Software Development Agreement with Residuals by pjt13416

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									MEASUREMENT ANALYSIS
AND ADJUSTMENT
Capital Project Skill Development Class
(CPSD)
G100398

                              By Jeremy Evans,
                              P.L.S.   Psomas
                              Supplemented by
                              Caltrans Staff
Introduction
  “The dark side of surveying is the
    belief that surveying is about
    measurements, precisions and
   adjustments. It is not and never
                will be.”
                             Dennis Mouland
                             P.O.B. Magazine
                                   July, 2002
Introduction
  Much has been written lately about least
  squares adjustment and the advantages it
  brings to the land surveyor. To take full
  advantage of a least squares adjustment
  package, the surveyor must have a basic
  understanding of the nature of
  measurements, the equipment he uses, the
  methods he employs, and the environment
  he works in.
Introduction
  Measurements and Adjustments:
          “War Stories”
Class Outline
  Survey Measurement Basics - A Review
  Measurement Analysis
  Error Propagation
  Introduction to Weighted and Least
  Squares Adjustments
  Least Squares Adjustment Software
  Sample Network Adjustments
Measure First,
Adjustment Last
  Adjustment programs assume that:
     Instruments are calibrated
     Measurements are carefully made
  Networks are stronger if:
     They include Redundancy
     They have Strength of Figure
   Adjust only after you have followed
  proper procedures!
Survey Measurement Basics
A Review of Plumb Bob 101
Surveying (Geospatial Services?)

  Surveying – “That discipline which
  encompasses all methods for measuring,
  processing, and disseminating information
  about the physical earth and our
  environment.” – Brinker & Wolf
  Surveyor - An expert in measuring,
  processing, and disseminating information
  about the physical earth and our
  environment.
Measurement vs. Enumeration
  A lot of statistical theory
  deals with enumeration,
  or counting. It’s a way
  to take a test sample
  instead of a census of
  the total population.
  The surveyor is
  concerned with
  Measurement. The true
  dimensions can never
  be known.
Instrument Testing
  Pointing error of typical total station
Instrument Specifications
Instrument Specifications
Instrument Specifications

  Distance Measurement
   sm = ±(0.01’ + 3ppm x D)
   What is the error in a 3500 foot
    measurement?
   sm= ±(0.01’+(3/1,000,000 x 3500)) =
    ± 0.021’
Calibration or “Don’t shoot
yourself in the foot.”
  Leica instruments should be serviced
  every 18 months.
  EDM’s should be calibrated every six
  months
  Tribrachs should be adjusted every six
  months, or more often as needed.
  Levels pegged every 90 days
Is It a Mistake or an Error?
  Mistake - Blunder in reading, recording or
  calculating a value.
  Error - The difference between a measured
  or calculated value and the true value.
Blunder


 a gross error or mistake resulting usually from
stupidity, ignorance, or carelessness.
Blunder
          • Setup over wrong point
          • Bad H.I.
          • Incorrect settings in equipment
Types of Errors
  Systematic
  Random

  An error is the difference between a
  measured value and the true value. Later we
  will compare this to the definition of residual
Systematic

   an error that is not determined by chance
  but is introduced by an inaccuracy (as of
  observation or measurement) inherent in
  the system
Systematic
                     • Glass with wrong offset
                     • Poorly repaired chain
                     • Imbalance between level
                       sightings




             Each measurement made
             with the tape is 0.1' shorter
             than recorded.
Random

 an error that has a random distribution
 and can be attributed to chance.
 without definite aim, direction, or
 method
 Random
• Poorly adjusted tribrach
• Inexperienced Instrument
  operator
• Inaccuracy in equipment
Nature of Random Errors
  A plus or minus error will occur with the
  same frequency
  Minor errors will occur more often than
  large ones
  Very large errors will rarely occur (see
  mistake)
Normal Distribution Curve #1




  A plus or minus error will occur with the same
  frequency, so
  Area within curve is equal on either side of the mean
Normal Distribution Curve #2




  Minor errors will occur more often than large
  ones, so
  The area within one standard deviation (s) of
  the mean is 68.3% of the total
Normal Distribution Curve #3




  Very large errors will rarely occur, so
  The total area within 2s of the mean is 95%
  of the sample population
     Histograms, Sigma, & Outliers




                                           MEAN
                2s              1s                  1s               2s


Histogram: Plot of the Residuals                         Bell shaped curve
                                \                        /

