READING - Stanford Universityweb.stanford.edu/class/engr1n/Day_5_Error_and_Stats.pdf · Std Dev 0.91 0.35 CV 0.231 0.112 Slope (rad) Slope (rad) Mean 0.069 0.054 Measurement Uncertainty

1

Stanford UniversitySchool of Engineering

ENGINEERING 1N

THE NATURE OF ENGINEERING

READING

• Taylor, Chapters 4 and 5

2

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

3

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)

0.00

1.00

2.00

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11.00

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

4

RELATIVE ERROR IN q

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

11.00

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

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

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)

5

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

6

Histograms

• Definedw Graphical presentation of a data set sorted by

magnitude, showing the relative proportion ofobservations of different magnitudes

0

5

10

15

20

25

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)

7

Raw Data Set 1

0.00

0.50

1.00

1.50

2.00

2.50

0 10 20 30 40 50 60 70 80 90 100Observation

Data Set 1 Histogram

0

2

4

6

8

10

12


Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033

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0

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16




0

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40



9


0

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60

1.35 1.55 1.75 1.95 2.15Bin Upper Limit (sec)


Histograms

• Design Issuesw Bin size

10

Raw Data Set 2

0.00

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0 10 20 30 40 50 60 70 80Observation


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11

Data Sets 1 & 2 Histograms

0

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35

40


Data Set 1 Data Set 2

Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033


Raw Data Sets 1 & 2

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0.50

1.00

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2.50

0 10 20 30 40 50 60 70 80 90 100Observation

12

Data Sets 1 & 2 Normalized Histograms

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0.45





Histograms

• Design Issuesw Bin sizew Normalization

13

Data Sets 1 & 2 Cumulative Histograms

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90

100





Data Sets 1 & 2 Normalized Cumulative Histograms

0

0.1

0.2

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0.7

0.8

0.9

1





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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

0.00

0.50

1.00

1.50

2.00

2.50

0 10 20 30 40 50 60 70 80 90 100Observation

15

Raw Data Sets 1 & 3

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2.50

0 10 20 30 40 50 60 70 80 90 100Observation


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Raw Data Set 4

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0 10 20 30 40 50 60 70 80 90 100Observation

Raw Data Sets 1 & 4

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3.00

0 10 20 30 40 50 60 70 80 90 100Observation

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Raw Data Set 5

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Raw Data Sets 1 & 5

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0 10 20 30 40 50 60 70 80 90 100Observation


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Raw Data Set 6

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Raw Data Sets 4 & 6

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0 10 20 30 40 50 60 70 80 90 100Observation

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0

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E1N - The Nature of Engineering

Terman Pond Bags of Salt Histogram (2001 & 2003)

0

1

2

3

4

5

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)

0

1

2

3

4

5

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

READING - Stanford Universityweb.stanford.edu/class/engr1n/Day_5_Error_and_Stats.pdf · Std Dev 0.91 0.35 CV 0.231 0.112 Slope (rad) Slope (rad) Mean 0.069 0.054 Measurement Uncertainty

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