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Software Tutorial --- Normal Statistics

The example session with the teaching software, PG2000, which is described below is intended as an example run to familiarise the user with the package. This documented example takes you through the following sequence of analyses:

Reading in a data file

Summary statistics and scatterplots of the data

Scatterplot of the data using transforms

Fitting a Normal distribution to the data

There are many other facilities within the package, which are given as alternative options on the menus. To start the tutorial, choose PG2000 from your Start menu or desktop icon . When you run PG2000, a record is kept of everything you do in that run. The default name for this file is ghost.lis and the default location for the file is the folder where your copy of PG2000 is kept. The first dialog you will see is:

You may change the name of the file, or accept the default. In some operating systems, the file extension may not be shown in the window or the File name box. The default file name is ghost.lis.

Note that, if you wish to change the name, you must type in the whole name including extension, since no default extension is offered in this case. For example, if you want to call your ghost file "myghost.lis" you need to type the whole name, not just "myghost".

If you already have a file with this name, Windows will issue a warning:

Click on  to specify a new name or  to overwrite previous copy of this file.

Your screen should now show something like:

The output above is the opening screen. To proceed to data analysis, use one of the menus at the top of the Window.

Reading in a data file

As you can see from the above I have elected to read in a set of sample data by clicking on the Read in a file  option and selecting Sample data  from the menu which appears. PG2000 will remember the last five data files accessed and include these in your options.

I have selected BROOMSBARN.DAT for my input data file. This is a set of 436 soil samples taken from a farm in England. The sample data are 40 metres apart. Co-ordinates on the data file are in grid spacing not metres.

Even if you select a file from the list of previously analysed data files, PG2000 will ask you to confirm your choice. This is actually a quick way of getting back to your working directory, since you can change your choice at this point. Be warned, though, that if you change which file you want to read it must be the same type of file – that is, if you are reading a standard Geostokos data file, you cannot change your mind at this point and read in a GeoEAS type file.

For this example, we will stick with BROOMSBARN. As your data is read in, it is stored on a working binary file. A progress bar will indicate how far the process has gone. When data input is complete, your Window should look like the table above.

The routine which has been used shows the first 10 lines of your data file so that you can check it is going in OK.

Scattergram or scatterplot

When the data has been read in you will see that the previously "greyed out" or inaccessible options on the main window toolbar will become activated. You can now select an option. Let us decide upon a statistical analysis. To do this, click on the Statistics option on the main toolbar.

If you choose the Scattergrams  option, you will display and summarize the data set and will enable you to get an idea of what the data set looks like in a simpler form than the full numerical listing.

The screen will switch to a dialog which will prompt you to choose the two variables for the axes of your graph.

 

The active screen in the top left hand corner contains the variables available for analysis in your data file. The bottom right box shows the variables already chosen (which at this point is none).

The Current working variables dialog box shows you that you are expected to select variables to be the X co-ordinate and the Y co-ordinate for your scattergram. The upper left dialog box Choosing variables  lists the variable names as they appeared in the data file, and is prompting you to choose the variable which will be the X co-ordinate on the graph. For this example, let us choose "K" (potassium) for the X co-ordinate. You need to check the box next to the "K" option.

transform_dialog

Upon selecting the "K" option, a new dialog box will appear asking you whether you wish to transform the variables to logarithms or rank transforms. In this case we do not wish to transform so we click on none_button. The dialog disappears and you will be asked for the Y co-ordinate:

I selected the "P" (phosphorus) option by clicking its check box. The transformation dialog again appears, from which we choose not to transform the variable by clicking on none_button.

The lower dialog moves up to the top left and displays your current working variables.

The accept_button  and change_button  buttons have now been activated. If you change your mind at this point, simply click on the change_button button and you will be returned to the original dialogs.

Clicking accept_button  will show you your scattergram. The scattergram is scaled to fit the whole of the display box or area.

Please note that even though you have chosen 'geographical' variables, the scale chosen is for the maximum display size. If you want points plotted on a 'geographical' scale (same for both axes) you must use the post-plotting routine which is available elsewhere in PG2000.

In the left-hand box of the graphical display, you will see the summary statistics for both variables plus the product moment correlation coefficient and the number of samples for which both variables were available. We can see from this graph that both variables tend to be "skewed" with a preponderance of lower values and a long scatter out into higher values.

When the graph is completed, you can select a new option from the main toolbar. You may wish to plot another graph in which case you must click on yes'>

Statistics and select the Scattergrams option again.