                                                                                  Outlier
                                                                                        \

 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5          0.5 1.0 1.5   2.0 2.5   3.0 3.5 4.0
  1 s : 68% of residuals
        must fall inside area        Residuals
  2 s 95 % of residuals
     must fall inside area
Measurement Components

  All measurements consist of two
  components: the measurement and
  the uncertainty statement.
            1,320.55’ ± 0.05’
  The uncertainty statement is not a
  guess, but is based on testing of
  equipment and methods.
Accuracy Vs. Precision
  Precision - agreement among readings of the
  same value (measurement). A measure of
  methods.
  Accuracy - agreement of observed values
  with the “true value”. A measure of results.
Measurement Analysis
Determining Measurement Uncertainties
Determining Uncertainty
  Uncertainty - the positive and negative range
  of values expected for a recorded or
  calculated value, i.e. the ± value (the second
  component of measurements).
Your Assignment
  Measure a line that is very close to 1000 feet
  long and determine the accuracy of your
  measurement.
  Equipment: 100’ tape and two plumb bobs.
  Terrain: Basically level with 2’ high brush.
  Environment: Sunny and warm.
  Personnel: You and me.
Planning the Project
  Test for errors in one tape length.
  Measure 1000 foot distance using same
  methods as used in testing.
  Determine accuracy of 1000 foot
  distance.
Test Data Set
Measured distances:
  99.96           100.02
  100.04          100.00
  100.00          99.98
  100.02          100.00
  99.98           100.00
Averages
  “Measures of Central Tendency”
     The value within a data set that tends to exist at
      the center.
  Arithmetic Mean
  Median
  Mode
Averages
  Most commonly used is Arithmetic Mean
  Considered the “most probable value”
          meas.
  mean 
           n
  n = number of observations
  Mean = 1000 / 10
  Mean = 100.00’
Residuals
  The difference between an individual reading
  in a set of repeated measurements and the
  mean
  Residual (n) = reading - mean
  Sum of the residuals squared (Sn2) is used in
  future calculations
Residuals
 Calculating Residuals (mean = 100.00’):
 Readings     residual     residual2
 99.96’         -0.04        0.0016
 100.02’       +0.02         0.0004
 100.04’       +0.04         0.0016
 100.00’          0             0
 100.00’          0             0
 99.98’         -0.02        0.0004
 100.02’       +0.02         0.0004
 100.00’          0             0
 99.98’         -0.02        0.0004
 100.00’          0             0
                    Sn2   = 0.0048
Standard Deviation
 The Standard Deviation is the ± range
 within which 68.3% of the residuals will fall
 or …
 Each residual has a 68.3% probability of
 falling within the Standard Deviation range
 or …
 If another measurement is made, the
 resulting residual has a 68.3% chance of
 falling within the Standard Deviation range.
Standard Deviation Formula
                               n 2
   Standard deviation (σ )  
                               n 1



         0.0048
     s         0.023'
            9
Standard Deviation
  Standard Deviation is a comparison of the
  individual readings (measurements) to the
  mean of the readings, therefore…
  Standard Deviation is a measure of….
Standard Deviation
  Standard Deviation is a comparison of the
  individual readings (measurements) to the
  mean of the readings, therefore…
  Standard Deviation is a measure of….

            PRECISION!
Standard Deviation of the
Mean
  This is an uncertainty statement regarding the mean
  and not a randomly selected individual reading as is
  the case with standard deviation.
  Since the individual measurements that make up the
  mean have error, the mean also has an associated
  error.
  The Standard Deviation of the Mean is the ± range
  within which the mean falls when compared to the
  “true value”, therefore the Standard Deviation of the
  Mean is a measure of ….
Standard Deviation of the
Mean
 This is an uncertainty statement regarding
 the mean and not a randomly selected
 individual reading as is the case with
 standard deviation.
 Since the individual measurements that
             ACCURACY!
 make up the mean have error, the mean
 also has an associated error.
 The Standard Error of the Mean is the ±
 range within which the mean falls when
 compared to the “true value”, therefore the
 Standard Deviation of the Mean is a
 measure of ….
Standard Deviation of the
Mean
                                        s
  Standard Error of the Mean (sm)  
                                        n
           0.023
  sm              0.007'
            10
  Distance = 100.00’±0.007’
          (1s Confidence level)
Probable Error
Besides the value of s =68.3%, other
 error values are used by statisticians
An error value of 50% is called Probable
 Error and is shown as “E” or “E50”

