*
Statistical Analysis - Graphical Techniques
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Systems Engineering Program
Department of Engineering Management, Information and
Systems
Stracener_EMIS 7370/STAT 5340_Sum 08_07.03.08
1.psd
*
Time Series Graph or Run Chart
Box Plot
Histogram and Relative Frequency Histogram
Frequency Distribution
Probability Plotting
*
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
Time Series Graph or Run Chart
*
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
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
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
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.
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
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
Box Plot
0
10
20
30
40
50
60
70
80
90
100
lower
extreme
upper
extreme
lower
quartile
median
mean
*
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
Histogram
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
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
Guidelines for Constructing Histograms Discrete Data
*
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.
Histogram and Relative Frequency Example
Sheet1
Car Battery Lives
2.24.13.54.53.23.732.6
3.41.63.13.33.83.14.73.7
2.54.33.43.62.93.33.93.1
3.33.13.74.43.24.11.93.4
4.73.83.22.63.934.23.5
Sheet2
Sheet3
*
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.
Histogram and Relative Frequency Example
Sheet1
Car Battery LivesRelative Frequency Distribution of
2.24.13.54.53.23.732.6Battery Lives
3.41.63.13.33.83.14.73.7ClassClassFrequencyRelative
2.54.33.43.62.93.33.93.1intervalmidpointffrequency
3.33.13.74.43.24.11.93.41.5-1.91.720.050
4.73.83.22.63.934.23.52.0-2.42.210.025
2.5-2.92.740.100
3.0-3.43.2150.375
3.5-3.93.7100.250
4.0-4.44.250.125
4.5-4.94.730.075
Total401.000
Sheet2
Sheet3
*
The following diagram is a relative frequency histogram of the
battery
lives with an approximate estimate of the probability density
function
superimposed.
Histogram and Relative Frequency
Chart1
1.7
2.2
2.7
3.2
3.7
4.2
4.7
Battery Lives (years)
Relative Frequency
Relative frequency histogram
0.05
0.025
0.1
0.375
0.25
0.125
0.075
Sheet1
Car Battery LivesRelative Frequency Distribution of
2.24.13.54.53.23.732.6Battery Lives
3.41.63.13.33.83.14.73.7ClassClassFrequencyRelative
2.54.33.43.62.93.33.93.1intervalmidpointffrequency
3.33.13.74.43.24.11.93.41.5-1.91.720.050
4.73.83.22.63.934.23.52.0-2.42.210.025
2.5-2.92.740.100
3.0-3.43.2150.375
3.5-3.93.7100.250
4.0-4.44.250.125
4.5-4.94.730.075
Sheet2
Sheet3
*
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
Probability Plotting
*
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.
Probability Plotting 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.
Probability Plotting General Procedure
*
If
the cumulative probability distribution function isWe now need
to linearize this function into the form
y = ax +b
Weibull Probability Plotting Paper
*
Then
which is the equation of a straight line of the form
y = ax +b
Weibull Probability Plotting Paper
*
where
and
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
Weibull Probability Plotting Paper
*
cumulative
probability(in %)
x
Weibull Probability Plotting Paper
*
To illustrate the process let 10, 20, 30, 40, 50, and 80 be a
random sample of size n = 6.
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.
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.
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.
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 x2=20,
^
^
Probability Plotting - Example
Sheet1
i
11010.9%
22026.6%
33042.2%
44057.8%
55073.4%
68089.1%
Sheet2
Sheet3
Sheet1
ixiF(xi)
11010.9%
22026.6%
33042.2%
44057.8%
55073.4%
68089.1%
Sheet2
Sheet3
*
cumulativeprobability
(in %)
x
Weibull Probability Plotting Paper
*
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
^
Probability Plotting - Example
*
The line represents the estimated relationship between x and
F(x):
x
Probability Plotting - Example
F(x)(in %)
^
*
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.
Probability Plotting - Example
*
Probability Plotting - Example
*
Probability Plotting - Example
*
Probability Paper - Normal
*
Probability Paper - Lognormal
*
Probability Paper - Exponential
*
Given the following random sample of size n=8, which probability
distribution provides the best fit?
