Measure

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					Intro to Measure.xls

This workbook demonstrates the Measurement Box model.

Measuring simulates the measurement process.
LiveSample draws a sample of 25 measurements.
The EstimatingSDBox sheet makes the point that the SD of the measurements is a good estimate of the SD of the box.
DeadSample is a single sample of 25 measurements with a computed 95% confidence interval.
The MCSim sheet, accessible from a button on LiveSample , performs Monte Carlo simulations of the process of taking 25 me
Q&A contains questions.
estimate of the SD of the box.

ations of the process of taking 25 measurements.
True Distance                 107.1165 miles             red means you CANNOT really see these values
Random Draw                 #NAME?     miles             black means you CAN observe these values
Observed Measurement        #NAME?     miles

Hit F9 to recalculate the sheet and note how the Observed Measurement changes because the Random Draw changes.

Click the Take a Measurement button to build your own sample of measured distances.
Note that the observed distances (starting in D18) are numbers, but they are composed of the fixed true distance plus a random




      Observation         True Distance Random Draw Observed Measurement
                      1        107.1165       2.2585               109.37
                      2        107.1165       0.9042               108.02
                      3        107.1165       0.9540               108.07
                      4        107.1165      -0.0859               107.03
                      5        107.1165       0.3234               107.44
                      6        107.1165       0.6566               107.77
                      7        107.1165      -1.4081               105.71
                      8        107.1165      -0.0729               107.04
                      9        107.1165       0.5386               107.66
                     10        107.1165      -0.4923               106.62
                     11        107.1165       1.0808               108.20
                     12        107.1165      -0.7814               106.34
                     13        107.1165      -0.0543               107.06
                     14        107.1165      -1.9024               105.21
                     15        107.1165      -0.7123               106.40
                     16        107.1165      -0.5331               106.58
                     17        107.1165       1.3162               108.43
                     18        107.1165       2.0596               109.18
                     19        107.1165      -0.7786               106.34
                     20        107.1165       1.3466               108.46
                     21        107.1165      -0.4181               106.70
                     22        107.1165      -0.7119               106.40
                     23        107.1165      -0.0627               107.05
                     24        107.1165       2.1900               109.31
                     25        107.1165       0.6085               107.72
dom Draw changes.


rue distance plus a random draw.
                                                    LiveSample


This worksheet is designed to illustrate the Measurement Box model applied to the
Distance Between Two Peaks Measurement Problem. 25 observations are
simulated, according to parameters set in red in the worksheet. The True Distance
being Measured and the SD of the Measuring Device are given in red. The actual
results of the experiments are given below in plain black.
In this spreadsheet, the Measurement Box is a correct model of the data
generation process. The simulated measurements are indeed unbiased,
independent of each other, and alike.

Scroll down (if needed) to begin.




The True Model
                                     Precision of
 True Distance                       Measuring
Being Measured        107.1165       Instrument            1.00
  Hit F9 to simulate another 100                        Measured        Residuals
  Observation       True Distance       Error        Distance (miles)    (miles)
        1             107.1165        #NAME?            #NAME?          #NAME?       sample average
        2             107.1165        #NAME?            #NAME?          #NAME?          sample SD
        3             107.1165        #NAME?            #NAME?          #NAME?
        4             107.1165        #NAME?            #NAME?          #NAME?      average of residuals
        5             107.1165        #NAME?            #NAME?          #NAME?        SD of residuals
        6             107.1165        #NAME?            #NAME?          #NAME?      Errors are not the same as residuals.
        7             107.1165        #NAME?            #NAME?          #NAME?      the error box model is the difference betw
        8             107.1165        #NAME?            #NAME?          #NAME?      observed value and the true value. Erro
                                                                                    CANNOT be observed. A residual is the
        9             107.1165        #NAME?            #NAME?          #NAME?      between the predicted value and the obs
       10             107.1165        #NAME?            #NAME?          #NAME?      value. In our example, the predicted valu
       11             107.1165        #NAME?            #NAME?          #NAME?      sample average. Residuals can be thou
                                                                                    estimates of the errors, but they are not t
       12             107.1165        #NAME?            #NAME?          #NAME?      thing.
       13             107.1165        #NAME?            #NAME?          #NAME?
       14             107.1165        #NAME?            #NAME?          #NAME?      But notice how the spread of the observa
                                                                                    equals the spread of the residuals and h
       15             107.1165        #NAME?            #NAME?          #NAME?      this estimated spread is to the true SD (i
       16             107.1165        #NAME?            #NAME?          #NAME?
       17             107.1165        #NAME?            #NAME?          #NAME?
       18             107.1165        #NAME?            #NAME?          #NAME?
       19             107.1165        #NAME?            #NAME?          #NAME?
       20             107.1165        #NAME?            #NAME?          #NAME?
       21             107.1165        #NAME?            #NAME?          #NAME?
       22             107.1165        #NAME?            #NAME?          #NAME?
       23             107.1165        #NAME?            #NAME?          #NAME?


                                                     Page 5
                         LiveSample


24   107.1165   #NAME?       #NAME?   #NAME?
25   107.1165   #NAME?       #NAME?   #NAME?




                          Page 6
                                                 LiveSample




                #NAME?
                #NAME?


