The histo application is used to calculate and display univariant statistics for differing data sets. For a sample population histo can be used to calculate basic statistics such as the mean, and variance. It may also be used to display the behavior of several different populations at once using stacked histograms and box and whisker plots. Histo can also be used to infer information about the population, i.e. is the population normal, using a ² (chi-squared) test or by plotting the frequency distribution as a probability plot.

The histo application is composed of three sections (Figure 6.1); the main menu- bar, the status and log text area, and the drawing or graph area. The menu-bar is used to select all histo commands, the log/status area is used by the program to report important messages or results, and the drawing area is the display area for the histograms.

Figure 6.1

The Main Menu:

[TOP] [SYNTAX]

File:

WARNING: if "histo.prf" already exists, you will be warned that it is about to be over- written. If you do not want the old version destroyed you must move it to a new file (e.g. the UNIX command mv histo.prf histo.old.prf would be sufficient). When you press OK, the old version will be over-written! This cannot be done from within the application. To rename the file you will have to execute the UNIX mv command from a UNIX prompt in another window.

If "histo.prf" does not exist in the current directory, it is created. This is an ASCII file and can be edited by the user. See Appendix C for details.

Print Setup:

[TOP] [SYNTAX]

Figure 6.2

[TOP] [SYNTAX]

Figure 6.3a and Figure 6.3b

Figure 6.4

Box and Whisker Plots:

Figure 6.5a and Figure 6.5b

Cumulative Distributions:

Figure 6.6a

1.0 - Cumulative Distributions:

Figure 6.6b

Probability Plots:

The Exceedence Type only affects the labeling on the X-axis. An Exceedence plot indicates the percentage of points which exceed a specified value. A Nonexceedence plot indicates the percentage of points which do not exceed a specified value. The appearance of the graphs otherwise is identical.

The Hazen and Weibull methods are two methods for determining the Rank Order of one data value within the data set. For further details see the histo Mathematics section (Equations 6-12 and 6-13), or refer to McCuen (1989).

It is common in nature that the distribution of a measured parameter has a log distribution (This is set with the Style:Transform Type:Log menu option discussed below). If this is the case, a Normal probability plot will show a curved line (Figure 6.7a). From the curved line, though one can say the data is not normally distributed, but little more. By Log transforming the data, if the line becomes "straight," the probability plot suggests that the data is log-normally distributed (Figure 6.7b).

Figure 6.7a and Figure 6.7b

Y-Axis Type:

NOTE: The transforms use the log base 10, not natural log.

Figure 6.8

[TOP] [SYNTAX]

Figure 6.9

Tabulated:

maximum, mean, median, variance, standard deviation, average deviation, skew, kurtosis, 2 test result

Figure 6.10

[TOP] [SYNTAX]

Line #1 = Mean Value Data Line
Line #2 = Standard Deviation Error-Bars
Line #3+ = Histogram bars

This dialog is described in the Graph:Style section of Chapter 5 (Figure 5.14).

Mesh:

[TOP] [SYNTAX]

[TOP] [SYNTAX]

When calculating the frequency distribution for the histogram, the column number (Pos.), X position, and frequency of occurrence within each histogram bar are reported to the log window. NOTE: All calculations are maintained in the log window, and the most recent are presented at the top of the log. An example is shown below for data1.dat using 10 columns:

	Calculation #6
 	   Number of Divisions = 10
	 Pos.       X       Frequency
	-------------------------------------
	     1:      -4.24          9
	     2:      -3.69         29
	     3:      -3.14         82
	     4:      -2.6         132
 	     5:      -2.05        138
 	     6:      -1.51         89
 	     7:      -0.959        56
	     8:      -0.413        27
	     9:       0.133         9
	    10:       0.68          5

[TOP] [SYNTAX]

[TOP] [SYNTAX]

[TOP]

There are three methods to load a file in histo. The first is to execute histo from the UNIX prompt and open the file from the menu, the second is to pass the file as a command line argument, and the third is to define the file name in the program preference file. To open a file from the main menu, execute histo from the UNIX prompt:

> histo

Once in the application, select the File:Open menu option. The pop-up dialog shown in Figure 5.2 will appear. Select the desired file. Once a file has been selected the graph of the data will be drawn. To open a file from the command line, enter at the UNIX prompt:

> histo [optional arguments] filename

For example:

> histo data1.dat

will open the graph file shown in Figure 6.8, and

> histo -tt 1 data1.dat

will open the same data file, but it will specify the x-axis is a log axis. NOTE: in both Figures 6.4 and 6.8, other variables then those passed on the command line were defined. These variables could have been set using the menus, or using a preference file (Appendix C). A preference file is used to define user preferred variable default values. Every time histo runs, it searches the current working directory for the file histo.prf. If it exists, histo reads the file and sets the variables as specified. This is the third way to open a file, because one of the arguments in the preference file is the name of the graph file.

[TOP]

Syntax:

histo [-d1090 #] [-d2575 #] [-dc1 to -d20 #] [-dive #] [-divn #] [-divr #] [-dm #] [-dme #] [-dmm #] [-dsd #] [-esp #] [-exceed #] [-fnt1 " "] [-fnt2 " "] [-fnt3 " "] [-fnt4 " "] [-fnt5 " "] [-fnt6 " "] [-fnts1 #.#] [-fnts2 #.#] [-fnts3 #.#] [-fnts4 #.#] [-fnts5 #.#] [-fnts6 #.#] [-ft #] [-gst #] [-help] [-lc {#}] [-lgf " "] [-lpbm #.#] [-lpc #] [-lpd #] [-lpf " "] [-lph #] [-lplm #.#] [-lppsext " "] [-lpo #] [-lpq " "] [-lpr] [-lprm #.#] [-lps #] [-lptm #.#] [-lty {#}] [-ltk {#.#}] [-md #] [-mox #.#] [-moy #.#] [-ms #] [-mx #.#] [-my #.#] [-nt #] [-prf " "] [-pt #] [-rfh #] [-ro #] [-sfl {#}] [-ssz {#.#}] [-sttl " "] [-sty {#}] [-tt #] [-ttl " "] [-xfmt " "] [-xlabel " "] [-xmax #.#] [-xmin #.#] [-xMt #.#] [-xmt #] [-xto #.#] [-xy #.#] [-yfmt " "] [-ylabel " "] [-ymax #.#] [-ymin #.#] [-yMt #.#] [-ymt #] [-ys #.#] [-yto #.#] [filename]

Meaning of flag symbols:

# = integer
#.# = float
" " = character string.
{} = variable is an array. Values must be seperated by a ',' and no spaces are allowed. Do not use the "{ }" symbols on the command line.

NOTES:

1). All parameters in [] brackets are optional.
2). Quotes must be used around character strings.
3). Filename, if given, must be listed last.
4). If no default is given, the feature is not currently supported on command line.

If no entry is required for flag, flag command executed.

Flag Definitions:

-d1090	=	draw 10-90 percentile	default = 1
		0 1 = = False True
-d2575	=	draw 25-75 percentile	default = 1
		0 1 = = False True
-dcl to -d20	=	active data column	default = a (1),0 (2-20)
		0 1 = = False True
-dive	=	histogram bar ending location	default = data maximum
-divn	=	number of divisions (histogram bars)	default = 10
-divr	=	division method	default = 0
		0 1 2 = = = number of divisions equal division width equal percentage divisions
-divs	=	histogram bar starting location	default = data minimum
-divw	=	division width (histogram bars)	default = data range / 10.0
-dm	=	draw mean	default = 1
		0 1 = = False True
-dme	=	draw median	default = 1
		0 1 = = False True
-dmm	=	draw minimum and maximum data extents	default = 1
		0 1 = = False True
-dsd	=	draw standard deviation	default = 1
		0 1 = = False True
-esp	=	exageration scale priority	default = 0
		0 1 = = favor y-exageration scale (-ys) favor x/y ratio
-exceed	=	exceedence on nonexceedence switch	default = 0
		0 1 = = Exceedence Nonexceedence
-fnt1	=	main title font	default = Helvetica-Bold
-fnt2	=	secondary title font	default = Helvetica-Bold
-fnt3	=	axes label font	default = Helvetica
-fnt4	=	division font	default = Helvetica
-fnt5	=	annotation font	default = Helvetica
-fnt6	=	mouse position font	default = Helvetica
-fnts1	=	main title font size	default = 24.0
-fnts2	=	main title font size	default = 15.0
-fnts3	=	main title font size	default = 15.0
-fnts4	=	main title font size	default = 12.0
-fnts5	=	main title font size	default = 10.0
-fnts6	=	main title font size	default = 12.0
-ft	=	frequency type	default = 0
		0 1 = = Count Percentage
-gst	=	frequency type	default = 0
		0 1 = = single stacked
-help	=	give this help menu
-lc {}	=	line color	default = variable
		0 1 2 3 4 5 6 7 = = = = = = = = Black White Red Green Blue Magenta Yellow Cyan
-lgf	=	log file name	defalut = "log.dat"
-lglp	=	line legend position	default = 1
		0 1 2 3 = = = = Top left Top right Bottom left Bottom right
-lgmw	=	maximum line legend width	default = 200
-lpbm	=	page bottom margin	default = 1.5
-lpc	=	number of copies to print	default = 1
-lpd	=	print destination	default = 0
		0 1 = = Printer File
-lpf	=	print filename	default = "junk.ps"
-lph	=	print header page	default = 0
		0 1 = = False True
-lplm	=	page left margin	default = 1.5
-lpo	=	print orientation	default = 0
		0 1 = = Portrait Landscape
-lppsext	=	search extention for postscript files	default = "*.ps"
-lpq	=	print queue	default = "ps"
-lpr	=	print file at specified orientations
-lprm	=	page right margin	default = 1.0
-lps	=	print output	default = 0
		0 1 = = Black & white Color
-lptm	=	page top margin	default = 1.5
-lsfl {}	=	fill line symbol	default = 0
		0 1 = = False True
-lsc {}	=	line symbol color	default = variable
		1 2 3 4 5 6 7 = = = = = = = Black White Red Green Blue Magenta Yellow Cyan
-lssz {}	=	line synbol size	default = 9.0
-lsty {}	=	line symbol type	default = 0
		-1 0 1 2 3 4 = = = = = = No Symbol Circle Cross Diamond Square X
-ltk {}	=	line thickness	default = 1.0
-lty {}	=	line type	default = 0
		-1 0 1 2 = = = = No Line Solid Dashed Double Dashed
-md	=	dash mesh	default = 0
		0 1 = = False True
-mox	=	X mesh origin	default = 0.0
-moy	=	Y mesh origin	default = 0.0
-ms	=	use mesh	default = 0
		0 1 = = False True
-mx	=	X mesh frequency	default = 1/10 DX
-my	=	Y mesh frequency	default = 1/10 DY
-nt	=	show normal curve(s)	default = 1
		0 1 = = Show Hide
-prf	=	preference file name	defalut = "histo.prf"
-pt	=	plot type	default = 0
		0 1 2 3 4 5 = = = = = = Histogram Box and Wisker Plot Cumulative Distribution Function 1 - CDF Probability
-rfh	=	screen refresh	default = 0
		0 1 = = On exposure On update
-ro	=	rank order option	default = 1
		0 1 = = Hazen (2i - 1) / 2n Weibull 1 / (n - 1)
-se	=	series file ending ID	default = last series ID
-ss	=	series file starting ID	default = 1
-sttl	=	Secondary title	default = " "
-tt	=	transform type	default = 0
		0 1 = = Normal Log (Base 10)
-ttl	=	Main title	default = Filename
-xfmt	=	Number of decimal places for X-axis	default = ".2f"
-xlabel	=	X-axis label	default = "X"
-xmax	=	Graph X-maximum	default = Data Maximum
-xmin	=	Graph X-minimum	default = Data Minimum
-xMt	=	X main tic frequency	default = 1/10 DX
-xmt	=	Number of minor X tics	default = 5
-xto	=	X axis label origin	default = 0.0
-xy	=	xy ratio	default = 1.5
-yfmt	=	Number of decimal places for X-axis	default = ".2f"
-ylabel	=	X-axis label	default = "Y"
-ymax	=	Graph Y-maximum	default = Data Maximum
-ymin	=	Graph Y-minimum	default = Data Minimum
-yMt	=	X main tic frequency	default = 1/10 DY
-ymt	=	Number of minor Y tics	default = 5
-ys	=	Y-axis exageration relative to X-axis	default = Calculated
-yto	=	X axis label origin	default = 0.0