Example - Probability Plotting
Sheet5
ixi
179.4096765982
288.1209305386
391.0639417067
498.7309365679
5104.1536168283
6105.1019
7106.5036374508
8112.035434338
Sheet4
Sheet1
Sheet2
Sheet3
*
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.
40 Specimens
Sheet1
ixixixix
148.51155.02153.13154.6
254.71255.72249.13249.9
347.81349.92355.63344.5
456.91454.82446.23452.9
554.81549.72552.03554.4
657.91658.92656.63660.2
744.91752.72752.93750.2
853.01857.82852.23857.4
954.71946.82954.13954.8
1046.72049.23042.34061.2
Sheet2
Sheet3
*
Time Series plot:
By visual inspection of the scatter plot, there seems to be no
trend.
40 Specimens
Chart1
52
53.9
50.9
50.1
54.1
52.8
49.2
54
53.2
53.6
54.2
53.4
52.6
51.6
50.2
52.8
52.2
52.8
53.1
52.2
54
52
52.2
51.8
50.2
51.6
51.8
51.7
51.9
52.1
51.6
51.5
52.4
53.4
50.5
51.6
51.2
52
50.6
49.6
Chart2
48.4988205647
54.7372948352
47.8297053126
56.8690708354
54.8171825761
57.8673435888
44.9246541797
53.0099142855
54.74485501
46.6513185023
55.0364617487
55.7413428799
49.9336109845
54.8472697977
49.6839092142
58.8838289735
52.7233211363
57.7531451603
46.7783749121
49.2002797222
53.0525013749
49.0763781221
55.6090398225
46.1983901408
52.0162572178
56.5687215788
52.8689767128
52.2033516466
54.1127220862
42.3493517195
54.6492955385
49.9327753863
44.5385736788
52.8546180652
54.4428857159
60.2144424596
50.2023496159
57.3826965996
54.8274996566
61.1844867711
Sheet1
48.5
54.7
47.8
56.9
54.8
57.9
44.9
53.0
54.7
46.7
55.0
55.7
49.9
54.8
49.7
58.9
52.7
57.8
46.8
49.2
53.1
49.1
55.6
46.2
52.0
56.6
52.9
52.2
54.1
42.3
54.6
49.9
44.5
52.9
54.4
60.2
50.2
57.4
54.8
61.2
Sheet2
Sheet3
*
40 Specimens
Using the descriptive statistics function in Excel, the
following were calculated:
Sheet4
Descriptive Statistics
Count40
Minimum42.35
Maximum61.18
Range18.84
Sum2104.82
Mean52.62
Median53.03
Sample Variance19.83
Standard Deviation4.45
Kurtosis2.51
Skewness-0.34
Sheet1
48.5-4.116.987626856876-70.016285897250288.579466228453
54.72.14.4811359804809.48597955368920.080579675551
47.852.6-4.822.950999465460-109.951815385863526.748376463549
56.94.218.05098758349176.692245040325325.838152739336
54.82.24.82574185689910.60098279729523.287784469425
57.95.227.530149920915144.448442893810757.909154668077
44.9-7.759.224895495179-455.7812494322673507.588246414870
53.00.40.1517015504840.0590860597560.023013360419
54.72.14.5132009414489.58797747540920.368982737883
46.7-6.035.630237791792-212.6807000667251269.513845099640
55.02.45.83723138103714.10296288165134.073270195759
55.73.19.74012558436330.39812767461394.870046399155
49.9-2.77.218971989155-19.39603838952052.113556580202
54.82.24.95883573453011.04255531063124.590051842056
49.7-2.98.623127622266-25.32195429427474.358329989881
58.96.339.230223657845245.7147209598801539.010448244510
52.70.10.0105875215980.0010894102670.000112095614
57.85.126.344811590985135.220533631175694.049097764518
46.8-5.834.129554524825-199.3865824794261164.826492063000
49.2-3.411.697396821757-40.006801804565136.829092405648
53.10.40.1866896021200.0806640819090.034853007540
49.1-3.512.560271485509-44.514196636877157.760419789698
55.63.08.93181584847526.69375318882479.777334351062
46.2-6.441.242537554041-264.8610325460901700.946903896490
52.0-0.60.365019256716-0.2205330402320.133239057773
56.63.915.58904229542761.550155296847243.018239688613
52.90.20.0617777286360.0153549322660.003816487755
52.2-0.40.173950560154-0.0725502222840.030258797378
54.11.52.2269492406613.3232688535494.959302920481
42.3-10.3105.494955644077-1083.54646795874011129.185666345800
54.62.04.1163138228998.35146597685816.944039488588
49.9-2.77.223462882354-19.41414049934252.178416012753
44.5-8.165.316328104624-527.8768821946994266.222717070830
52.90.20.0548461795010.0128445700340.003008103406
54.41.83.3213614091266.05304918620211.031441610032
60.27.657.669094292588437.9400827981623325.724436527380
50.2-2.45.847090755130-14.13870887371634.188470298723
57.44.822.679226471064108.004625843982514.347313325788
54.82.24.87117653930310.75104808009023.728360877053
61.28.673.343146256475628.1152017869815379.217102798720
19.335314995007-27.723493035941937.352335997336
skew-0.326077679
kurt2.5072657191
Sheet2
Sheet3
*
40 Specimens
From looking at the Histogram and the Normal Probability Plot,
we see that the tensile strength can be estimated by a normal
distribution.
