Transcript BASIC CONCEPTS OF PROBABILITY

Certified Quality Engineer
Programme (CQE)
Module 6
Quantitative Methods Part 1
By
Associate Professor Dr Sha’ri M. Yusof
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia, Skudai, Johor
Basic Concepts Of Probability
Probability is a measure that describes
the chance that an event will occur.
Dimensionless number ranges from zero
to one - with 0 meaning an impossible
event and 1 refer to event that is
certain to occur.
Probability of 0.5 means the event is
just as likely to occur as not.
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Basic Concepts Of Statistics
 The word statistics has two generally
accepted meaning:


A collection of quantitative data
pertaining to any subject or group,
especially when the data are
systematically gathered and collated.
The science that deals with the collection,
tabulation, analysis, interpretation and
presentation of quantitative data.
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Basic Concepts Of Statistics
 The use of statistics in quality engineering
deals with the second meaning and involves





Collecting
Tabulating
Analyzing
Interpreting
Presenting data
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Collecting And Summarizing
Data
Descriptive Statistics


to describe and analyze a subject or
group
analytical techniques summarize data by
computing
a measure of central tendency
 a measure of the dispersion.

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Measure of Central Tendency
A measure of central tendency of a
distribution is a number that describes
the central position of the data or how
the data tend to build up in the center.
Three measures commonly used :
1) average
2) median
3) mode
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Average


It is the sum of the all the observations
divided by the number of observations
3 different techniques available for
calculating the average
1) ungrouped data
2) grouped data
3) weighted average
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Average
Ungrouped data.
This method is used when the data are unorganized.
The average is represented by the symbol x, which is read as
“x bar” and is given by the formula;
x =  xi / n = (x1 + x2 +….+xn)/n
where x = average
n = observed values
x1, x2,...,xn = observed value identified by
the subscripts 1,2,..n or general subscript i
 = symbol meaning “sum of ”
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Example
 A food inspector examined a random sample
of 7 cans of tuna to determine the percent of
foreign impurities. The following data were
recorded :
1.8, 2.1, 1.7, 1.6, 0.9, 2.7 and 1.8
 Compute the sample mean.
 x =  xi / n =
(1.8+2.1+1.7+1.6+0.9+2.7+1.8)/7 = 1.8%
impurities
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Exercise
 In studying the drying time of a new acrylic
paint, the data in hours, were coded by
subtracting 5.0 from the observation.
 Find the sample mean and sample standard
deviation (s) for the drying times of 10 panels
of wood using the paint if the coded
measurements are :
1.4 , 0.8, 2.4, 0.5, 1.3, 2.8, 3.6, 3.2, 2.0, 1.9
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Grouped data.
When data have been grouped into frequency
distribution, the following technique is applicable.
Formula for the average of grouped data
x = (fiXi)/n = (fiX1 + f2X2 + …+fhXh) / (f1 + f2+…+fh )
where n = sum of the frequency
fi = frequency in a cell or frequency of an observed value
xi = cell midpoint or an observed value
h = no. of cells or no. of observed values
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Example
 Frequency Distribution for Weights of 50 components
Class
Interval
Weight (g)
Class
Boundary
Class
midpoint
(xi)
No of
pieces
(fi)
7–9
6.5 –9.5
8
2
10 – 12
9.5 – 12.5
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8
13 – 15
12.5 – 15.5
14
14
16 – 18
15.5 – 18.5
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19
19- 21
18.5 – 21.5
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Totals ()
50
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fixi
fixi2
Weighted average
When a number of averages combined with different
frequencies, a weighted average can be computed
The formula for the weighted average is given by :
xw = wixi
wi
where xw = weighted average
wi = weight of the i th average
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Example – weighted average
 On a trip a family bought 21.3 litres of
gasoline at 1.21 per litre, 18.7 litres at 1.29
cents per litre, and 23.5 litres at 1.25. Find
the mean price per litre.
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Median