Scattergram 2, using transformations

To illustrate the use of the transformations for the variables, we draw another graph using the variables on this file "Log10 K" and "Log10 P". It is obvious that the person constructing this data file was aware of the "skewness" of the K and P measurements as illustrated in the previous scattergram. By adding a column of logarithms, the analyst hopes to make the scatter more symmetric.

Upon selecting the Scattergrams option your screen should show:

PG2000 will remember your previous selection. Since you are redefining your variables, you must click on change_button  to redefine your variables. You will again be asked to select the X co-ordinate and the Y co-ordinate. For the first variable we simply take logarithms. For the second we add a constant to the variable so that the transformation actually becomes Normal (Gaussian) - the determination of such a constant is described later in this demonstration run.

For the X co-ordinate check the corresponding box of the "Log10 K" option.

transform_dialog

The transformation dialog again appears, from which we choose not to transform the variable by clicking on none_button. For the vertical axis, choose "log10 P" and no transformation.

transform_dialog

Verify your choices as prompted:

A scattergram of these two variables will be produced, with a table of statistics on the left hand side.

Of course, we could produce a virtually identical plot without using the extra columns in the data file, by using the logarithmic transform available within the software. The major difference is that PG2000 uses natural logarithms – loge or ln where e=2.718282 – not logarithms to the base 10. Repeat the above sequence of actions, but selecting the basic variables and logarithmic transform.

Select the Scattergramsoption. Your screen should show:

PG2000 remembers your previous selection. Since you are redefining your variables, you must click on change_button  to redefine your variables. You will again be asked to select the X co-ordinate and the Y co-ordinate.

For the X co-ordinate check the corresponding box of the "K" option and then click on "take natural logarithms" in the

 dialog.

PG2000 also allows a variation of logarithmic transform which includes an "additive constant". If you are interested in this option, please refer to the Tutorial on lognormal statistics. Click on  to confirm that you want logarithmic transform. You can still cancel this option by clicking on none_button  instead of .   For the vertical axis, choose "P" and logarithmic transformation.

Note the difference in the names supplied for your variables. Verify your choices as prompted:

A scattergram of these two variables will be produced, with a table of statistics on the left hand side.

Note that the overall picture is identical whether you use log10 or natural logarithms. The values will be different by a factor of log10(e) or loge(10) {2.302585 or 0.434294} depending on which way round you look at them. The correlations are identical for both logarithmic transforms.

(i) (ii) (iii)
(i) Statistics from original variables K and P (ii) Statistics from original variable log10 K and log10 P (iii) Statistics from logarithms of original Variables K and P

Looking at descriptive statistics and histograms

To illustrate the use of histograms and descriptive statistics, we use the "K" variable in the BroomsBarn data set. Select the  menu and click on the option :

Choose the "measurement to be analysed" in the same way as previously:

 

Click in the box next to "K" and the transformation options will be offered:

 

For the moment we will make no transformation of the values, so click on . As usual, you will be asked to confirm your choice of variable:

Click on . Various summary statistics will be shown in two dialogs:

AND 

The table in the top left hand corner shows the usual descriptive statistics, with one small exception. The 'higher order' statistics – standard deviation, skewness and kurtosis – are divided by (n-1) and not n, the number of samples. That is:

statistic

formula

Arithmetic mean

= sum of values divided by n

Variance (square of standard deviation)

= Sum of each (sample value – average)2/(n-1)

skewness

= Sum of each (sample value – average)3/(n-1) divided by standard deviation cubed

kurtosis

= Sum of each (sample value – average)2 /(n-1) divided by standard deviation to the power 4
(or variance squared)

Coeff. Of variation

= arithmetic mean divided by standard deviation

In an ideal universe, where the population would follow a Normal distribution, the mean and standard deviation (divided by n-1) of the samples are 'best' estimates for the mean and standard deviation of that population. The skewness statistic would be:

#   zero for a symmetrical data set

#   positive if there were more samples in the lower values and a long tail to the high values

#   negative if there the samples are concentrated in the high values with a long tail to the lower values

We standardise by the standard deviation cubed, to remove the original variability of the samples and to obtain a statistic which actually reflects shape rather than spread. Similarly with the kurtosis. An ideal Normal distribution has a kurtosis of 3. A value less than 3 suggests that the shape of the histogram will be flatter than the ideal Normal. A value greater than 3 suggests a more 'peaked' shape.

Note: some software packages subtract the 3 from the kurtosis statistic, so that negative values may be encountered!

The coefficient of variation is also a (more empirical) measure of skewness BUT only for positive skewness and only if the values cannot take negative values. This implies that the statistic is, for example, useless when using a logarithmic transform where values can be negative.