Using the histogram feature of excel the following data was
calculated:
and the graph:
Sheet4
Descriptive Statistics
Count40
Sum2104.82
Mean52.62
Standard Error0.70
Median53.03
Standard Deviation4.45
Sample Variance19.83
Kurtosis-0.39
Skewness-0.34
Range18.84
Minimum42.35
Maximum61.18
Sheet5
BinFrequency
42.34935171951
45.48854089482
48.627730075
51.76691924536
54.906108420615
58.04529759588
More3
Sheet6
BinFrequency
400
453
5010
5516
609
More2
Sheet1
48.540
54.745
47.850
56.955
54.860
57.9
44.9
53.0
54.7
46.7
55.0
55.7
49.9
54.8
49.7
58.9
52.7
57.8
46.8
49.2
53.1
49.1
55.6
46.2
52.0
56.6
52.9
52.2
54.1
42.3
54.6
49.9
44.5
52.9
54.4
60.2
50.2
57.4
54.8
61.2
Sheet2
Sheet3
Chart1
40
45
50
55
60
More
Histogram of Tensile Strengths
0
3
10
16
9
2
Sheet4
Descriptive Statistics
Count40
Sum2104.82
Mean52.62
Standard Error0.70
Median53.03
Standard Deviation4.45
Sample Variance19.83
Kurtosis-0.39
Skewness-0.34
Range18.84
Minimum42.35
Maximum61.18
Sheet5
BinFrequency
42.34935171951
45.48854089482
48.627730075
51.76691924536
54.906108420615
58.04529759588
More3
Sheet6
BinFrequency
400
453
5010
5516
609
More2
Sheet6
Histogram of Tensile Strengths
Sheet1
48.540
54.745
47.850
56.955
54.860
57.9
44.9
53.0
54.7
46.7
55.0
55.7
49.9
54.8
49.7
58.9
52.7
57.8
46.8
49.2
53.1
49.1
55.6
46.2
52.0
56.6
52.9
52.2
54.1
42.3
54.6
49.9
44.5
52.9
54.4
60.2
50.2
57.4
54.8
61.2
Sheet2
Sheet3
*
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
40
45
50
55
60
65
lower
extreme
upper
extreme
lower
quartile
upper
quartile
median
mean
*
40 Specimens
Chart1
4042.349351719542.3493517195
4044.538573678861.1844863892
4044.9246541797
4046.1983901408
4046.6513185023
4046.7783749121
4047.8297053126
4048.4988205647
4049.0763781221
4049.2002797222
4049.6839092142
4049.9327753863
4049.9336109845
4050.2023496159
4052.0162572178
52.2033516466
52.7233211363
52.8546180652
52.8689767128
53.0099142855
53.0525013749
54.1127220862
54.4428857159
54.6492955385
54.7372948352
54.74485501
54.8171825761
54.8274996566
54.8472697977
55.0364617487
55.6090398225
55.7413428799
56.5687215788
56.8690708354
57.3826965996
57.7531451603
57.8673435888
58.8838289735
60.2144424596
61.1844867711
Normal Probability Plot
0.10%
1%
5%
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
99%
99.90%
12.9170633852
23.103258526
22.6964484019
22.0838968642
28.6344415043
72.7658067418
30.2617639943
31.5905357536
34.6213904675
33.7242648937
39.900543369
35.4399356991
43.7071879685
36.9005478104
46.9598411553
38.1891835685
50
39.3542462814
53.0401588447
40.426676959
56.2928120315
41.4275940682
60.099456631
42.372113411
65.3786095325
43.2715581963
69.7382360057
44.1346868582
77.9161031358
44.9684570311
87.0829366148
45.7785848892
46.5699134918
47.3465764924
48.1122709869
48.8703530107
49.6239466782
50.3760533218
51.1296469893
51.8877290131
52.6534235076
53.4300865082
54.2214151108
55.0315429689
55.8653131418
56.7284418037
57.627886589
58.5724059318
59.573323041
60.6457537186
61.8108164315
63.0994521896