           E50= (0.6745)s
90% & 95% Probable Error
A 50% level of certainty for a measure of
  precision or accuracy is usually unacceptable.
90% or 95% level of certainty is normal for
  surveying applications

 E90  (1.6449 s)         E 95  (1.96s )

              E90                       E 95
 E90 m                    E95 m  
                 n                        n
95% Probable Error
 E 95  (2s )  (2  0.023)  0.046'

            E 95    0.046
 E 95 m                0.015 '
              n       10

 Distance = 100.00’±0.015’
       (2s Confidence Level)
Meaning of E95

  “If a measurement falls outside
   of two standard deviations, it
     isn’t a random error, it’s a
               mistake!”

                         Francis H. Moffitt
How Errors Propagate

  Error in a Series
  Errors in a Sum
  Error in Redundant
  Measurement
Error in a Series
  Describes the error of multiple measurements
  with identical standard deviations, such as
  measuring a 1000’ line with using a 100’
  chain.



          Eseries   E n
Error in a Sum


 Esum   E1  E2  E3  ...  En
                   2        2        2               2




  Esum is the square root of the sum the errors of each
  of the individual measurements squared
  It is used when there are several measurements with
  differing standard deviations
Exercise for Errors in a Sum
  Assume a typical single point occupation. The
  instrument is occupying one point, with
  tripods occupying the backsight and
  foresight.
  How many sources of random error are there
  in this scenario?
Exercise for Errors in a Sum
  There are three tribrachs, each with its own
  centering error that affects angle and distance
  Each of the two distance measurements have errors
  The angle turned by the instrument has several
  sources of error, including poor leveling and parallax
Error in Redundant
Measurements
 If a measurement is repeated multiple
 times, the accuracy increases, even if
 the measurements have the same value

                         E
        Ered.meas.  
                         n
Sample of Redundancy
                 Expected accuracy of a given number of unique observations at a given baseline
                            length, at the 95% confidence interval and stated in mm.
Base                       Horizontal                     Base                        Vertical
Line   Shots                                              Line   Shots
(KM)   1 2        3   4    5    6    7    8    9    10    (KM)   1 2       3    4    5    6    7    8    9    10
  0    20   14   12   10    9    8    8    7    7    6       0   40   28   23   20   18   16   15   14   13   13
  1    22   16   13   11   10    9    8    8    7    7       1   42   30   24   21   19   17   16   15   14   13
  2    24   17   14   12   11   10    9    8    8    8       2   44   31   25   22   20   18   17   16   15   14
  3    26   18   15   13   12   11   10    9    9    8       3   46   33   27   23   21   19   17   16   15   15
  4    28   20   16   14   13   11   11   10    9    9       4   48   34   28   24   21   20   18   17   16   15
  5    30   21   17   15   13   12   11   11   10    9       5   50   35   29   25   22   20   19   18   17   16
  6    32   23   18   16   14   13   12   11   11   10       6   52   37   30   26   23   21   20   18   17   16
  7    34   24   20   17   15   14   13   12   11   11       7   54   38   31   27   24   22   20   19   18   17
  8    36   25   21   18   16   15   14   13   12   11       8   56   40   32   28   25   23   21   20   19   18
  9    38   27   22   19   17   16   14   13   13   12       9   58   41   33   29   26   24   22   21   19   18
 10    40   28   23   20   18   16   15   14   13   13      10   60   42   35   30   27   24   23   21   20   19
Eternal Battle of Good Vs. Evil
   With Errors of a Sum (or Series), each
  additional variable increases the total
  error of the network
  With Errors of Redundant
  Measurement, each redundant
  measurement decreases the error of
  the network.
Sum vs. Redundancy
 Therefore, as the network becomes
 more complicated, accuracy can be
 maintained by increasing the number of
 redundant measurements
Error Ellipses
 Used to described the accuracy of a
 measured survey point.
 Error Ellipse is defined by the dimensions of
 the semi-major and semi-minor axis and the
 orientation of the semi-major axis
 Assuming standard errors, the
 measurements have a 39.4% chance of
 falling within the Error Ellipse
 E95 = ± 2.447s
Coordinate Standard
Deviations and Error Ellipses
Coordinate Standard Deviations and Error Ellipses:

Point     Northing            Easting                N SDev   E SDev
12        583,511.320         2,068,582.469          0.021    0.017




         Northing Standard Deviation
                                       {   }
                                               Easting Standard Deviation
Positional Accuracy vs.
Precision Ratio
Or, “How good is one error ellipse
compared to all those others?
Introduction to Adjustments
Adjustment - “A process designed to remove
  inconsistencies in measured or computed
  quantities by applying derived corrections to
  compensate for random, or accidental errors,
  such errors not being subject to systematic
  corrections”.