                #NAME?
                #NAME?
 are not the same as residuals. An error in
or box model is the difference between the
ed value and the true value. Errors
OT be observed. A residual is the difference
 n the predicted value and the observed
 In our example, the predicted value is the
  average. Residuals can be thought of as
 es of the errors, but they are not the same


 ce how the spread of the observations
the spread of the residuals and how close
imated spread is to the true SD (in cell D16).




                                                  Page 7
      Parameters
True Value       4.00
SD Errors        1.00

Sample
Average        #NAME?


Observation    Error   Measurement Residual   The point of this worksheet is that the SD of the measuremen
           1    #NAME?   #NAME?     #NAME?    of the SD of the box. The argument proceeds in two steps.
           2    #NAME?   #NAME?     #NAME?    (1) In the Measurement Model, the Sample SD of the measu
           3    #NAME?   #NAME?     #NAME?    Sample SD of the errors themselves. This is due to a propert
           4    #NAME?   #NAME?     #NAME?    SDs of two lists, differ only by a constant, are the same. In th
           5    #NAME?   #NAME?     #NAME?
           6    #NAME?   #NAME?     #NAME?    Measurement = Error + True Value
           7    #NAME?   #NAME?     #NAME?    Measurement = Error + 4.00.
           8    #NAME?   #NAME?     #NAME?
           9    #NAME?   #NAME?     #NAME?    Thus every Measurement in the list is equal to a constant plus
          10    #NAME?   #NAME?     #NAME?    Since the numbers in the Measurement list and the numbers i
          11    #NAME?   #NAME?     #NAME?    differ by the constant 4.00, they have the same SD.
          12    #NAME?   #NAME?     #NAME?
          13    #NAME?   #NAME?     #NAME?    (2) The Sample SD of the errors is a good estimate of the (po
          14    #NAME?   #NAME?     #NAME?    This latter point can be demonstrated via a Monte Carlo simul
          15    #NAME?   #NAME?     #NAME?
          16    #NAME?   #NAME?     #NAME?
          17    #NAME?   #NAME?     #NAME?    NOTE: It is also the case that the SD of the Residuals is the s
          18    #NAME?   #NAME?     #NAME?    The reason is that the Residuals differ from their respective M
          19    #NAME?   #NAME?     #NAME?    namely the Sample Average:
          20    #NAME?   #NAME?     #NAME?
    SD          #NAME?   #NAME?    #NAME?     Residual = Measurement - Sample Average
                                               #NAME?

                                              Thus the SD of the Residuals is the same as the SD of the Me

                                              The calculations in Row 27 confirm this claims by example.
 t the SD of the measurements is a good estimate
ent proceeds in two steps.
the Sample SD of the measurements is exactly equal to the
ves. This is due to a property of the SD which says that the
 onstant, are the same. In this example, for every observation




 ist is equal to a constant plus the Error.
rement list and the numbers in the Error list
have the same SD.

is a good estimate of the (population) SD of the Errors.
ated via a Monte Carlo simulation.


e SD of the Residuals is the same as the SD of the Measurments.
 differ from their respective Measurements by a constant,




he same as the SD of the Measurements.

rm this claims by example.
                                           DeadSample


The Data from ONE SIMPLE RANDOM SAMPLE
                                The data in this sheet are dead. They are the same as the example in Section 11
              Distance Measured
Observation         (miles)                    APPLYING THE BOX MODEL
     1              107.23
     2              106.41             106.652 sample average
     3              105.97               1.043 sample SD
     4              106.13
     5              108.35               0.209 estimated SE of the sample average
     6              105.60
     7              105.55
     8              105.64      The estimate for the True Distance is the sample average,
     9              106.80             106.652 miles.
    10              105.57      The typical discrepancy between this estimate and the
    11              108.77      unobserved True Distance is
    12              108.56               0.209 miles.
    13              108.65
    14              105.99
    15              105.48
    16              106.83
    17              107.12
    18              105.51
    19              106.19
    20              106.71
    21              106.59
    22              107.71
    23              106.82
    24              106.18
    25              105.95




                                             Page 10
                                     DeadSample



me as the example in Section 11.2.




                                      Page 11
Q&A for Measure.xls

1. In the text, we say that the measurement box model contains three important assumptions
about the data generating process:
            i. The measurement process is unbiased.
            ii. Each measurement is independent of the other measurements.
            iii. The measurements are all alike, that is they are identically distributed.

In the LiveSample sheet you will notice that we simulate the errors using the formula
=NormalRandom(0,Error_SD)
Every error for each of the 25 observations is produced this way.
a) How could you change the LiveSample sheet to violate assumption i. above?
b) In terms of the language of the box model, what have you changed to violate assumption i.?
c) How could you change the LiveSample sheet to violate assumption iii. above?
d) In terms of the language of the box model, what have you changed to violate assumption iii.?

2. Verify for yourself that the SE of the Sample Average depends directly on the precision of the measuring instrument.
Go to the LiveSample sheet and set the value in cell D16 (labeled as "Precision of Measuring Instrument")
to 4. Run a Monte Carlo simulation in the MCSim sheet and take a picture of the results. Return to the
MCSim sheet and change the precision to 1. Run another Monte Carlo simulation and compare the empirical SDs
from the two simulations. The empriical SDs approximate the exact SE, the spread of the probability histogram
for the Sample Average. What seems to be the relationship between the spread of the error box and the
spread of the probability histogram?

				
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posted:12/1/2011
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