READING - Stanford Universityweb.stanford.edu/class/engr1n/Day_5_Error_and_Stats.pdf · Std Dev...
21
1 Stanford University School of Engineering ENGINEERING 1N THE NATURE OF ENGINEERING READING • Taylor, Chapters 4 and 5
Transcript of READING - Stanford Universityweb.stanford.edu/class/engr1n/Day_5_Error_and_Stats.pdf · Std Dev...
1
Stanford UniversitySchool of Engineering
ENGINEERING 1N
THE NATURE OF ENGINEERING
READING
• Taylor, Chapters 4 and 5
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Bike RampsBike Ramp Angles
Upper Ramp Lower RampTechnique Slope (°) Slope (°)tan 3.70 2.34tan* 2.12 3.23tan** 4.39 3.29tan* 4.57 3.26tan 3.576 3.327tan, sin, cos** 4.56 3.06tan** 4.60 3.22
Count 7 7Max 4.60 3.33Min 2.1 2.3
Mean 3.93 3.10Std Dev 0.91 0.35
C V 0.231 0.112
Slope (rad) Slope (rad)Mean 0.069 0.054
x
yz
q
q = tan-1 y
xÊ Ë
ˆ ¯
∂q∂x
=-y
x2 + y2 ∂q∂y
=x
x2 + y2
Error Propagation
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Error Propagation
x
yz
q
dq =
-yx2 + y2
È
Î Í ˘
˚ ˙
2
dx 2 +x
x2 + y2È
Î Í ˘
˚ ˙
2
dy 2
dq £
-yx2 + y2 dx +
xx2 + y2 dy Bound
Independent Errors
RELATIVE ERROR IN TAN-1(y/x)
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0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50Tan-1(y/x) [rad]
dx/x = 0.01
dx/x = 0.1
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RELATIVE ERROR IN q
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0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80q (rad)
—— dx/x = 0.1 dx/x = 0.01
cos-1(x/z)
sin-1(y/z)
tan-1(y/x)
RELATIVE ERROR IN q
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0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80q (rad)
—— dx/x = 0.1 dx/x = 0.01
cos-1(x/z)
sin-1(y/z)
tan-1(y/x)
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Bike RampsBike Ramp Angles
Upper Ramp Lower RampTechnique Slope (°) Slope (°)tan 3.70 2.34tan* 2.12 3.23tan** 4.39 3.29tan* 4.57 3.26tan 3.576 3.327tan, sin, cos** 4.56 3.06tan** 4.60 3.22
Count 7 7Max 4.60 3.33Min 2.1 2.3
Mean 3.93 3.10Std Dev 0.91 0.35
C V 0.231 0.112
Slope (rad) Slope (rad)Mean 0.069 0.054
Measurement Uncertainty
• Quantificationw Bounds
Maximum/minimum values, instrument scale divisions
w Statistical measuresStandard deviation, s or s
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Histograms
• Definedw Graphical presentation of a data set sorted by
magnitude, showing the relative proportion ofobservations of different magnitudes
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30
35
40
Bin Upper Limit (sec)
True Value = 1.700
Descriptive Statistics
• Central Tendencyw Mean (Average)w Median Midpointw Mode Most Frequent
• Spreadw Standard Deviationw Variancew Coefficient of Variation
• Asymmetryw (Coefficient of) Skew
x = 1
nxi
i =1
n
Â
s =
1n -1
xi - x [ ]2
i =1
n
Â
CV =
sx
g =
n2
n - 1[ ] n - 2[ ]
xi - x [ ]3
i= 1
n
Âs3
†
Var = s2 Most commonuncertainty
index(as a multiple)
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Raw Data Set 1
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Data Set 1 Histogram
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Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
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Data Set 1 Histogram
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Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 1 Histogram
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Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
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Data Set 1 Histogram
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1.35 1.55 1.75 1.95 2.15Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Histograms
• Design Issuesw Bin size
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Raw Data Set 2
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Data Set 2 Histogram
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Bin Upper Limit (sec)
Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
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Data Sets 1 & 2 Histograms
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Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
Raw Data Sets 1 & 2
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Data Sets 1 & 2 Normalized Histograms
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Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
Histograms
• Design Issuesw Bin sizew Normalization
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Data Sets 1 & 2 Cumulative Histograms
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Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
Data Sets 1 & 2 Normalized Cumulative Histograms
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Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
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Cumulative Histograms
• The information in a histogram can be rearranged to createa cumulative histogram.w Bar height = Fraction of observations ≤ bin limitw Bar height = Sum of all histogram bar heights for all bins ≤ bin limitw There is a reciprocal relationship between the histogram and the
cumulative histogram, analogous to the relationship between thederivative and the integral
Histogram ¤ Cumulative histogram
Raw Data Set 3
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Raw Data Sets 1 & 3
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Data Sets 1 & 3 Histograms
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Bin Upper Limit (sec)
Data Set 1 Data Set 3
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 3Max = 1.984Min = 1.410Median = 1.689Mean = 1.689Std Dev = 0.110CV = 0.065Skew = 0.277
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Raw Data Set 4
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Raw Data Sets 1 & 4
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Data Sets 1 & 4 Histograms
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Data Set 1 Data Set 4
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 4Max = 2.728Min = 0.793Median = 1.707Mean = 1.692Std Dev = 0.330CV = 0.195Skew = 0.034
Raw Data Set 5
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Raw Data Sets 1 & 5
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Data Sets 1 & 5 Histograms
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Bin Upper Limit (sec)
Data Set 1 Data Set 5
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 5Max = 1.843Min = 1.198Median = 1.502Mean = 1.497Std Dev = 0.110CV = 0.073Skew = 0.038
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Raw Data Set 6
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Raw Data Sets 4 & 6
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Data Sets 4 & 6 Histograms
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Bin Upper Limit (sec)
Data Set 4 Data Set 6
Data Set 4Max = 2.728Min = 0.793Median = 1.707Mean = 1.692Std Dev = 0.330CV = 0.195Skew = 0.034
Data Set 6Max = 3.051Min = 0.986Median = 1.681Mean = 1.697Std Dev = 0.334CV = 0.197Skew = 0.938
E1N - The Nature of Engineering
Terman Pond Bags of Salt Histogram (2001 & 2003)
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90 100 110 120 130 140 150 160 32050 Lb. Bags of Salt [Bin upper limit]
Max = 320Min = 99
Average = 136.2 BagsSt Dev = 54.6 Bags
CV = 0.40
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E1N - The Nature of Engineering
Terman Pond Bags of Salt Histogram (2001 & 2003)
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90 100 110 120 130 140 150 16050 Lb. Bags of Salt [Bin upper limit]
Max = 160Min = 99
Average = 123.1 BagsSt Dev = 20.8 Bags
CV = 0.17