                               Definitions of Surveying and
                                          Associated Terms,
                                                1989 Reprint
Introduction to Adjustments
  Common Adjustment methods:
    Compass Rule

    Transit Rule

    Crandall's Rule

    Rotation and Scale (Grant Line Adjustment)

    Least Squares Adjustment
Weighted Adjustments
  Weight - “The relative reliability (or worth) of
  a quantity as compared with other values of
  the same quantity.”


                               Definitions of Surveying and
                                          Associated Terms,
                                                1989 Reprint
Weighted Adjustments
  The concept of weighting measurements to
  account for different error sources, etc. is
  fundamental to a least squares adjustment.
  Weighting can be based on error sources, if
  the error of each measurement is different, or
  the quantity of readings that make up a
  reading, if the error sources are equal.
Weighted Adjustments
  Formulas:
              W  (1  E2) (Error Sources)

              C  (1  W) (Correction)

              W  n (repeated measurements of
               the same value)

              W  (1  n) (a series of
                      measurements)
Weighted Adjustments
             A = 4324’36”, 2x
             B = 4712’34”, 4x
   A         C = 8922’20”, 8x
             Perform a weighted
             adjustment based on the
             above data



   C         B
ANGLE     No. Meas    Mean ValueRel. Corr.      Corrections                  Adjusted Value


 A        2           43 24’ 36”             4/
                                                   4 or
                                                          4/
                                                               7
                                                                   4/
                                                                        7X   30” = 17”   43 24’ 53”
 B        4           47 12’ 34”             2/
                                                   4 or
                                                          2/
                                                               7
                                                                   2/
                                                                        7X   30” = 09”   47 12’ 43”
 C        8           89 22’ 20”             1/
                                                   4 or
                                                          1/
                                                               7
                                                                   1/
                                                                        7X   30” = 04”   89 22’ 24”
          TOTALS      17959’ 30”             7/
                                                   4 or
                                                          7/
                                                               7                = 30” 180 00’ 00”




 The relative correction for the three angles are 1 : 2 : 4, the inverse proportion to
 the number of turned angles. This is the first set of relative corrections.
 The sum of the relative corrections is 1 + 2 + 4 = 7 , This is used as the
 denominator for the second set of corrections. The sum of the second set of
 relative corrections shall always equal 1. The second set is used for corrections.
Weighted Adjustments
  BM “B”
  Elev. = 102.0       +7.8’, 2 mi.

                                  BM “NEW”

                                             +6.2’, 10 mi.

                  +10.0’, 4 mi.

    BM “A”                                            BM “C”
    Elev. = 100.0’                                    Elev. = 104.0’
Introduction to Least Squares
Adjustment
Simple Examples
What Least Squares Is ...
  A rigorous statistical adjustment of survey
  data based on the laws of probability and
  statistics
  Provides simultaneous adjustment of all
  measurements
  Measurements can be individually weighted
  to account for different error sources and
  values
  Minimal adjustment of field measurements
What is Least Squares?
 A Least Squares adjustment distributes random errors
 according to the principle that the Most Probable Solution
 is the one that minimizes the sums of the squares of the
 residuals.
 This method works to keep the amount of adjustment to
 the observations and, ultimately the ‘movement’ of the
 coordinates to a minimum.
Least Squares Example
  Arithmetic Mean
  Using Least squares to prove a simple
  arithmetic mean solution
Least Squares Example
A point is measured for location 3 times. The measurements give the
following NE coordinates:

a. 0,0            b. 0,5         c.5, 0


c 5,0    
                                          What is the best solution for
                                          an average?
              ?                           How can you prove it?

                      
a 0,0                  b 0,5
Student exercise
  GROUP #1               GROUP #2
  Determine the sum      Determine sum of
  of the squares from    the squares from

X=2.5, Y=2.5            Mean X, Mean Y
                        (1.667, 1.667)
Solution
If ?= 1.667, 1.667, then Distance a-?= 2.357, b-?= 3.727, c-?=3.727


c 5,0   
                                   N= (0 + 0 +5)  3 = 1.667
                                   E= (0 + 5 +0)  3 = 1.667
                                   2.357² + 3.727² + 3.727² = 33.333
                  ?


