36
ONTINUOUS PROBABILITY DISTRIBUTIONS
ne component (such as lifetime). If there are additional components of interest (s
ot fit.
ributions). In either case, a test of the model selected is desirable to check its
in the past, always been adequately fitted by a Weibull model (though with parame
Sec 44 BASIC STATISTICAL METHODS
37
ONTINUOUS PROBABILITY DISTRIBUTIONS
which a plot on probability paper follows a straight line. These convenient method
rejected (e.g., because of a poor probability paper fit), an alternative is to fit
Sec 44 BASIC STATISTICAL METHODS
38
STATISTICAL ESTIMATION
ngle number (a point estimate) or a pair of numbers (an interval estimate); there a
Sec 44 BASIC STATISTICAL METHODS
39
STATISTICAL ESTIMATION
one can be 95 percent sure that at least 99 percent of the population will be in
mple, if we observe that 15 of 100 items chosen at random from a very large lot ar
Sec 44 BASIC STATISTICAL METHODS
40
STATISTICAL ESTIMATION
lected randomly and if the sample size is less than 10 percent of the population
y value. The variability is known as σ = 10.0.
Sec 44 BASIC STATISTICAL METHODS
41
STATISTICAL ESTIMATION
ind when we take a future item from the population. For example, in the example of
Sec 44 BASIC STATISTICAL METHODS
42
STATISTICAL ESTIMATION
etermination of engineering tolerance limits (which specify the allowable limits
dimensions because they determine the overall assembly length. The conventional m Nominal value of the result = nominal valueA + nominal valueB + nominal valueC
Sec 44 BASIC STATISTICAL METHODS
43
STATISTICAL ESTIMATION
Tolerance T of the result = TA + TB + TC Nominal value of assembly length = 1.000 + 0.500 + 2.000 = 3.500 Tolerance of assembly length = 0.0010 + 0.0005 + 0.0020 = ±0.0035
up the assembly. If the component tolerances are met, then all assemblies will me
Sec 44 BASIC STATISTICAL METHODS
44
STATISTICAL ESTIMATION
f defectives p in a lot may be a random variable about which we can fit a distrib
ere a random variable with that distribution. This is called the personal probabil
Sec 44 BASIC STATISTICAL METHODS
45
1
THE STATISTICAL TOOL KIT
the collection, analysis, and interpretation of data or, more broadly, as “the scie
Sec 44 BASIC STATISTICAL METHODS
2
SOURCES AND SUMMARIZATION OF DATA
tion. Investigators using historical data are like blind people probing an elephan
tion, analysis of the data to draw statistical conclusions, and making the transit
tement that can be evaluated by statistical methods.
ive than attributes data (go or no-go data), but the information is much more usef d the hazards of historical data sets. ic consequences of a wrong decision.
Sec 44 BASIC STATISTICAL METHODS
3
SOURCES AND SUMMARIZATION OF DATA
a.
f a parameter, define the precision needed for the estimate. large enough to influence the sample size or the method of data analysis; laborat te the required sample size. dering the desired precision of the result, statistical risk, variability of the d he order of measurements when time is a key parameter. g data in groups defined so as to reflect the different conditions that are to be ny assumptions required. grams that will be needed.
Sec 44 BASIC STATISTICAL METHODS
4
SOURCES AND SUMMARIZATION OF DATA
a.
selected in a random manner. present at the time of each observation. he process shows sufficient stability to make predictions valid for the future.
ed for determining the sample size and for analyzing the data. Take corrective st the original problem. re needed. key sample estimates and other factors in the analysis and noting the effect on f
Sec 44 BASIC STATISTICAL METHODS
5
SOURCES AND SUMMARIZATION OF DATA
Data.
sis to determine if the original technical problem has been evaluated or if it ha
e summary. form by emphasizing results in terms of the original problem rather than the stat where appropriate. Use simple statistical methods in the body of the report, and p ific problem apply to other problems or if the data and calculations could be a u
Sec 44 BASIC STATISTICAL METHODS
6
SOURCES AND SUMMARIZATION OF DATA
ed during the production process, for example. If a satisfactory process goes out
analysis is an indication of the most important variables to include in the desig
Sec 44 BASIC STATISTICAL METHODS
7
SOURCES AND SUMMARIZATION OF DATA
bles that could conceivably have an effect on the output (xk +1 ,…, xm)
reason (such as equipment malfunction), and where the basic model does not change
performing this examination are called data screening methods. Among the most pow
Sec 44 BASIC STATISTICAL METHODS
8
SOURCES AND SUMMARIZATION OF DATA
smaller L is, the more good observations one will wrongly detect as potential ou
of the data points) should be subjected to examination to see if there are proble
Sec 44 BASIC STATISTICAL METHODS
9
SOURCES AND SUMMARIZATION OF DATA
often, two or more methods will be used to attain the clarity of description that
ence of the various values. The following are the steps taken to construct a freq
Sec 44 BASIC STATISTICAL METHODS
10
SOURCES AND SUMMARIZATION OF DATA
without rhyme or reason to it. These characteristics of the data may relate to a
concepts in all statistical analysis.
