Statistical Analysis - Graphical Techniques

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Systems Engineering Program
Department of Engineering Management, Information
and Systems
EMIS 7370/5370 STAT 5340 PROBABILITY AND
STATISTICS FOR SCIENTISTS AND ENGINEERS
Statistical Analysis - Graphical Techniques
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
Stracener_EMIS 7370/STAT 5340_Sum 08_07.03.08
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• Time Series Graph or Run Chart
• Box Plot
• Histogram and Relative Frequency Histogram
• Frequency Distribution
• Probability Plotting

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Time Series Graph or Run Chart

• A plot of the data set x1, x2, , xn in the
  order
• in which the data were obtained
• Used to detect trends or patterns in the data
• over time

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Box Plot

• A pictorial summary used to describe the
• most prominent statistical features of the data
• set, x1, x2, , xn, including its
• - Center or location
• - Spread or variability
• - Extent and nature of any deviation from
  symmetry
• - Identification of outliers

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Box Plot

• Shows only certain statistics rather than all
  the
• data, namely
• - median
• - quartiles
• - smallest and greatest values in the sample
• Immediate visuals of a box plot are the center,
• the spread, and the overall range of the data

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Box Plot
Given the following random sample of size
25 38, 10, 60, 90, 88, 96, 1, 41, 86, 14, 25,
5, 16, 22, 29, 34, 55, 36, 37, 36, 91, 47, 43,
30, 98 Arranged in order from least to
greatest 1, 5, 10, 14, 16, 22, 25, 29, 30, 34,
36, 36, 37, 38, 41, 43, 47, 55, 60, 86, 88, 90,
91, 96, 98
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Box Plot

• First, find the median, the value exactly in the
• middle of an ordered set of numbers.
• The median is 37
• Next, we consider only the values to the left of
• the median
• 1, 5, 10, 14, 16, 22, 25, 29, 30, 34, 36, 36
• We now find the median of this set of numbers.
• The median for this group is (22 25)/2 23.5,
• which is the lower quartile.

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Box Plot

• Now consider the values to the right of the
• median.
• 38, 41, 43, 47, 55, 60, 86, 88, 90, 91, 96, 98
• The median for this set is (60 86)/2 73,
  which
• is the upper quartile.
• We are now ready to find the interquartile range
• (IQR), which is the difference between the upper
• and lower quartiles, 73 - 23.5 49.5
• 49.5 is the interquartile range

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Box Plot
The lower quartile 23.5 The median is 37 The
upper quartile 73 The interquartile range is
49.5 The mean is 45.1
upper quartile
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Histogram
A graph of the observed frequencies in the
data set, x1, x2, , xn versus data magnitude
to visually indicate its statistical properties,
including - shape - location or central
tendency - scatter or variability
Guidelines for Constructing Histograms Discrete
Data
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Guidelines for Constructing Histograms Discrete
Data

• If the data x1, x2, , xn are from a discrete
• random variable with possible values y1, y2, ,
  yk
• count the number of occurrences of each value
• of y and associate the frequency fi with yi,
• for i 1, , k,
• Note that

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Guidelines for Constructing Histograms Discrete
Data

• If the data x1, x2, , xn are from a continuous
• random variable
• - select the number of intervals or cells, r,
• to be a number between 3 and 20, as an
• initial value use r (n)1/2, where n is the
• number of observations
• - establish r intervals of equal width, starting
• just below the smallest value of x
• - count the number of values of x within
• each interval to obtain the frequency
• associated with each interval
• - construct graph by plotting (fi, i) for
• i 1, 2, , k

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Histogram and Relative Frequency Example
To illustrate the construction of a relative
frequency distribution, consider the following
data which represent the lives of 40
carbatteries of a given type recorded to the
nearest tenth of a year.The batteries were
guaranteed to last 3 years.
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Histogram and Relative Frequency Example
For this example, using the guidelines for
constructing a histogram, the number of classes
selected is 7 with a class width of 0.5.
The frequency and relative frequency distribution
for the data are shown in the following table.
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Histogram and Relative Frequency
The following diagram is a relative frequency
histogram of the battery lives with an
approximate estimate of the probability density
function superimposed.
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Probability Plotting