64.5600643009
66.2757351063
68.4094642464
71.3655584957
76.896741474
Sheet1
48.5Standard Deviation4.6282284525
54.7Mean52.620426178
47.8
56.9
54.8
57.9
44.9
53.0
54.7
46.7
55.0
55.7
49.9
54.8
49.7
58.9
52.7
57.8
46.8
49.2
53.1
49.1
55.6
46.2
52.0
56.6
52.9
52.2
54.1
42.3
54.6
49.9
44.5
52.9
54.4
60.2
50.2
57.4
54.8
61.2
Sheet1
4042.349351719542.3493517195
4044.538573678861.1844863892
4044.9246541797
4046.1983901408
4046.6513185023
4046.7783749121
4047.8297053126
4048.4988205647
4049.0763781221
4049.2002797222
4049.6839092142
4049.9327753863
4049.9336109845
4050.2023496159
4052.0162572178
52.2033516466
52.7233211363
52.8546180652
52.8689767128
53.0099142855
53.0525013749
54.1127220862
54.4428857159
54.6492955385
54.7372948352
54.74485501
54.8171825761
54.8274996566
54.8472697977
55.0364617487
55.6090398225
55.7413428799
56.5687215788
56.8690708354
57.3826965996
57.7531451603
57.8673435888
58.8838289735
60.2144424596
61.1844867711
Normal Probability Plot
99.90%
99%
95%
90%
80%
70%
60%
50%
40%
30%
20%
10%
5%
1%
0.10%
12.9170633852
23.103258526
22.6964484019
22.0838968642
28.6344415043
72.7658067418
30.2617639943
31.5905357536
34.6213904675
33.7242648937
39.900543369
35.4399356991
43.7071879685
36.9005478104
46.9598411553
38.1891835685
50
39.3542462814
53.0401588447
40.426676959
56.2928120315
41.4275940682
60.099456631
42.372113411
65.3786095325
43.2715581963
69.7382360057
44.1346868582
77.9161031358
44.9684570311
87.0829366148
45.7785848892
46.5699134918
47.3465764924
48.1122709869
48.8703530107
49.6239466782
50.3760533218
51.1296469893
51.8877290131
52.6534235076
53.4300865082
54.2214151108
55.0315429689
55.8653131418
56.7284418037
57.627886589
58.5724059318
59.573323041
60.6457537186
61.8108164315
63.0994521896
64.5600643009
66.2757351063
68.4094642464
71.3655584957
76.896741474
Sheet2
4012.917063385242.323.10325852622.696448401942.3493517195
4022.083896864244.528.634441504372.765806741861.1844863892
4030.261763994344.931.5905357536
4034.621390467546.233.7242648937
4039.90054336946.735.4399356991
4043.707187968546.836.9005478104
4046.959841155347.838.1891835685
405048.539.3542462814
4053.040158844749.140.426676959
4056.292812031549.241.4275940682
4060.09945663149.742.372113411
4065.378609532549.943.2715581963
4069.738236005749.944.1346868582
4077.916103135850.244.9684570311
4087.082936614852.045.7785848892
52.246.5699134918
52.747.3465764924
52.948.1122709869
52.948.8703530107
53.049.6239466782
53.150.3760533218
54.151.1296469893
54.451.8877290131
54.652.6534235076
54.753.4300865082
54.754.2214151108
54.855.0315429689
54.855.8653131418
54.856.7284418037
55.057.627886589
55.658.5724059318
55.759.573323041
56.660.6457537186
56.961.8108164315
57.463.0994521896
57.864.5600643009
57.966.2757351063
58.968.4094642464
60.271.3655584957
61.276.896741474
182.324100554160.5347589852
182.916556771561.4499744086
183.528555756362.4371945276
184.087656255663.516691979
184.206856904964.7183436638
184.813774088166.0890522238