Defining the necessary parameters for a histogram

A histogram is a graph which shows how the values vary amongst our samples. The graph shows value along the horizontal axis, which should (therefore!) reflect the range of our data values. The vertical axis is, technically, "frequency density". This is not the actual number of samples within a defined interval, but the number divided by the width of the interval. The difference is pretty academic if your histogram intervals are all the same size.

The software offers you default parameters for constructing the histogram, based on a simplistic assumption of basic Normality of the population. The average value is placed at the centre of the horizontal axis (values). The number of intervals is calculated as n/10 – or 12 if this comes out smaller than 12. The width of the intervals is selected to give a range of around 2 or so standard deviations either side of the average value. If the first interval falls lower than the lowest sample value, this is adjusted to be a little more sensible. For the "K" values in the Brooms Barn data set, the basic statistics convert to histogram parameters as follows:

==>

Of course, there is no guarantee that these default parameters are at all sensible. For example, a brief inspection of the Brooms Barn data shows that "K" is only measured in whole numbers. It seems a little silly, then, to choose a histogram interval of 1.2! If we amend this width to 2, we will have 43 intervals of 2 added to the lowest interval value of 14. The highest value shown on the histogram will be 57 – but the data values go up to 96. For a more sensible interval on this run, choose 2. I have also adjusted the lowest interval to 13 instead of 14. Our final options look like:

Accepting these parameters results in a new menu bar appearing at the top of your screen:

 and choose :

This will result in a full screen picture of the histogram, with the associated statistics still in the top left hand corner:

We can see that the histogram is skewed towards the left hand side of the graph, with more values squashed between 12 and 26 than between 26 and 96. This shape gives the positive skewness of just over 2 and a kurtosis around 4 times that of the ideal Normal distribution. If we look at the logarithms of the sample values, we hope to stretch out the lower end and squash in the upper end. With this data set, we can look at the column "Log10 K" – alternatively we could use the natural logarithm transform available within the software.

In any case, we need to start again by selecting , then select the  menu and click on the option . Using the same procedure as before, change from "K" to "log10 K". The new summary statistics are shown, as before. Note that the skewness is now less that 0.4 and the kurtosis is just under 3.6.

The suggested histogram parameters are 43 intervals, starting at 1.2 and with a width of 0.016. The logarithmic (base 10) values vary from 1.0792 to 1.9823. The default histogram is shown as the first graph below. Alongside, we show three alternative histograms just changing the interval width each time.

(i) (ii)
(i) accepting default histogram parameters (ii) choosing an interval width of 0.1
.
(iii) (iv)
(iii) interval width at 0.05 (iv) interval width of 0.025

Of the four above, the interval width at 0.05 seems to compromise between "lots of intervals, lots of detail" and getting a real idea of what the shape of the population might be. If we want to do any statistical inference or estimation, it is the shape of that population we have to predict. In an ideal world, we would want to have a Normal population, so that we may use most statistical theory. In this case, we need to decide whether the sample values of  "Log10 K" look like they come from a Normal distribution. From the  menu, select and click on:

You will be offered a set of choices:

For now, just click on . A new dialog appears showing the mean and standard deviation as estimated from your sample data:

.

If you click on , the software will superimpose a perfect Normal distribution with this mean and standard deviation (see upper graph on the next page). Note the chi-squared goodness of fit statistic of 16.72 with 10 degrees of freedom. Checking with any table of chi-squared statistics – for example, Table 3 in Practical Geostatistics – shows that, with 10 degrees of freedom, a statistic of 15.99 and over would be encountered 1 time in 10 if this Normal model is the true population distribution. Most statisticians choose a 5% level of 'significance'. With 10 degrees of freedom, we would need a chi-squared statistic of over 18.31 before we could legitimately doubt the fitted model.

If you click on , the software will find the mean and standard deviation of the Normal distribution which most closely fits your histogram (see lower graph on the next page). The software had 5 'iterations' (attempts) at fitting a better model, before coming up with this solution.


above: Normal model with mean and standard deviation estimated from samples

below: best fit (least squares) Normal model

The arithmetic mean of the Normal model has changed hardly at all and the standard deviation has dropped by around 6%. This time the chi-squared goodness of fit statistic has dropped to under 14.5, a value significantly below the 5% value of 18.31. We also have an added measurement of goodness of fit. The "root mean square percentage" difference between model and data histogram is 1.3. In crude terms, the difference between the model value for each block in the histogram and the actual data averages around 1.3%.

