Next Page Previous Page Home Tools & Aids Search Handbook
1. Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
1.4.2.6. Filter Transmittance

1.4.2.6.3.

Quantitative Output and Interpretation
Summary Statistics As a first step in the analysis, a table of summary statistics is computed from the data. The following table, generated by Dataplot, shows a typical set of statistics. 
 
                                SUMMARY
 
                     NUMBER OF OBSERVATIONS =       50
 
 
***********************************************************************
*        LOCATION MEASURES         *       DISPERSION MEASURES        *
***********************************************************************
*  MIDRANGE     =   0.2002000E+01  *  RANGE        =   0.1399994E-02  *
*  MEAN         =   0.2001856E+01  *  STAND. DEV.  =   0.4291329E-03  *
*  MIDMEAN      =   0.2001638E+01  *  AV. AB. DEV. =   0.3480196E-03  *
*  MEDIAN       =   0.2001800E+01  *  MINIMUM      =   0.2001300E+01  *
*               =                  *  LOWER QUART. =   0.2001500E+01  *
*               =                  *  LOWER HINGE  =   0.2001500E+01  *
*               =                  *  UPPER HINGE  =   0.2002100E+01  *
*               =                  *  UPPER QUART. =   0.2002175E+01  *
*               =                  *  MAXIMUM      =   0.2002700E+01  *
***********************************************************************
*       RANDOMNESS MEASURES        *     DISTRIBUTIONAL MEASURES      *
***********************************************************************
*  AUTOCO COEF  =   0.9379919E+00  *  ST. 3RD MOM. =   0.6191616E+00  *
*               =   0.0000000E+00  *  ST. 4TH MOM. =   0.2098746E+01  *
*               =   0.0000000E+00  *  ST. WILK-SHA =  -0.4995516E+01  *
*               =                  *  UNIFORM PPCC =   0.9666610E+00  *
*               =                  *  NORMAL  PPCC =   0.9558001E+00  *
*               =                  *  TUK -.5 PPCC =   0.8462552E+00  *
*               =                  *  CAUCHY  PPCC =   0.6822084E+00  *
***********************************************************************
Location One way to quantify a change in location over time is to fit a straight line to the data set using the index variable X = 1, 2, ..., N, with N denoting the number of observations. If there is no significant drift in the location, the slope parameter should be zero. For this data set, Dataplot generates the following output: 
 LEAST SQUARES MULTILINEAR FIT
       SAMPLE SIZE N       =       50
       NUMBER OF VARIABLES =        1
       NO REPLICATION CASE
  
  
               PARAMETER ESTIMATES           (APPROX. ST. DEV.)    T VALUE
        1  A0                   2.00138       (0.9695E-04)       0.2064E+05
        2  A1       X          0.184685E-04   (0.3309E-05)        5.582
  
       RESIDUAL    STANDARD DEVIATION =        0.3376404E-03
       RESIDUAL    DEGREES OF FREEDOM =          48
The slope parameter, A1, has a t value of 5.6, which is statistically significant. The value of the slope parameter is 0.0000185. Although this number is nearly zero, we need to take into account that the original scale of the data is from about 2.0012 to 2.0028. In this case, we conclude that there is a drift in location, although by a relatively minor amount.
Variation One simple way to detect a change in variation is with a Bartlett test after dividing the data set into several equal sized intervals. However, the Bartlett test is not robust for non-normality. Since the normality assumption is questionable for these data, we use the alternative Levene test. In partiuclar, we use the Levene test based on the median rather the mean. The choice of the number of intervals is somewhat arbitrary, although values of 4 or 8 are reasonable. Dataplot generated the following output for the Levene test. 
               LEVENE F-TEST FOR SHIFT IN VARIATION
                      (ASSUMPTION: NORMALITY)
  
 1. STATISTICS
       NUMBER OF OBSERVATIONS    =       50
       NUMBER OF GROUPS          =        4
       LEVENE F TEST STATISTIC   =   0.9714893
  