Median is the middle value for a set of data arranged in an
increasing or decreasing order
Case 1 - when the number of data in the series is odd – middle
value
Case 2 - when the number of data is even - median is the
average of the two middle numbers
Example (case 1) – 5 test results 82, 93, 86, 92, 79
What is the median? Arrange data. Answer = 86
Example (Case 2) – The nicotine contents for a random sample
of 6 cigarettes of a certain brand are found to be 2.3, 2.7, 2.5,
2.9, 3.1 and 1.9
If we arrange in increasing order of magnitude , we get = 1.9
2.3 2.5 2.7 2.9 3.1 , and the median is the mean of 2.5
and 2.7.
Therefore, x = (2.5+2.7)/2 = 2.6 milligrams
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Median
Grouped Data
When data grouped into frequency distribution, the median is
obtained by finding the cell that has the middle number and
then interpolate within the cell.
Formula for computing median :
Md = Lm + n/2 –cfm
I
fm
where Md = median
Lm = lower boundary of the cell with the median
n = total no. of observations
fm = frequency of median cell
cfm = cumulative frequency of all cells below Lm
I = cell interval
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Mode
 Mode of a set of numbers (data) is the value that
occurs with the highest frequency
 Possible for mode to be nonexistent in a series of
numbers or to have more than one value.
 A series of numbers is referred to as unimodal if it
has one mode, bimodal if it has two modes and
multimodal if there are more than two modes.
 Data grouped into frequency distribution, the
midpoint of the cell with the highest frequency is the
mode, since this point represents the highest point
(highest frequency) of the histogram
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Measure of Dispersion
Measures of dispersion describe how
the data are spread out from the
average or scattered on each side of
the central value.
Common measures



Range (simplest)
Standard deviation
Variance
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Range
 Range of a series of numbers is the difference
between the largest and smallest values or
observations.
R = Xh - Xl
where R = range
Xh = highest observation in a series
Xl = lowest observation in a series
 Example – The temperature for a process
recorded 40.2 , 38.7, 42.5, 39.6, 40.9. What
is the value of range?
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Standard Deviation