histo -lpr -xmin 0.0 -ymin 0.0 -ymax 12.0 -md 1 -ms 1 -mx 1.0 -my 1.0 -xMt 1.0 - yMt 1.0 -ttl "Semivariogram of Elevation Data" -sttl "UNCERT histo module" - xlabel "distance (feet)" -ylabel "gamma h" -xfmt ".1f" -yfmt ".1f" -esp 1 -xy 1.0 water.out

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		1.0	23.23	0.123	1.45
		2.21	12.34	0.00123	1.56
		3.31	12.98	0.231	2.34
		4.56	8.21	0.345	1.76
		5.12	10.92	0.456	1.43
		.	.	.	.
		.	.	.	.
		.	.	.	.

This data set has four columns. Each column could be analyzed by selecting and plotting the data using the Data Column option under Data:Modify.

NOTE: This is the only file format available in histo which allows character data. See the notes in Chapter 5 about file format restrictions.

	VARIABLE LENGTH SETS
	2		Number of data sets
	4		Points in data set #1
	12.3
	15.5
	1.3
	99.5
	3		Points in data set #2
	0.012
	0.435
	0.098

The format is:

LINE 1 : VARIABLE LENGTH SETS
LINE 2 : Number of data sets

For each set:

LINE S1 : Number of points in set
LINE S2-... : Sample number.

GEO-EAS/GSLIB:

[TOP]

For a data set with n samples, the sample mean () is calculated as:

(6-1)

where xi = an individual sample value. The median (M) is calculated by (Press et al, 1992):

(6-2)

The unbiased sample variance (s2) may be calculated as (McCuen, 1989):

(6-3)

and the standard deviation (s) is defined as:

(6-4)

The standardized skew (g) is defined as (Press et al, 1992):

(6-5)

where the skew is a measure of symmetry. A symmetric distribution will have a skew of zero, and non-symmetric skews will be positive or negative, as shown in Figure 6.11a (McCuen, 1989). The 4th order moment, kurtosis (k), is defined (Press et al, 1992):

(6-6)

The kurtosis is a measure of the peakedness or flatness of the distribution relative to a normal distribution. A positive kurtosis reflects a peaked distribution (leptokurtic) and a negative kurtosis is relatively flat (platykuric) (Figure 6.11b, Press et al, 1992).

Figure 6.11a and Figure 6.11b

In addition to these statistical terms, the 10th, 25th, 75th, and 90th percentiles are often calculated. These are simply read from the sorted data set.

10th = xn*0.1

25th = xn*0.25

75th = xn*0.75

90th = xn*0.9

Calculation of Normal-Distribution and the Development of a Probability Distribution Graph Axis:

(6-7)

where Oi = the observed frequency of values over a range, and Ei = the expected frequency of values over a range. To determine the expected frequency, the normal distribution itself must be evaluated. The expected frequency is based on the probability y samples will occur between two values. It is calculated by:

(6-8)

(6-9)

(6-10)

Note that Equation 6-9 cannot be directly integrated and must be estimated numerically or evaluated from tabulated values. Below is a table of calculated probabilities for given z values (The table used by the software uses double precision values (non-truncated at the fourth decimal place)). These values agree with tables by McCuen (1989) to the fourth significant digit; the differences at the fifth place and beyond may only be in rounding.