185.097331798867.709498934
185.890979612969.7382360057
186.780670067572.5694748224
196.20799192477.9161031358
Sheet3
*
40 Specimens
Chart1
1042.3493517195
1044.5385736788
1044.9246541797
1046.1983901408
1046.6513185023
1046.7783749121
1047.8297053126
1048.4988205647
1049.0763781221
1049.2002797222
1049.6839092142
1049.9327753863
1049.9336109845
1050.2023496159
1052.0162572178
52.2033516466
52.7233211363
52.8546180652
52.8689767128
53.0099142855
53.0525013749
54.1127220862
54.4428857159
54.6492955385
54.7372948352
54.74485501
54.8171825761
54.8274996566
54.8472697977
55.0364617487
55.6090398225
55.7413428799
56.5687215788
56.8690708354
57.3826965996
57.7531451603
57.8673435888
58.8838289735
60.2144424596
61.1844867711
LogNormal Probability Plot
0.10%
1%
5%
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
99%
99.90%
12.9170633852
23.103258526
22.0838968642
28.6344415043
30.2617639943
31.5905357536
34.6213904675
33.7242648937
39.900543369
35.4399356991
43.7071879685
36.9005478104
46.9598411553
38.1891835685
50
39.3542462814
53.0401588447
40.426676959
56.2928120315
41.4275940682
60.099456631
42.372113411
65.3786095325
43.2715581963
69.7382360057
44.1346868582
77.9161031358
44.9684570311
87.0829366148
45.7785848892
46.5699134918
47.3465764924
48.1122709869
48.8703530107
49.6239466782
50.3760533218
51.1296469893
51.8877290131
52.6534235076
53.4300865082
54.2214151108
55.0315429689
55.8653131418
56.7284418037
57.627886589
58.5724059318
59.573323041
60.6457537186
61.8108164315
63.0994521896
64.5600643009
66.2757351063
68.4094642464
71.3655584957
76.896741474
Sheet1
48.5Standard Deviation4.7453095531
54.7Mean52.4394683838
47.8
56.9
54.8
57.9
44.9
53.0
54.7
46.7
55.0
55.7
49.9
54.8
49.7
58.9
52.7
57.8
46.8
49.2
53.1
49.1
55.6
46.2
52.0
56.6
52.9
52.2
54.1
42.3
54.6
49.9
44.5
52.9
54.4
60.2
50.2
57.4
54.8
61.2
Sheet1
1042.3493517195
1044.5385736788
1044.9246541797
1046.1983901408
1046.6513185023
1046.7783749121
1047.8297053126
1048.4988205647
1049.0763781221
1049.2002797222
1049.6839092142
1049.9327753863
1049.9336109845
1050.2023496159
1052.0162572178
52.2033516466
52.7233211363
52.8546180652
52.8689767128
53.0099142855
53.0525013749
54.1127220862
54.4428857159
54.6492955385
54.7372948352
54.74485501
54.8171825761
54.8274996566
54.8472697977
55.0364617487
55.6090398225
55.7413428799
56.5687215788
56.8690708354
57.3826965996
57.7531451603
57.8673435888
58.8838289735
60.2144424596
61.1844867711
LogNormal Probability Plot
99.90%
99%
95%
90%
80%
70%
60%
50%
40%
30%
20%
10%
5%
1%
0.10%
12.9170633852
23.103258526
22.0838968642
28.6344415043
30.2617639943
31.5905357536
34.6213904675
33.7242648937
39.900543369
35.4399356991
43.7071879685
36.9005478104
46.9598411553
38.1891835685
50
39.3542462814
53.0401588447
40.426676959
56.2928120315
41.4275940682
60.099456631
42.372113411
65.3786095325
43.2715581963
69.7382360057
44.1346868582
77.9161031358
44.9684570311
87.0829366148
45.7785848892
46.5699134918
47.3465764924
48.1122709869