Standard deviation - numerical value in the units of the
observed values that measures the spreading (variation) of
the data.
Large standard deviation - greater variability of the data
than smaller standard deviation, given by the formula:
s =   (xi – x)2 / (n-1)
where s = sample standard deviation
xi = observed value ith
x = average
n = number of data (observed values)
It is reference value that measures the dispersion in the data
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Exercise
 A car manufacturer tested a random sample
of 10 steel-belted tyres of a certain brand and
recorded the following tread wear: 48000,
53000, 45000, 61000, 59000, 56000, 63000,
49000, 53000 and 54000 kilometers. Find the
standard deviation of this set of data.
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Collecting And
Summarizing Data
Consider the data below which represents the lives of 40
similar car batteries recorded to nearest tenth of a year. What
can you learn from these numbers?
2.2
4.1
3.5
4.5
3.2
3.7
3.0
2.6
3.4
1.6
3.1
3.3
3.8
3.1
4.7
3.7
2.5
4.3
3.4
3.6
2.9
3.3
3.9
3.1
3.3
3.1
3.7
4.4
3.2
4.1
1.9
3.4
4.7
3.8
3.2
2.6
3.9
3.0
4.2
3.5
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Frequency distribution
 Group large number of data into different classes
(groups) and determining the number of observations
that fall into each group
 Decide no of classes – too few lose information, too
many also no meaning
 Usually choose 5 – 20 classes
 Let us choose 7 classes – class width must be
enough to put in all the data
 Approximate width – find range divide by no of
classes = (4.7-1.6)/7 = 0.443 should have same no
of significant places as data, therefore choose the
value 0.5
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Frequency distribution
 Decide where to start bottom interval – start at 1.5
and lower boundary is 1.45. Then add width 1.45
+0.5 = 1.95 continue for the others
 Midpoint is (1.5+1.9)/2 = 1.7
 Count the no of observations and record in the table
 Total the frequency to check all data has been
counted
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Frequency distribution
Class interval
Class
boundaries
Class
midpoint
Frequency
1.5 – 1.9
1.45 – 1.95
1.7
2
2.0 – 2.4
1.95 –2.45
2.2
1
2.5 – 2.9
2.45 – 2.95
2.7
4
3.0 – 3.4
2.95 – 3.45
3.2
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3.5 – 3.9
3.45 – 3.95
3.7
10
4.0 – 4.4
3.95 – 4.45
4.2
5
4.5 – 4.9
4.45 – 4.95
4.7
3
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Graphical Representation
Frequency
Frequency Histogram
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14
12
10
8
6
4
2
0
1.45 –
1.95
1.95
–2.45
2.45 –
2.95
2.95 –
3.45
Battery lives
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3.45 –
3.95
3.95 –
4.45
4.45 –
4.95
General steps for Constructing FD
1.
2.
3.
4.
5.
6.
7.
8.
Decide number of class intervals (groups) required
Determine the range
Divide the range by no. of classes to estimate approximate
width of interval
List lower class limit of bottom interval and lower class
boundary. Add lower class width to lower class boundary to
get upper class boundary
List all the class limits and class boundaries by adding class
width to the limits and boundaries of previous interval
Determine the class marks (midpoint) by averaging the class
limits or class boundaries
Tally the frequencies for each class
Sum the frequency column and check against total no. of
observations
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PROBABILITY DISTRIBUTION
1) Discrete Distribution
 Specific values such as the integers 0,
1, 2, 3 are used.
 Typical discrete probability
distributions are, binomial and
Poisson.
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Binomial Probability Distribution
 Applicable to discrete probability problems that have an infinite
number of items or that have a steady stream of items coming
from a work center.
 The binomial is applied to problems that have attributes such as
conforming or nonconforming, success or failure, pass or fail
and heads or tails.
 It corresponds to successive terms in the binomial expansion
which is,
(p+q)n = pn + npn-1 +[ n(n-1)/2]pn-2q2 +….+qn
Where p = probability of an event such as nonconforming unit
(proportion nonconforming)
q = 1- p = probability of a nonevent such as conforming
unit
(proportion conforming)
n = number of trials or the sample size.
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 The binomial formula for a single term is
P (d) =
n!
podqon-d
d! (n – d)!
Where P (d) = probability of d nonconforming units
n = number in the sample
d = number nonconforming in the sample.
po = proportion (fraction) nonconforming in the population
qo = proportion (fraction) conforming (1-po) in the
population
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Poisson Probability Distribution
 Applicable to many situations that involve observations per unit
of times or observations per unit of amount.
 Applicable when n is quite large and Po is small. The formula for
Poisson distribution is ;
P (c) = (nPo)c e-npo
c!
Where c = count or number of events of a given classification
occurring
in a sample, such as count of nonconformities,
cars, customers
or machine breakdowns.
nPo = average count or average number of events of a given
classification occurring in a sample.
e = 2.718281
 The Poisson distribution can be used as an approximation for
the binomial in some situations, then the symbol c has the same
meaning as d.
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Probability Distribution
2) Continuous Distributions
 When measurable data such as
meters, kilograms and ohms are used.
 Only the normal distribution is of
sufficient importance in quality
control.
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Normal Probability Distribution
 All normal distributions of continuous variables can be converted
to the standardized normal distribution by using the
standardized normal value, z.
 The formula for the standardized normal curve is ;
f (z) = 1 e -z2/2 = 0.3989e –z2/2
√2π
Where π = 3.14159
e = 2.71828
z = xi – μ
σ
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i.
Relationship to the Mean and Standard Deviation

There is a definite relationship among the mean, the
standard deviation and the normal curve.
μ,mean is the value at which the center of the mountain is
located.
σ, is called standard deviation which is a lateral length of the
mountain from the center at approximately ⅔ of its height.
The larger the standard deviation, the flatter the curve (data
are widely dispersed) and the smaller the standard deviation,
the more peaked the curve (data are narrowly dispersed).
If the standard deviation is zero, all values are identical to
the mean and there is no curve. Refer to the figure below;




σ
Approx.⅔
0
μ
x
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ii.
What is 4-sigma control

A relationship exist between the standard deviation and the
area under the normal curve as shown in figure below
Relation of σ with
mountain
σ
x
±σ(68.3%)
± 2σ(95%)
± 3σ (99.7%)
± 4σ (99.99%)