------+-------------------------------------------------------------------------------
  Z   |  0.00    0.01    0.02    0.03    0.04    0.05    0.06    0.07    0.08    0.09
------+-------------------------------------------------------------------------------
-3.90 | 0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000 
-3.80 | 0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000   
-3.70 | 0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0000  0.0000  
-3.60 | 0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  
-3.50 | 0.0002  0.0002  0.0002  0.0002  0.0002  0.0002  0.0002  0.0001  0.0001  0.0001  
      |
-3.40 | 0.0003  0.0003  0.0003  0.0003  0.0003  0.0002  0.0002  0.0002  0.0002  0.0002  
-3.30 | 0.0005  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0003  0.0003  0.0003  
-3.20 | 0.0007  0.0006  0.0006  0.0006  0.0006  0.0005  0.0005  0.0005  0.0005  0.0005  
-3.10 | 0.0009  0.0009  0.0009  0.0008  0.0008  0.0008  0.0008  0.0007  0.0007  0.0007  
-3.00 | 0.0013  0.0013  0.0012  0.0012  0.0012  0.0011  0.0011  0.0010  0.0010  0.0010  
      |
-2.90 | 0.0018  0.0018  0.0017  0.0017  0.0016  0.0016  0.0015  0.0015  0.0014  0.0014  
-2.80 | 0.0025  0.0024  0.0024  0.0023  0.0022  0.0022  0.0021  0.0020  0.0020  0.0019  
-2.70 | 0.0034  0.0033  0.0032  0.0031  0.0030  0.0030  0.0029  0.0028  0.0027  0.0026  
-2.60 | 0.0046  0.0045  0.0044  0.0042  0.0041  0.0040  0.0039  0.0038  0.0037  0.0035  
-2.50 | 0.0062  0.0060  0.0058  0.0057  0.0055  0.0054  0.0052  0.0051  0.0049  0.0048  
      |
-2.40 | 0.0082  0.0080  0.0077  0.0075  0.0073  0.0071  0.0069  0.0067  0.0065  0.0064  
-2.30 | 0.0107  0.0104  0.0102  0.0099  0.0096  0.0094  0.0091  0.0089  0.0086  0.0084  
-2.20 | 0.0139  0.0135  0.0132  0.0129  0.0125  0.0122  0.0119  0.0116  0.0113  0.0110  
-2.10 | 0.0179  0.0174  0.0170  0.0166  0.0162  0.0158  0.0154  0.0150  0.0146  0.0142  
-2.00 | 0.0227  0.0222  0.0217  0.0212  0.0207  0.0202  0.0197  0.0192  0.0188  0.0183  
      |
-1.90 | 0.0287  0.0281  0.0274  0.0268  0.0262  0.0256  0.0250  0.0244  0.0238  0.0233  
-1.80 | 0.0359  0.0351  0.0344  0.0336  0.0329  0.0322  0.0314  0.0307  0.0301  0.0294  
-1.70 | 0.0446  0.0436  0.0427  0.0418  0.0409  0.0401  0.0392  0.0384  0.0375  0.0367  
-1.60 | 0.0548  0.0537  0.0526  0.0516  0.0505  0.0495  0.0485  0.0475  0.0465  0.0455  
-1.50 | 0.0668  0.0655  0.0643  0.0630  0.0618  0.0606  0.0594  0.0582  0.0571  0.0559  
      |
-1.40 | 0.0808  0.0793  0.0778  0.0764  0.0750  0.0736  0.0722  0.0708  0.0695  0.0681  
-1.30 | 0.0968  0.0951  0.0935  0.0918  0.0902  0.0885  0.0869  0.0854  0.0838  0.0823  
-1.20 | 0.1151  0.1132  0.1113  0.1094  0.1075  0.1057  0.1039  0.1021  0.1003  0.0986  
-1.10 | 0.1357  0.1336  0.1314  0.1293  0.1272  0.1251  0.1231  0.1211  0.1191  0.1171  