  
    FOR LEVENE TEST STATISTIC
       0          % POINT    =   0.0000000E+00
       50         % POINT    =   0.8004835
       75         % POINT    =    1.416631
       90         % POINT    =    2.206890
       95         % POINT    =    2.806845
       99         % POINT    =    4.238307
       99.9       % POINT    =    6.424733
  
  
          58.56597       % Point:    0.9714893
  
 3. CONCLUSION (AT THE 5% LEVEL):
       THERE IS NO SHIFT IN VARIATION.
       THUS: HOMOGENEOUS WITH RESPECT TO VARIATION.
In this case, since the Levene test statistic value of 0.971 is less than the critical value of 2.806 at the 5% level, we conclude that there is no evidence of a change in variation.
Randomness There are many ways in which data can be non-random. However, most common forms of non-randomness can be detected with a few simple tests. The lag plot in the 4-plot in the previous seciton is a simple graphical technique.

One check is an autocorrelation plot that shows the autocorrelations for various lags. Confidence bands can be plotted at the 95% and 99% confidence levels. Points outside this band indicate statistically significant values (lag 0 is always 1). Dataplot generated the following autocorrelation plot.

autocorrelation plot

The lag 1 autocorrelation, which is generally the one of most interest, is 0.93. The critical values at the 5% level are -0.277 and 0.277. This indicates that the lag 1 autocorrelation is statistically significant, so there is strong evidence of non-randomness.

A common test for randomness is the runs test. 

  
                      RUNS UP
 
           STATISTIC = NUMBER OF RUNS UP
               OF LENGTH EXACTLY I
 
   I         STAT     EXP(STAT)    SD(STAT)       Z
 
   1         1.0     10.4583      3.2170       -2.94
   2         3.0      4.4667      1.6539       -0.89
   3         1.0      1.2542      0.9997       -0.25
   4         0.0      0.2671      0.5003       -0.53
   5         0.0      0.0461      0.2132       -0.22
   6         0.0      0.0067      0.0818       -0.08
   7         0.0      0.0008      0.0291       -0.03
   8         1.0      0.0001      0.0097      103.06
   9         0.0      0.0000      0.0031        0.00
  10         1.0      0.0000      0.0009     1087.63
 
 
           STATISTIC = NUMBER OF RUNS UP
               OF LENGTH I OR MORE
 
   I         STAT     EXP(STAT)    SD(STAT)       Z
 
   1         7.0     16.5000      2.0696       -4.59
   2         6.0      6.0417      1.3962       -0.03
   3         3.0      1.5750      1.0622        1.34
   4         2.0      0.3208      0.5433        3.09
   5         2.0      0.0538      0.2299        8.47
   6         2.0      0.0077      0.0874       22.79
   7         2.0      0.0010      0.0308       64.85
   8         2.0      0.0001      0.0102      195.70
   9         1.0      0.0000      0.0032      311.64
  10         1.0      0.0000      0.0010     1042.19
 
 
                     RUNS DOWN
 
           STATISTIC = NUMBER OF RUNS DOWN
               OF LENGTH EXACTLY I
 
   I         STAT     EXP(STAT)    SD(STAT)       Z
 
   1         3.0     10.4583      3.2170       -2.32
   2         0.0      4.4667      1.6539       -2.70
   3         3.0      1.2542      0.9997        1.75
   4         1.0      0.2671      0.5003        1.46
   5         1.0      0.0461      0.2132        4.47
   6         0.0      0.0067      0.0818       -0.08
   7         0.0      0.0008      0.0291       -0.03
   8         0.0      0.0001      0.0097       -0.01
   9         0.0      0.0000      0.0031        0.00
  10         0.0      0.0000      0.0009        0.00
 