48.8703530107
49.6239466782
50.3760533218
51.1296469893
51.8877290131
52.6534235076
53.4300865082
54.2214151108
55.0315429689
55.8653131418
56.7284418037
57.627886589
58.5724059318
59.573323041
60.6457537186
61.8108164315
63.0994521896
64.5600643009
66.2757351063
68.4094642464
71.3655584957
76.896741474
Sheet2
1012.917063385242.323.10325852643.819745992823.9743649032
1022.083896864244.528.634441504388.74773325594789.8973559713
1030.261763994344.931.5905357536
1034.621390467546.233.7242648937
1039.90054336946.735.4399356991
1043.707187968546.836.9005478104
1046.959841155347.838.1891835685
105048.539.3542462814
1053.040158844749.140.426676959
1056.292812031549.241.4275940682
1060.09945663149.742.372113411
1065.378609532549.943.2715581963
1069.738236005749.944.1346868582
1077.916103135850.244.9684570311
1087.082936614852.045.7785848892
52.246.5699134918
52.747.3465764924
52.948.1122709869
52.948.8703530107
53.049.6239466782
53.150.3760533218
54.151.1296469893
54.451.8877290131
54.652.6534235076
54.753.4300865082
54.754.2214151108
54.855.0315429689
54.855.8653131418
54.856.7284418037
55.057.627886589
55.658.5724059318
55.759.573323041
56.660.6457537186
56.961.8108164315
57.463.0994521896
57.864.5600643009
57.966.2757351063
58.968.4094642464
60.271.3655584957
61.276.896741474
Sheet3
*
40 Specimens
Chart1
3.70849358223.74595311333.7084935822
3.70849358223.79635563793.7895736004
3.70849358223.80498673493.8706536186
3.70849358223.8329449523.9517336368
3.70849358223.84270119064.032813655
3.70849358223.84542102164.1138936733
3.70849358223.8676468965
3.70849358223.8815394794
3.70849358223.8933778217
3.70849358223.8958993089
3.70849358223.9056811224
3.70849358223.9106776085
3.70849358223.9106943428
3.70849358223.9160618307
3.70849358223.9515563085
3.70849358223.9551467006
3.70849358223.965057884
3.70849358223.9675450891
3.70849358223.9678167153
3.70849358223.970478958
3.9712820152
3.9910693169
3.9971521837
4.0009363241
4.002545284
4.0026833919
4.0040036955
4.0041918866
4.0045524097
4.0079959065
4.0183457748
4.0207221136
4.0354562102
4.0407516228
4.0497428049
4.0561778096
4.0581532115
4.0755665024
4.0979122315
4.1138936733
Weibull Probability Plot
0.10%
0.20%
0.30%
0.50%
1%
2%
3%
5%
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
99%
99.90%
41
44
48
52
56
61
-6.9072550036
-4.3757438213
-7.2526177538
-6.2136072637
-3.2643645676
-7.2526177538
-5.8076411106
-2.7404930065
-7.2526177538
-5.2958121984
-2.3906822386
-7.2526177538
-4.6001492222
-2.1257221216
-7.2526177538
-3.9019386489
-1.9110827892
-7.2526177538
-3.4913669807
-1.7297202729
-2.9701952395
-1.5719525273
-2.2503673296
-1.4317440643
-1.4999400026
-1.3050729026
-1.0309303623
-1.1891197438
-0.6717269721
-1.0818278631
-0.3665129178
-0.9816470555
-0.087421476
-0.8873759959
0.1856267689
-0.7980610834
0.475885014
-0.7129289212
0.8340323556
-0.6313392162
1.0971886321
-0.5527521431
1.5271798468
-0.4767035757
1.9326466116
-0.4027868111
-0.3306383372
-0.2599258342
-0.1903393255