Sec 44 BASIC STATISTICAL METHODS
11
SOURCES AND SUMMARIZATION OF DATA
ion.
e range is based on only two values, it is most useful when the number of observa
Sec 44 BASIC STATISTICAL METHODS
12
SOURCES AND SUMMARIZATION OF DATA
nts. Construction of computer programs to perform the calculations is a task that
Sec 44 BASIC STATISTICAL METHODS
13
PROBABILITY MODELS FOR EXPERIMENTS
pulation is a large source of measurements from which the sample is taken. (Note t
pulation. Figures on slides (16-18) summarizes some distributions.
Sec 44 BASIC STATISTICAL METHODS
14
PROBABILITY MODELS FOR EXPERIMENTS
tinuous probability distribution. Experience has shown that most continuous chara
ete probability distribution. The common discrete distributions are the Poisson, b
Sec 44 BASIC STATISTICAL METHODS
15
PROBABILITY MODELS FOR EXPERIMENTS
Sec 44 BASIC STATISTICAL METHODS
16
PROBABILITY MODELS FOR EXPERIMENTS
Sec 44 BASIC STATISTICAL METHODS
17
PROBABILITY MODELS FOR EXPERIMENTS
Sec 44 BASIC STATISTICAL METHODS
18
PROBABILITY MODELS FOR EXPERIMENTS
le outcomes of the experiment of interest to us, that set is called the sample space o
or simplicity let us denote these outcomes, respectively, by e1, e2, e3, e4, e5, e6, e7, e
or example, the probability of HHH in our experiment of tossing three coins is usually
large number of experiments, we also must have probabilities that sum to 1 when all ou
Sec 44 BASIC STATISTICAL METHODS
19
PROBABILITY MODELS FOR EXPERIMENTS
) occurs in our example of tossing three coins. The frequency with which we find “
Sec 44 BASIC STATISTICAL METHODS
20
PROBABILITY MODELS FOR EXPERIMENTS
0.0 (impossibility of occurrence), and the most intuitive definition of probability
Sec 44 BASIC STATISTICAL METHODS
21
PROBABILITY MODELS FOR EXPERIMENTS
has occurred, then on those trials of the experiment where A2 has occurred, how of P(A1|A2)
Sec 44 BASIC STATISTICAL METHODS
22
DISCRETE PROBABILITY DISTRIBUTIONS
ure or success or 0, 1, 2, 3,…as a number of occurrences of some event of interest
set of values x1,…, xn. In this case the probability of xi is 1/n. Since the probabi
the same number of times. (This makes values equally likely to occur in the sample
through 1499. Then the chance that an item selected at random from the lot will h
Sec 44 BASIC STATISTICAL METHODS
23
DISCRETE PROBABILITY DISTRIBUTIONS
en the probability of r occurrences in n trials is
robability p of occurrence of an event of interest (commonly termed a success), th
Sec 44 BASIC STATISTICAL METHODS
24
DISCRETE PROBABILITY DISTRIBUTIONS
e probability of exactly r occurrences in n trials from a lot of N items having d
rrence of the event of interest changes from trial to trial because of depletion
Sec 44 BASIC STATISTICAL METHODS
25
DISCRETE PROBABILITY DISTRIBUTIONS
10 times the sample size, and the probability of occurrence p on each trial is le
me, or in space, or in location, for example) with a probability of occurrence roug
hits will follow a Poisson distribution, and the number of shells fired may be se
Sec 44 BASIC STATISTICAL METHODS
26
DISCRETE PROBABILITY DISTRIBUTIONS
ributions in situations where the sample size is not set in advance but rather is
of occurrence of an event is constant from trial to trial and we make trials unt
Sec 44 BASIC STATISTICAL METHODS
27
DISCRETE PROBABILITY DISTRIBUTIONS
is, where the outcome of interest relates to one variable’s value (such as the numb here are any number of categories into which the items may be classified.
ectively p1,…, pk (with p1+ … pk = 1 so that one of them must occur), then the pr
Sec 44 BASIC STATISTICAL METHODS
28
DISCRETE PROBABILITY DISTRIBUTIONS
bility. In either case, a test of the model selected is desirable to check its val
practical situation. For example, if one draws 50 items at random from a large lo
cell totals with those predicted by the model using the chi square test discussed
n be fitted to the data using the relative frequencies observed in the past.