-1.00 | 0.1587  0.1563  0.1539  0.1516  0.1492  0.1469  0.1446  0.1424  0.1401  0.1379  
      |
-0.90 | 0.1841  0.1815  0.1789  0.1763  0.1737  0.1711  0.1686  0.1661  0.1636  0.1612  
-0.80 | 0.2119  0.2091  0.2062  0.2033  0.2005  0.1977  0.1950  0.1922  0.1895  0.1868  
-0.70 | 0.2421  0.2389  0.2359  0.2328  0.2297  0.2267  0.2237  0.2207  0.2178  0.2148  
-0.60 | 0.2743  0.2710  0.2677  0.2644  0.2612  0.2579  0.2547  0.2515  0.2483  0.2452  
-0.50 | 0.3086  0.3051  0.3016  0.2982  0.2947  0.2913  0.2878  0.2844  0.2811  0.2777  
      |
-0.40 | 0.3447  0.3410  0.3373  0.3337  0.3301  0.3265  0.3229  0.3193  0.3157  0.3122  
-0.30 | 0.3822  0.3784  0.3746  0.3708  0.3670  0.3633  0.3595  0.3558  0.3521  0.3484  
-0.20 | 0.4208  0.4169  0.4130  0.4091  0.4053  0.4014  0.3975  0.3937  0.3898  0.3860  
-0.10 | 0.4603  0.4563  0.4523  0.4484  0.4444  0.4405  0.4365  0.4326  0.4287  0.4248  
 0.00 | 0.5000  0.4961  0.4921  0.4881  0.4841  0.4802  0.4762  0.4722  0.4682  0.4642  
      |
 0.00 | 0.5000  0.5039  0.5079  0.5119  0.5159  0.5198  0.5238  0.5278  0.5318  0.5358  
 0.10 | 0.5397  0.5437  0.5477  0.5516  0.5556  0.5595  0.5635  0.5674  0.5713  0.5752  
 0.20 | 0.5792  0.5831  0.5870  0.5909  0.5947  0.5986  0.6025  0.6063  0.6102  0.6140  
 0.30 | 0.6178  0.6216  0.6254  0.6292  0.6330  0.6367  0.6405  0.6442  0.6479  0.6516  
 0.40 | 0.6553  0.6590  0.6627  0.6663  0.6699  0.6735  0.6771  0.6807  0.6843  0.6878  
      |
 0.50 | 0.6914  0.6949  0.6984  0.7018  0.7053  0.7087  0.7122  0.7156  0.7189  0.7223  
 0.60 | 0.7257  0.7290  0.7323  0.7356  0.7388  0.7421  0.7453  0.7485  0.7517  0.7548  
 0.70 | 0.7579  0.7611  0.7641  0.7672  0.7703  0.7733  0.7763  0.7793  0.7822  0.7852  
 0.80 | 0.7881  0.7909  0.7938  0.7967  0.7995  0.8023  0.8050  0.8078  0.8105  0.8132  
 0.90 | 0.8159  0.8185  0.8211  0.8237  0.8263  0.8289  0.8314  0.8339  0.8364  0.8388  
      |
 1.00 | 0.8413  0.8437  0.8461  0.8484  0.8508  0.8531  0.8554  0.8576  0.8599  0.8621  
 1.10 | 0.8643  0.8664  0.8686  0.8707  0.8728  0.8749  0.8769  0.8789  0.8809  0.8829  
 1.20 | 0.8849  0.8868  0.8887  0.8906  0.8925  0.8943  0.8961  0.8979  0.8997  0.9014  
 1.30 | 0.9032  0.9049  0.9065  0.9082  0.9098  0.9115  0.9131  0.9146  0.9162  0.9177  
 1.40 | 0.9192  0.9207  0.9222  0.9236  0.9250  0.9264  0.9278  0.9292  0.9305  0.9319  
      |
 1.50 | 0.9332  0.9345  0.9357  0.9370  0.9382  0.9394  0.9406  0.9418  0.9429  0.9441  
 1.60 | 0.9452  0.9463  0.9474  0.9484  0.9495  0.9505  0.9515  0.9525  0.9535  0.9545  
 1.70 | 0.9554  0.9564  0.9573  0.9582  0.9591  0.9599  0.9608  0.9616  0.9625  0.9633  
 1.80 | 0.9641  0.9649  0.9656  0.9664  0.9671  0.9678  0.9686  0.9693  0.9699  0.9706  
 1.90 | 0.9713  0.9719  0.9726  0.9732  0.9738  0.9744  0.9750  0.9756  0.9762  0.9767  
      |