• Data are plotted on special graph paper
• designed for a particular distribution
• - Normal - Weibull
• - Lognormal - Exponential
• If the assumed model is adequate, the plotted
• points will tend to fall in a straight line
• If the model is inadequate, the plot will not
• be linear and the type extent of departures
• can be seen
• Once a model appears to fit the data
• reasonably will, percentiles and parameters can
• be estimated from the plot

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Probability Plotting Procedure

• Step 1 Obtain special graph paper, known
  asprobability paper, designed for the
  distribution under
• examination. Weibull, Lognormal and Normal paper
• are available at
• http//www.weibull.com/GPaper/index.htm
• Step 2 Rank the sample values from smallest
• to largest in magnitude i.e., X1 ? X2 ? ..., Xn.

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Probability Plotting General Procedure

• Step 3
• Plot the Xis on the paper versus
  or
• , depending on whether the
  marked axis
• on the paper refers to the or the proportion
• of observations. The axis of the graph paper on
• which the Xis are plotted will be referred to as
  
• the observational scale, and the axis for
• as the cumulative scale.
• Step 4 If a straight line appears to fit the
  data,
• draw a line on the graph, by eye.
• Step 5 Estimate the model parameters from
• the graph.

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Weibull Probability Plotting Paper
If the cumulative probability
distribution function isWe now need to
linearize this function into the form y ax b
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Weibull Probability Plotting Paper
Then which is the equation of a
straight line of the form y ax b
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Weibull Probability Plotting Paper
where and
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Weibull Probability Plotting Paper
which is a linear equation with a slope of b
and an intercept of . Now the x- and
y-axes of the Weibull probability plotting paper
can be constructed. The x-axis is simply
logarithmic, since x ln(T) and
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Weibull Probability Plotting Paper
cumulative probability(in )
x
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Probability Plotting - Example
To illustrate the process let 10, 20, 30, 40,
50, and 80 be a random sample of size n 6.
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Probability Plotting - Example
We need value estimates corresponding to each
of the sample values in order to plot the data on
the Weibull probability paper. These estimates
are accomplished with what are called median
ranks.
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Probability Plotting - Example
Median ranks represent the 50 confidence level
(best guess) estimate for the true value of
F(t), based on the total sample size and the
order number (first, second, etc.) of the data.
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Probability Plotting - Example
There is an approximation that can be used to
estimate median ranks, called Benards
approximation. It has the form where n is
the sample size and i is the sample order number.
Tables of median ranks can be found in may
statistics and reliability texts.
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Probability Plotting - Example
Based on Benards approximation, we can now
calculate F(t) for each observed value of X.
These are shown in the following table For
example, for x220,


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Weibull Probability Plotting Paper
cumulativeprobability (in )
x
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Probability Plotting - Example
Now that we have y-coordinate values to go with
the x-coordinate sample values so we can plot the
points on Weibull probability paper.

F(x)(in )
x
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Probability Plotting - Example
The line represents the estimated relationship
between x and F(x)
x
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Probability Plotting - Example
In this example, the points on Weibull
probability paper fall in a fairly linear
fashion, indicating that the Weibull distribution
provides a good fit to the data. If the points
did not seem to follow a straight line, we might
want to consider using another probability
distribution to analyze the data.
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Probability Plotting - Example
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Probability Plotting - Example
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Probability Paper - Normal
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Probability Paper - Lognormal
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Probability Paper - Exponential
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Example - Probability Plotting
Given the following random sample of size n8,
which probability distribution provides the best
fit?
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40 Specimens
40 specimens are cut from a plate for tensile
tests. The tensile tests were made, resulting in
Tensile Strength, x, as follows
Perform a statistical analysis of the tensile
strength data.
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40 Specimens
Time Series plot By visual
inspection of the scatter plot, there seems to be
no trend.
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40 Specimens
Using the descriptive statistics function in
Excel, the following were calculated
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40 Specimens
Using the histogram feature of excel the
following data was calculated and the graph
From looking at the Histogram and the Normal
Probability Plot, we see that the tensile
strength can be estimated by a normal
distribution.
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40 Specimens
Box Plot
The lower quartile 49.45 The median is 53.03 The
mean 52.6 The upper quartile 55.3 The
interquartile range is 5.86
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40 Specimens
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40 Specimens
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40 Specimens
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40 Specimens
The tensile strength distribution can be
estimated by