Sec 44 BASIC STATISTICAL METHODS
29
ONTINUOUS PROBABILITY DISTRIBUTIONS
all values greater than zero for the failure time of a motor that is run continuo
are proportional to its length, then the uniform distribution is appropriate. The
value between a and b. For example, if a valve on a water line is spun at random
omputer simulation models and are of great importance in simulation studies of qu
Sec 44 BASIC STATISTICAL METHODS
30
ONTINUOUS PROBABILITY DISTRIBUTIONS
ent below it. In an exponential population, 36.8 percent are above the mean and 63.
Sec 44 BASIC STATISTICAL METHODS
31
ONTINUOUS PROBABILITY DISTRIBUTIONS
gives a good discussion of this and other assumptions in reliability calculation
Sec 44 BASIC STATISTICAL METHODS
32
ONTINUOUS PROBABILITY DISTRIBUTIONS
ely approximates the normal distribution. In practice, βvaries from about 1/3 to
Sec 44 BASIC STATISTICAL METHODS
33
ONTINUOUS PROBABILITY DISTRIBUTIONS
istribution . Many engineering characteristics can be approximated by the
18,
π =3.141, μ=population average,
norma
σ=population standard deviation.
Sec 44 BASIC STATISTICAL METHODS
34
ONTINUOUS PROBABILITY DISTRIBUTIONS
s 1 standard deviation of the population, 95.46 percent of the population will fal
om a normally distributed population plots approximately as a straight line on no
Sec 44 BASIC STATISTICAL METHODS
35
ONTINUOUS PROBABILITY DISTRIBUTIONS
re Z has a normal distribution, Y is said to have a lognormal distribution (since
mixture distribution if Y results from source i a percentage 100pi of the time (i
Sec 44 BASIC STATISTICAL METHODS
36
ONTINUOUS PROBABILITY DISTRIBUTIONS
ne component (such as lifetime). If there are additional components of interest (s
ot fit.
ributions). In either case, a test of the model selected is desirable to check its
in the past, always been adequately fitted by a Weibull model (though with parame
Sec 44 BASIC STATISTICAL METHODS
37
ONTINUOUS PROBABILITY DISTRIBUTIONS
which a plot on probability paper follows a straight line. These convenient method
rejected (e.g., because of a poor probability paper fit), an alternative is to fit
Sec 44 BASIC STATISTICAL METHODS
38
STATISTICAL ESTIMATION
ngle number (a point estimate) or a pair of numbers (an interval estimate); there a
Sec 44 BASIC STATISTICAL METHODS
39
STATISTICAL ESTIMATION
one can be 95 percent sure that at least 99 percent of the population will be in
mple, if we observe that 15 of 100 items chosen at random from a very large lot ar
Sec 44 BASIC STATISTICAL METHODS
40
STATISTICAL ESTIMATION
lected randomly and if the sample size is less than 10 percent of the population
y value. The variability is known as σ = 10.0.
Sec 44 BASIC STATISTICAL METHODS
41
STATISTICAL ESTIMATION
ind when we take a future item from the population. For example, in the example of
Sec 44 BASIC STATISTICAL METHODS
42
STATISTICAL ESTIMATION
etermination of engineering tolerance limits (which specify the allowable limits
dimensions because they determine the overall assembly length. The conventional m Nominal value of the result = nominal valueA + nominal valueB + nominal valueC
Sec 44 BASIC STATISTICAL METHODS
43
STATISTICAL ESTIMATION
Tolerance T of the result = TA + TB + TC Nominal value of assembly length = 1.000 + 0.500 + 2.000 = 3.500 Tolerance of assembly length = 0.0010 + 0.0005 + 0.0020 = ±0.0035
up the assembly. If the component tolerances are met, then all assemblies will me
Sec 44 BASIC STATISTICAL METHODS
44
STATISTICAL ESTIMATION
f defectives p in a lot may be a random variable about which we can fit a distrib
ere a random variable with that distribution. This is called the personal probabil
Sec 44 BASIC STATISTICAL METHODS
45
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