 2.00 | 0.9773  0.9778  0.9783  0.9788  0.9793  0.9798  0.9803  0.9808  0.9812  0.9817  
 2.10 | 0.9821  0.9826  0.9830  0.9834  0.9838  0.9842  0.9846  0.9850  0.9854  0.9858  
 2.20 | 0.9861  0.9865  0.9868  0.9871  0.9875  0.9878  0.9881  0.9884  0.9887  0.9890  
 2.30 | 0.9893  0.9896  0.9898  0.9901  0.9904  0.9906  0.9909  0.9911  0.9914  0.9916  
 2.40 | 0.9918  0.9920  0.9923  0.9925  0.9927  0.9929  0.9931  0.9933  0.9935  0.9936  
      |
 2.50 | 0.9938  0.9940  0.9942  0.9943  0.9945  0.9946  0.9948  0.9949  0.9951  0.9952  
 2.60 | 0.9954  0.9955  0.9956  0.9958  0.9959  0.9960  0.9961  0.9962  0.9963  0.9965  
 2.70 | 0.9966  0.9967  0.9968  0.9969  0.9970  0.9970  0.9971  0.9972  0.9973  0.9974  
 2.80 | 0.9975  0.9976  0.9976  0.9977  0.9978  0.9978  0.9979  0.9980  0.9980  0.9981  
 2.90 | 0.9982  0.9982  0.9983  0.9983  0.9984  0.9984  0.9985  0.9985  0.9986  0.9986  
      |
 3.00 | 0.9987  0.9987  0.9988  0.9988  0.9988  0.9989  0.9989  0.9990  0.9990  0.9990  
 3.10 | 0.9991  0.9991  0.9991  0.9992  0.9992  0.9992  0.9992  0.9993  0.9993  0.9993  
 3.20 | 0.9993  0.9994  0.9994  0.9994  0.9994  0.9995  0.9995  0.9995  0.9995  0.9995  
 3.30 | 0.9995  0.9996  0.9996  0.9996  0.9996  0.9996  0.9996  0.9997  0.9997  0.9997  
 3.40 | 0.9997  0.9997  0.9997  0.9997  0.9997  0.9998  0.9998  0.9998  0.9998  0.9998  
      |
 3.50 | 0.9998  0.9998  0.9998  0.9998  0.9998  0.9998  0.9998  0.9999  0.9999  0.9999  
 3.60 | 0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  
 3.70 | 0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  0.9999  1.0000  1.0000  
 3.80 | 1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  
 3.90 | 1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 

(6-11)

From this relationship, not only can we evaluate normality based on the 2 test, but this distribution can be used to develop probability plots (NOTE: On probability paper, the probability axis is non-linear and non-logarithmic. Its scale can be determined as a function of z and p(z)).

To develop the probability plot, the data must be rank ordered (sorted). Two common methods are presented by Weibull (pw) and Hazen (ph) (McCuen, 1989) and the expected value for a given rank is calculated as:

(6-12)

(6-13)

where n is the number of samples and i is the rank-order of the given sample. These methods generate slightly different results, but either method is valid. Which method is used is largely a matter of user preference, and the user's impression of what works best for a particular data set.

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Press, W.H., S.A. Teukolsky, W.T. Vettering, and B.P. Flannery, 1992, Numerical Recipes in C, The Art of Scientific Computing, Second Edition, Cambridge University Press, New York, pps. 612-614.

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