-0.1215813478
-0.0533591521
0.0146233279
0.0826760122
0.1511325382
0.2203655649
0.2908053716
0.3629664347
0.4374890912
0.5152018941
0.5972292379
0.685181261
0.7815251843
0.890405279
1.0197814405
1.1888836465
1.477511537
Sheet1
48.5
54.7
47.8
56.9
54.8
57.9
44.9
53.0
54.7
46.7
55.0
55.7
49.9
54.8
49.7
58.9
52.7
57.8
46.8
49.2
53.1
49.1
55.6
46.2
52.0
56.6
52.9
52.2
54.1
42.3
54.6
49.9
44.5
52.9
54.4
60.2
50.2
57.4
54.8
61.2
Sheet1
3.70849358223.74595311333.7084935822
3.70849358223.79635563793.7895736004
3.70849358223.80498673493.8706536186
3.70849358223.8329449523.9517336368
3.70849358223.84270119064.032813655
3.70849358223.84542102164.1138936733
3.70849358223.8676468965
3.70849358223.8815394794
3.70849358223.8933778217
3.70849358223.8958993089
3.70849358223.9056811224
3.70849358223.9106776085
3.70849358223.9106943428
3.70849358223.9160618307
3.70849358223.9515563085
3.70849358223.9551467006
3.70849358223.965057884
3.70849358223.9675450891
3.70849358223.9678167153
3.70849358223.970478958
3.9712820152
3.9910693169
3.9971521837
4.0009363241
4.002545284
4.0026833919
4.0040036955
4.0041918866
4.0045524097
4.0079959065
4.0183457748
4.0207221136
4.0354562102
4.0407516228
4.0497428049
4.0561778096
4.0581532115
4.0755665024
4.0979122315
4.1138936733
Weibull Probability Plot
99.90%
99%
95%
90%
80%
70%
60%
50%
40%
30%
20%
10%
5%
3%
2%
1%
0.50%
0.30%
0.20%
0.10%
61
56
52
48
44
41
-6.9072550036
-4.3757438213
-7.2526177538
-6.2136072637
-3.2643645676
-7.2526177538
-5.8076411106
-2.7404930065
-7.2526177538
-5.2958121984
-2.3906822386
-7.2526177538
-4.6001492222
-2.1257221216
-7.2526177538
-3.9019386489
-1.9110827892
-7.2526177538
-3.4913669807
-1.7297202729
-2.9701952395
-1.5719525273
-2.2503673296
-1.4317440643
-1.4999400026
-1.3050729026
-1.0309303623
-1.1891197438
-0.6717269721
-1.0818278631
-0.3665129178
-0.9816470555
-0.087421476
-0.8873759959
0.1856267689
-0.7980610834
0.475885014
-0.7129289212
0.8340323556
-0.6313392162
1.0971886321
-0.5527521431
1.5271798468
-0.4767035757
1.9326466116
-0.4027868111
-0.3306383372
-0.2599258342
-0.1903393255
-0.1215813478
-0.0533591521
0.0146233279
0.0826760122
0.1511325382
0.2203655649
0.2908053716
0.3629664347
0.4374890912
0.5152018941
0.5972292379
0.685181261
0.7815251843
0.890405279
1.0197814405
1.1888836465
1.477511537
Plot Data
-6.90725500363.70849358223.7-4.37574382133.7084935822-7.2526177538
-6.21360726373.70849358223.8-3.26436456763.7895736004-7.2526177538
-5.80764111063.70849358223.8-2.74049300653.8706536186-7.2526177538
-5.29581219843.70849358223.8-2.39068223863.9517336368-7.2526177538
-4.60014922223.70849358223.8-2.12572212164.032813655-7.2526177538
-3.90193864893.70849358223.8-1.91108278924.1138936733-7.2526177538
-3.49136698073.70849358223.9-1.72972027294.1138936733-7.2526177538
-2.97019523953.70849358223.9-1.5719525273
-2.25036732963.70849358223.9-1.4317440643
-1.49994000263.70849358223.9-1.3050729026
-1.03093036233.70849358223.9-1.1891197438
-0.67172697213.70849358223.9-1.0818278631