                              
a 0,0                              b 0,5
What Least Squares Isn’t ...
  A way to correct a weak strength of figure
  A cure for sloppy surveying - Garbage in /
  Garbage out
  The only adjustment available to the land
  surveyor
Least Squares
  Least Squares Should Be Used for
  The Adjustment Of:      Collected By:
  Conventional Traverse   Theodolite & Chain
  Control Networks        Total Stations
  GPS Networks            GPS Receivers
  Level Networks          Levels
  Resections              EDMs
 Least Squares
                       A                       B
       Observed                    E
       1st Iteration

       2nd Iteration           G
                                       F

What happens?
                                           C
 Iterative Process
                           D
 Each iteration applies adjustments to
 observations, working for best solution
 Adjustments become smaller with each
 successive iteration
Least Squares
The Iterative Process
1 Creates a calculated observation for each field
    observation by inversing between approximate
    coordinates.
2   Calculates a "best fit" solution of observations and
    compares them to field observations to compute
    residuals.
3   Updates approximate coordinate values.
4   Calculates the amount of movement between the
    coordinate positions prior to iteration and after
    iteration.
5   Repeats steps 1 - 4 until coordinate movement is no
    greater than selected threshold.
Least Squares
Four component that need to be addressed
prior to performing least squares adjustment
1   Errors
2   Coordinates
3   Observations
4   Weights
Errors
  Blunder - Must be removed
  Systematic - Must be Corrected
  Random - No action needed
Coordinates
Because the Least Squares process begins by
calculating inversed observations approximate
coordinate values are needed.
     1 Dimensional Network (Level Network) - Only
      1 Point.
     2 Dimensional Network - All Points Need
      Northing and Easting.
     3 Dimensional Network - All Points Need
      Northing, Easting, and Elevation. (Except for
      adjustments of GPS baselines.)
Weights
 •   Each Observation Requires an Associated Weight
       Weight = Influence of the Observation on Final
        Solution
          • Larger Weight - Larger Influence

       Weight = 1/σ2
          • σ = Standard Deviation of the Observation

       The Smaller the Standard Deviation the Greater the
        Weight



      σ = 0.8  Weight = 1/0.82 = 1.56            More
                                                  Influence
      σ = 2.2  Weight = 1/2.22 = 0.21            Less
                                                  Influence
     Methods of Establishing Weights
            •   Observational Group
                       •   Least Desirable Method
 Good for combining
 Observations from •        Example: All Angles Weighted at the Accuracy of
 different classes of      the Total Station
 instruments.
            •   Each Observation Individually Weighted
                       •   Best Method
Good for projects
where standard         •    Standard Deviation of Field Observations Used as
deviation is calculated    the Weight of the Mean Observation
for each observation.
            •   Combination of Types
                       •   Assigns the Least weight possible for each observation
Least Squares
 If you remember nothing else about least squares today,
                    remember this!
     Least Squares Adjustment Is a Two Part Process
    1   - Unconstrained Adjustment
          Analyze the Observations, Observations
           Weights, and the Network
    2   - Constrained Adjustment
          Place Coordinate Values on All Points in the
           Network
Unconstrained Adjustment
   •   Also Called
        Minimally Constrained Adjustment
        Free Adjustment
   •   Used to Evaluate
        Observations
        Observation Weights
        Relationship of All Observations
   •   Only fix the minimum required points
       Flow Chart
         Start
Field Observations
                                                            Edit Field Data               Perform
  Setup                      Field Data
                                                            • Remove Blunders        Unconstrained Least
   Observation                 Needs            Yes
                                                            • Correct Systematic     Squares Adjustment
   Standard Deviation         Editing?
                                                              Errors


                                                No


                              Print out
                           Unconstrained                        Statistics                Analyze
                        Adjustment Statistics   No               Indicate                Adjustment
                                                                Problems                  Statistics



                                                                   Yes
       Perform
                          Constrain Fixed                                               Modify Input
   Constrained Least
                          Control Points                                                   Data
  Squares Adjustment