Its relation with ± σ tells that product within the range of
finished
μ±σ are 68.3% of all produced.
The relation of the mountain with ± 4σ means that products
within the range of μ ± 4σ are 99.99% off all produced.
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STATISTICAL DECISION MAKING
1) Hypothesis Testing
• The hypothesis may be concerned
with a parameter or with the type
population.
• We are concerned with one (or more)
parameters and compare the observed
sample statistics with the
hypothesized parameter.
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
1.
2.
3.
4.
Element of Testing a Hypothesis on One
Parameter, for example, µ
Basic assumptions are made which are assumed
true and not open to question in the test.
Commonly the type of population is assumed, for
example, that is normal.
Although in a sense a null hypothesis is an
assumption. Instead the hypothesis is under test
and may be rejected, whereas we never use our
test reject.
Rejecting a hypothesis when it is actually true is
committing an error of the first kind.
Because of variability it is in general impossible or
infeasible to make a test for which the probability α
of an error of the first kind is zero. Nevertheless
we do want to keep the risk α at some specified
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low value, perhaps 0.01.
5.
6.
7.
8.
9.
A sample statistics (an estimator) for the
parameter in question is chosen.
An alternative hypothesis is next chosen,
containing other values of the parameter
considered possible, or of economic or scientific
interest.
The chosen risk α in 4, the type of alternative
hypothesis in 6 and knowledge of the distribution
of the statistic in 5 enables us to set a critical
region or rejection region for the statistic. The
critical region will have two parts for alternatives
such as µ ≠ 100 but one part for those such as µ
< 100 or µ > 100.
An error of the second kind is committed when we
accept the null hypothesis when in fact it is not
true. The probability of an error of the second kind
is called β.
In general, it is well to 38draw an operating
characteristic (oc) curve giving the probability of
STATISTICAL DECISION MAKING
2) Analysis of Variances
 Theoretical Formulas
•
•
•
If z = ax + by where a and b are constant
coefficients, the mean of z is given by z = ax
+by where x and y are the means of x and y.
If x and y are independent, the variance of z is
given by
z2 = a2 x2 + b2 y2 where x2 and y2 are the
variances of x and y.
The variance becomes a sum even when the
particular case involves a difference of random
variables. This characteristic is called the additivity
of variances.
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The Expectation and The Variance Of
Sample Means.
 When n measurements are taken from a population
with population mean  and population variance 2
and the values of the measurement are x1, x2,…, xn
and their mean is y, then
y = [1/n] xi = [1/n] x1 +[ 1/n] x2 +….+[1/n] xn
 The expectation and the variance of y are obtained
by
y =  and y2 = (1/n)2n 2 = 2/n.
 This is a well known formula for the distribution of
the sample mean.
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
When Random Variables Are Not
Independent.
 The additivity of variances works well when the
random variables are mutually independent.
 Two random variables are said to be independent
when the value of one variable varies without any
relation to the other variable.
 If x and y are independent, the mean and the
variance z will be z = x – y and z2 = x2 + y2
 When x and y are not independent, the mean and the
variance ofz = ax + by are given by
z = ax + by and z2 = a2x2 + b2 y2 + 2ab
x y
  is the correlation coefficient which shows the
degree of relationship between two variables. The
values of  is between –1 and +1.
 The stronger is the relationship between the two
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variables, the closer is the
absolute value of  to 1.
MODEL RELATIONSHIPS
BETWEEN VARIABLES
1) Simple Linear Regression
 Such a straight line is generally called a
regression line, where y is the response
variable (or dependent variable) and x is
the explanatory (or independent) variable.
 Also  is a constant and  is called a
regression coefficient.
 The quantitative way of grasping the
relation between42 x and y in a regression
form of x and y is called regression analysis.
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Various Scatter Diagram Having The Same
Regression Line.
Figure 1
Figure 2
Figure 3
Figure 4
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Model Relationship Between
Variables
2) Simple Linear Correlation
 Many types of scattering patterns and some
representative types are as follows;
Positive correlation
Negative correlation
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 When y increases with x, this is a positive
correlation and the opposite of the positive
correlation, since as x increases, y decreases;
this is called a negative correlation.
 The method of judging the existence of
correlation by making a scatter diagram and
calculating the correlation coefficient is called
correlation analysis.
 For either correlation analysis or regression
analysis, the starting point is a scatter
diagram.
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