-0.36651291783.70849358223.9-0.9816470555
-0.0874214763.70849358223.9-0.8873759959
0.18562676893.70849358224.0-0.7980610834
0.4758850143.70849358224.0-0.7129289212
0.83403235563.70849358224.0-0.6313392162
1.09718863213.70849358224.0-0.5527521431
1.52717984683.70849358224.0-0.4767035757
1.93264661163.70849358224.0-0.4027868111
4.0-0.3306383372
4.0-0.2599258342
4.0-0.1903393255
4.0-0.1215813478
4.0-0.0533591521
4.00.0146233279
4.00.0826760122
4.00.1511325382
4.00.2203655649
4.00.2908053716
4.00.3629664347
4.00.4374890912
4.00.5152018941
4.00.5972292379
4.00.685181261
4.10.7815251843
4.10.890405279
4.11.0197814405
4.11.1888836465
4.11.477511537
*
The tensile strength distribution can be estimated by
40 Specimens
f(x)
F(x)
^
^
=
=
k
i
i
n
f
1
2.24.13.54.53.23.732.6
3.41.63.13.33.83.14.73.7
2.54.33.43.62.93.33.93.1
3.33.13.74.43.24.11.93.4
4.73.83.22.63.934.23.5
Car Battery Lives
ClassClassFrequencyRelative
intervalmidpointffrequency
1.5-1.91.720.050
2.0-2.42.210.025
2.5-2.92.740.100
3.0-3.43.2150.375
3.5-3.93.7100.250
4.0-4.44.250.125
4.5-4.94.730.075
Total401.000
Relative Frequency Distribution of
Battery Lives
Relative frequency histogram
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
1.72.22.73.23.74.24.7
Battery Lives (years)
Relative Frequency
+
-
=
4
.
0
3
.
0
100
)
(
n
i
x
F
i
4
.
0
n
3
.
0
i
)
x
(
F
^
i
+
-
=
+
-
=
4
.
0
n
3
.
0
i
100
)
x
(
F
^
i
b
q
-
-
=
t
e
t
1
)
(
F
(
)
,
W
~
T
(
)
(
)
(
)
[
]
(
)
(
)
q
b
b
q
b
q
b
q
b
ln
ln
)
T
(
F
1
1
ln
ln
ln
)
T
(
F
1
ln
ln
)
T
(
F
1
ln
ln
)
T
(
F
1
ln
-
=
-
=
-
-
-
=
-
=
-
-
x
x
x
e
x
-
-
=
)
t
(
F
1
1
ln
ln
y
b
=
a
(
)
t
x
ln
=
(
)
i.e.,
,
ln
q
b
-
=
b
(
)
q
b
b
ln
-
=
x
y
(
)
q
b
ln
-
-
=
)
t
(
F
1
1
ln
ln
y
(
)
%)
100
(
0.4
n
0.3
i
MR
x
F
i
i
+
-
@
=
(
)
%
6
.
26
%
100
*
0.4
6
0.3
2
20
F
=
+
-
=
i
x
i
F(x
i
)
11010.9%
22026.6%
33042.2%
44057.8%
55073.4%
68089.1%
(
)
(
)
x
F
,
x
i x
i
179.40968
288.12093
391.06394
498.73094
5104.1536
6105.1019
7106.5036
8112.0354
148.51155.02153.13154.6
254.71255.72249.13249.9
347.81349.92355.63344.5
456.91454.82446.23452.9
554.81549.72552.03554.4
657.91658.92656.63660.2
744.91752.72752.93750.2
853.01857.82852.23857.4
954.71946.82954.13954.8
1046.72049.23042.34061.2
30.0
35.0
40.0
45.0
50.0
55.0
60.0
65.0
0510152025303540
Descriptive Statistics
Count40
Minimum42.35
Maximum61.18
Range18.84
Sum2104.82
Mean52.62
Median53.03
Sample Variance19.83
Standard Deviation4.45
Kurtosis2.51
Skewness-0.34
BinFrequency
400
453
5010
5516
609
More2
Histogram of Tensile Strengths
0
2
4
6
8
10
12
14
16
18
4045505560More
Normal Probability Plot
0.10%
1%
5%
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
99%
99.90%
404550556065
LogNormal Probability Plot
0.10%
1%
5%
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
99%
99.90%
10100
Weibull Probability Plot
0.10%
0.20%
0.30%
0.50%
1%
2%
3%
5%
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
99%
99.90%
414448525661
(
)
45
.
4
,
62
.
52
N
~
X
=
=
s
0
0.2
0.4
0.6
0.8
1
49505152535455