    Print out Final                                                   Least Square
  Coordinate Values            Performed by
                                                Decision Step         Adjustment
   for All Points in               User
                                                                       Software
     Adjustment

             Finish
Analyze the Statistical Results

There are 4 main statistical areas that need to be looked
  at:
   1.   Standard deviation of unit weight
   2.   Observation residuals
   3.   Coordinate standard deviations and error ellipses
   4.   Relative errors

A 5th statistic that is sometimes available that should be
  looked at:
   Chi-square Test
Standard Deviation of
Unit Weight
Also Called
   Standard Error of Unit Weight
   Error Total
   Network Reference Factor
The Closer This Value Is to 1.0 the Better
   The Acceptable Range Is ? to ?
   > 1.0 - Observations Are Not As Good As
    Weighted
   < 1.0 - Observations Are Better Than Weighted
Observation Residuals
•   Amount of adjustment applied to observation to
    obtain best fit
                     This is the residual that is being minimized
•   Used to analyze each observation
•   Usually flags excessive adjustments (Outliers)
     (Star*net flags observations adjusted more
      than 3 times the observations weight)
•   Large residuals may indicate blunders
    Observation Residuals
                Site      Observation          Residual        S Dev.     Flag
             10-11-12    214  33’ 17.2”         1.7”           1.2”
             11-12-13    174  16’ 43.8”         7.2”           1.9”       *
             12-13-14    337  26’ 08.6          2.1”           1.3”




                                                                                  Outlier


-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5    0     0.5 1.0 1.5    2.0 2.5   3.0 3.5 4.0
Coordinate Standard
Deviations and Error Ellipses
  •   Coordinate standard deviations represent the
      accuracy of the coordinates
  •   Error ellipses are a graphical representation of the
      standard deviations
  •   The better the network the rounder the error
      ellipses
  •   High standard deviations can be found in networks
      with a good standard deviation of unit weight and
      well weighted observations due to effects of the
      network geometry
Relative Errors

                  Predicted amount of error
                  that can be expected to
                  occur between points when
                  an observation is made in
                  the network.
Chi-square Test
 noun: (ki'skwâr) a statistic that is a sum of terms
 each of which is a quotient obtained by dividing the
 square of the difference between the observed and
 theoretical values of a quantity by the theoretical
 value
 In other words: A statistical analysis of the statistics.
 10 coins 6 to 4 (6-5) or 100 coins 60-40 (60-50)
Least Squares Examples
  Straight Line Best Fit
Straight Line Best Fit
Straight Line Best Fit
Straight Line Best Fit
Straight Line Best Fit
Least Squares “Rules”
  Redundancy of survey data strengthens
  adjustment
  Error Sources must be determined correctly
  Each adjustment consists of two “parts”:
     Minimally Constrained Adjustment
     Fully Constrained Adjustment
Star*Net Adjustment Software
   A Tour of the Software Package




 Star*Net
Sample Network Adjustment
   A Simple 2D Network Adjustment




Star*Net
Sample Network Adjustments
    A 3D “Grid” Adjustment using GPS and
    Conventional Data




 Star*Net
Beyond Control Surveys
  Other Uses for Least Squares
  Adjustments / Analysis
Questions & Discussion
Systematic vs. Random Error
  Systematic Error - An error whose
  magnitude and algebraic sign can be
  determined or corrected by procedure.
  Example: temperature correction for steel
  tape or balanced level distances.
  Random Error - An error whose magnitude
  and algebraic sign cannot be determined.
  They tend to be small and compensating.
  Measurement Analysis is the study of
  random errors.
What Least Squares Is...
  Adjustment report provides details of survey
  measurements
  A TOOL to be used by the Surveyor to
  complement his knowledge of measurements
Random Error Propagation
   Error in a Sum (Esum) =
   ±(E12 + E22 + E32 + ….. + En2)1/2
   Error in a Series (Eseries) =
   ±(E (n)1/2)
   Error in Redundant Measurement
       (Ered.) = ±(E / (n)1/2)

 Esum   E1 2  E2 2  E3 2  ...  En 2
  Eseries   E n
                   E
  Ered.meas.  
                    n
Instrument Specifications
  Angle Measurement:
   Stated Accuracy vs. Display
   What is DIN 18723?
   What is the True Accuracy of a
    Measured Angle?

								
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