 
           STATISTIC = NUMBER OF RUNS DOWN
               OF LENGTH I OR MORE
 
 
   I         STAT     EXP(STAT)    SD(STAT)       Z
 
   1         8.0     16.5000      2.0696       -4.11
   2         5.0      6.0417      1.3962       -0.75
   3         5.0      1.5750      1.0622        3.22
   4         2.0      0.3208      0.5433        3.09
   5         1.0      0.0538      0.2299        4.12
   6         0.0      0.0077      0.0874       -0.09
   7         0.0      0.0010      0.0308       -0.03
   8         0.0      0.0001      0.0102       -0.01
   9         0.0      0.0000      0.0032        0.00
  10         0.0      0.0000      0.0010        0.00
 
 
           RUNS TOTAL = RUNS UP + RUNS DOWN
 
         STATISTIC = NUMBER OF RUNS TOTAL
              OF LENGTH EXACTLY I
 
   I         STAT     EXP(STAT)    SD(STAT)       Z
 
   1         4.0     20.9167      4.5496       -3.72
   2         3.0      8.9333      2.3389       -2.54
   3         4.0      2.5083      1.4138        1.06
   4         1.0      0.5341      0.7076        0.66
   5         1.0      0.0922      0.3015        3.01
   6         0.0      0.0134      0.1157       -0.12
   7         0.0      0.0017      0.0411       -0.04
   8         1.0      0.0002      0.0137       72.86
   9         0.0      0.0000      0.0043        0.00
  10         1.0      0.0000      0.0013      769.07
 
 
         STATISTIC = NUMBER OF RUNS TOTAL
               OF LENGTH I OR MORE
 
   I         STAT     EXP(STAT)    SD(STAT)       Z
 
   1        15.0     33.0000      2.9269       -6.15
   2        11.0     12.0833      1.9745       -0.55
   3         8.0      3.1500      1.5022        3.23
   4         4.0      0.6417      0.7684        4.37
   5         3.0      0.1075      0.3251        8.90
   6         2.0      0.0153      0.1236       16.05
   7         2.0      0.0019      0.0436       45.83
   8         2.0      0.0002      0.0145      138.37
   9         1.0      0.0000      0.0045      220.36
  10         1.0      0.0000      0.0014      736.94
 
 
          LENGTH OF THE LONGEST RUN UP         =    10
          LENGTH OF THE LONGEST RUN DOWN       =     5
          LENGTH OF THE LONGEST RUN UP OR DOWN =    10
 
          NUMBER OF POSITIVE DIFFERENCES =    23
          NUMBER OF NEGATIVE DIFFERENCES =    18
          NUMBER OF ZERO     DIFFERENCES =     8
Values in the column labeled "Z" greater than 1.96 or less than -1.96 are statistically significant at the 5% level. Due to the number of values that are much larger than the 1.96 cut-off, we conclude that the data are not random.
Distributional Analysis Since we rejected the randomness assumption, the distributional tests are not meaningful. Therefore, these quantitative tests are omitted. We also omit Grubbs' outlier test since it also assumes the data are approximately normally distributed.
Univariate Report It is sometimes useful and convenient to summarize the above results in a report. 
  
 Analysis for filter transmittance data
  
 1: Sample Size                           = 50
  
 2: Location
    Mean                                  = 2.001857
    Standard Deviation of Mean            = 0.00006
    95% Confidence Interval for Mean      = (2.001735,2.001979)
    Drift with respect to location?       = NO
  
 3: Variation
    Standard Deviation                    = 0.00043
    95% Confidence Interval for SD        = (0.000359,0.000535)
    Change in variation?
    (based on Levene's test on quarters
    of the data)                          = NO
  
 4: Distribution
    Distributional tests omitted due to
    non-randomness of the data
  
 5: Randomness
    Lag One Autocorrelation               = 0.937998
    Data are Random?
      (as measured by autocorrelation)    = NO
  
 6: Statistical Control
    (i.e., no drift in location or scale,
    data are random, distribution is 
    fixed, here we are testing only for
    normal)
    Data Set is in Statistical Control?   = NO
  
 7: Outliers?
    (Grubbs' test omitted)                = NO
Home Tools & Aids Search Handbook Previous Page Next Page