Bear in mind, that we do not expect an exact match between data and model – unless we have very many samples in our data set. To use statistical inference, we assume that the samples we have are drawn from a much larger population "at random and independently". If this is true, the chi-squared goodness of fit statistic should vary around the value of the 'degrees of freedom'. A chi-squared goodness of fit statistic which is too small is just as worrying as one which is too large, suggesting that sampling has been influenced to produce an idealised histogram.

Another way to illustrate the difference between data and model is to use a 'probability plot'. From the model option bar, select , then change the display by using the menu bar:

The display will switch to the following, remembering the model we have already fitted:

This type of graph was first used in the 1940s and has a special scale along the horizontal axis. The vertical axis is scaled to the values of our samples – in this case "log10 K". The horizontal axis is 'the percentage of the sampled values which fell below a given value. This is a "cumulative" graph rather than an interval one like the histogram. For example, 70% of our samples have values which lie at or below 1.45:

Notice, that if we read the value from the line (Normal model) rather than the symbol (data) we get a slightly higher value on the vertical axis. This is the difference between the model and the data and this is what the software minimises to get the "best fit".

One of the advantages of the probability plot is that we do not have to group our data into intervals as we do with the histogram. If we have fewer than 500 samples (software restriction), we can get far more detail in our probability plot by posting every sample separately. Click on , then change the display by using the menu bar:

This option returns you to the basic statistical summary and the histogram parameter dialog with all the original defaults:

If you have not noticed it before, there is a large 'button' which says:

You can use this for any data set with less than 500 samples, for larger data sets it is greyed out and you have to specify histogram intervals. Clicking on this button, returns us to the menu bar:

Notice that the two histogram options have greyed out because we did not group the data into intervals.

Selecting  now results in the graph on the next page. Each symbol now represents one sample, rather than one histogram interval. Notice the rounding on the original data, which results in many samples with exactly the same sample value. Notice also the deviations in the 'tails' of the graph:

      {#}   at the lower end the measured values are a little higher than ideal suggesting that there may be problems with measuring low concentrations;

      {#}   at the upper end the measured values are also a little higher than ideal with a noticeable break between 45 and a little over 50 in the original "K" units. Referring back to our scattergrams, you can clearly see this blank space in the graph of "K" versus "P".  

It would definitely be worth reviewing the data set for those samples with "K" over 45 to check why there is this gap between the rest of the samples and those ones.

Otherwise, the main body of the data seems to conform nicely to the Normal distribution and we can be confident that any statistical inference based on Normality assumptions can be applied to "log10 K".

You might want to try repeating this exercise with the "P" values and with "pH". Be prepared for surprises with "pH"!

Finishing up

Clicking on the  button will pass you back to the main menu. To finish this run of the program, select:

Clicking on this menu item or on close_window  will end your run with the software. You will see the closing down dialog box:

The above Tutorial session should serve only to illustrate a possible use of the various routines from PG2000. Try running the program again, choosing your own responses. Try reading in one of the other data files which are provided, say, samples.dat.

General Notes

There are a few points which you may have noted in following the Tutorial session above. Most of the routines communicate between themselves, without you having to worry about getting the right information from one to the other. For example, after you read in the complete contents of the data file, the routines ask which of the variables you actually want to analysis. This information is then stored internally and may be accessed by any of the other routines. When we went from plotting graphs of one variable against another to fitting a distribution, the routines knew that you had selected some variables, but that these were inappropriate for the new analysis. On the other hand, repeating the scattergram request, the routine suggested that you could continue to use the same choice of variables. This is a feature of most of PG2000, in that it will recall what you chose previously and ask whether this is to change or not.

PG2000 does not distinguish between upper and lower case letters, so you may type in whatever you find most pleasing. When the program requires a numerical answer, your input will be checked to make sure that it is actually a number. If you type in any illegal characters and press ENTER, the checking routine will filter out the unacceptable characters which you type. It should be noted that, if the routine is expecting a whole number then a decimal point is unacceptable. Much of the numerical input is checked for valid values.

A copy of this run should have been made on a file called 10.0pt;font-family:"Courier New";mso-bidi-font-family:"Times New Roman"'>GHOST.LIS unless you changed the name at the beginning of the run. Send this file to your printer if you want a record of the analysis or look at it with Wordpad or Notepad.

PG2000 — like any computer software — is not completely error-free. Neither is it fool-proof. You can always get out of the software by right clicking on the Taskbar. This will invoke the 'End Task' facility to close the Window without damaging the rest of your system. If you cannot figure out what went wrong, note down as much information as you can about the program you were running, the data you were using and exactly where it broke down. Contact your supplier locally or Geostokos direct for assistance, software@kriging.com. Send us the ghost.lis file and (if you can) the data you were analysing at the time.

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