Manual InnerSoft STATS

InnerSoft STATS
Application Manual v0.4

APPLICATION MANUAL V0.4 InnerSoft STATS
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INDEX
1. Installation and Activation
1.1. Installation
1.2. Trial Version
1.3. Activation
1.4. License
1.4.1. Upgrades
1.4.2. License Renewal
2. Introduction to ISSTATS
2.1. Open and Save files
2.2. Technical Limitations and Recommendations
3. Working in the Main window
3.1. File Menu
3.2. Analyze Menu
3.3. Help Menu
4. Working in the Worksheet window
4.1. File Menu
4.2. Edit Menu
4.3. Options Menu
5. Working in the Output window
6. Descriptive Statistics
7. One Sample Test
7.1. Z Test
7.2. T Test
7.3. Variance Test
8. Two-Sample Test
8.1. 2-Sample t-Test
8.2. Paired t-Test
8.3. 2 Variances F-Test
9. One-Way ANOVA
10. Homoscedasticity Tests
11. Methods and Formulas Help

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1. – Installation and Activation
1.1. – Installation
Click on setup.exe to install the application.
1.2. – Trial Version
InnerSoft STATS works initially as limited trial version. You will not be allowed to use following tools:
 Open or Save any Project or Worksheet.
 Using Paste or Insert when editing cells.
1.3. – Activation
In order to use InnerSoft STATS Full Version, user must purchase a license before activating the product.
To purchase InnerSoft STATS Full Version, you must first contact with us by email
innersoft@itspanish.org
innersoft@gmail.com
You can also call us (Spain number phone):
There are different payment methods that you can find at:
http://isstats.itspanish.org/index.htm#buy
Once product has been paid, you can activate it. Just enter Help/Activate ISSTATS and follow instructions.
For a successful activation process, InnerSoft requires permissions to read/write files in its installation
folder, usually: 'C:Program FilesISSTATS'.
So may you have to login to Windows as Administrator or have to modify User Account Control in order
to activate successfully InnerSoft STATS.

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1.4. – License
Registering data is chained to a PC Computer. When you pay a license, you can run InnerSoft STATS in
3 different computers. Thus you can ask for 3 Activation Codes.
You will not lose the license when formatting the hard drives or changing Windows S.O., BUT if you
make changes of hardware the activation code may become invalid.
Extra Activation Codes (more than 3) for same user have an additional cost.
1.4.1. – Upgrades
You will receive the Activation Codes for the product upgrades at no cost during one year. If you do not
renew the license after one year, will not lose the license of the product version that you purchased, BUT
will not receive new Activation Codes for the new software versions.
1.4.2. – License Renewal
The renewal will let you receive the Activation Codes for the new versions of the software during a year.

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2. – Introduction to ISSTATS
When you start ISSTATS, a new, empty project is opened for you. You will see three windows:
 Main window
o Project Manager window
o Worksheet window
o Output window
Controlling Windows
You can open and minimize ISSTATS windows just as you would other windows in your operating
system. You can also control ISSTATS windows with commands on the Window menu.
ISSTATS Environment
The project contains:
A Worksheet that contains your data. You can NOT have multiple worksheets in one project. Your data
will be displayed as columns. There is a Data window for your worksheet in the project. You can enter
and edit data directly in the Data window.
A Project Manager, which contains:
 A Root folder for setting the project information.
 A Settings folder for setting some options.
 A Variables folder for your worksheet. This folder displays a summary of the columns used in the
worksheet.
An Output window that displays your results.
Multiple Toolbars for issuing commands and a Status Bar.
2.1. – Open and Save files
Use the file menu to open, close or save the three file types that ISSTATS can use.
 Projects—contain a worksheet, along with project settings and project information
 Worksheets—contain only your columns data
 Outputs—contain the computed outputs
When you open a project file, the worksheet that were inside that project when you last saved is available
to you. When you save a project, the worksheet is saved within that project file.
The output results must be saved apart from the project file.
ISSTATS does not support multiple Worksheets or multiple Output documents. If you create a new table
or document, remember to save the previous data before it disappears.
You can save a project or worksheet in XML or Binary file format. The advantages and disadvantages of
using each format are:

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XML
 XML format is human readable and editable in plain text
 Can be imported by lot of applications (Excel…)
 Takes high size: storing a variable with 1’5 millions of values will take around 50 Mb in a XML
file
 Has a high compression rate: a XML file of 50 Mb will get only 1 Mb of size once compressed in
zip format
 Opening and saving is fast
Binary
 Binary format is not human readable
 Can be only open by ISSTATS
 Takes less size than XML: storing a variable with 1’5 millions of values will take around 30 Mb
in a Binary file.
 Has a low compression rate: a Binary file of 50 Mb will get 6 Mb of size once compressed in zip
format.
 Opening and saving is slow.
Thus XML compressed in zip is the better way of storing large amounts of data.
2.2. – Technical Limitations and Recommendations
ISSTATS uses decimal data type instead of double data type. Decimal is specially used for financial and
monetary calculation which requires higher accuracy. It is not adequate for scientific applications because
of its lower range: the largest value is +/–7.9228162514264337593543950335, and the smallest nonzero
value is +/–0.0000000000000000000000000001 (+/–1E–28). So if you need to analyses social or
scientific data with large numbers (astronomy distances, human stupidity…), just use a different statistical
package.
ISSTATS work with on–memory data (no database storage). Thus it can compute quickly. In the other
hand, memory usage is huge.
ISSTATS can handle millions of data. It can compute all descriptive parameters of a variable with 1’5
million values in only a few seconds (Processor: Pentium R Dual Core CPU E 6600 @ 3.06 GHz 3.07
GHz RAM: 8.0 GB SO: Windows 7 64 Bits). In the other hand, the Data table can easily crash if you try
to manipulate this huge data. Operations like copying, pasting, removing and selecting cells in such a big
table can hang the application. It really depends on the total number of cells in the table. To avoid this
problem, limit the number of table columns when operating with huge data. If you want to handle
variables with millions of values, set the number of table columns to minimum in the Worksheet
OPTIONS menu: if you are using only 4 variables, set only 4 columns.
You should perform operations with columns as much as possible. If you want to clear a variable, select
the column and clear the column; avoid selecting all cells in the column and clearing these cells. If you
want to copy a variable then copy the column, not the cells.
Actually, undo/redo history has no limits. A limit will be implemented in a future. If you feel that
ISSTATS is consuming too much memory in your PC, you should save your project, restart the
application and open your project again. This will clear the undo/redo history.
Please, remember this is 0.4 version.

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3. – Working in the Main window
3.1. – File Menu
New Project
Creating a new project closes the current project. If your current project has been changed since it was
last saved, ISSTATS will prompt you to save all or part of the project before closing it.
Open Project
Opening a new project closes the current project. If your current project has been changed since it was
last saved, ISSTATS will prompt you to save all or part of the project before closing it.
Save Project
Saves the current project in a XML or Binary file. If the project has been saved before, Save Project
saves the file with the current file name without displaying a dialog box. If the project has not been saved
before, Save Project opens the Save Project As dialog box.
When you save the project, you save the input information about your work:
 the contents of project settings and project information
 the columns of data in the Worksheet window
The output document will not be saved in your project file.
Save Project As
Saves the current project in a XML or Binary file with a different name. If the project has not been saved
before, choosing File > Save Project also opens this dialog box.
Dialog box items
 Save in: Choose a drive and folder.
 File name: Enter a file name.
 Save as type: Choose a file type from the list.
o XML files (*.xml)
o DAT files (*.dat)
Exit
Exits you from ISSTATS. ISSTATS will prompt you to save the changes before closing the application.
3.2. – Analyze Menu
Descriptive Statistics: See Chapter 6. – Descriptive Statistics
One Sample Test: See Chapter 7. – One Sample Test
Two-Sample Test: See Chapter 8. – Two-Sample Test

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One-Way ANOVA: See Chapter 9. – One-Way ANOVA
Homoscedasticity Tests: See Chapter 10. – Homoscedasticity Tests
3.3. – Help Menu
Contents
Open this PDF.
Buy ISSTATS
Open a web page with information about buying InnerSoft STATS.
Activate ISSTATS
Open the menu to activate ISSTATS in order to use all features.
About
Shows ISSTATS version and contact information.

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4. – Working in the Worksheet window
4.1. – File Menu
New Worksheet
Creating a new worksheet adds an empty worksheet to the current project. An empty Data window will
appear.
Open Worksheet
Copies data from a file into the current project. When you open a file, you copy the contents of the file
into the current project. Any changes you make to the file while in the project will not affect the original
file.
Save Worksheet
Saves the current worksheet in the same file and format as the file name displayed in the window title. If
the current worksheet is untitled, File > Save Worksheet brings up a dialog box where you can specify a
file name and location in which to save your data. See File > Save Worksheet As for a description of this
dialog box.
Choose File > Save Worksheet As to rename your worksheet or save it to a new location.
Save Worksheet As
Saves the worksheet data in a file. Use this command if you want to rename your worksheet or save it to a
new location.
Dialog box items
 Save in: Choose a drive and folder.
 File name: Enter a file name.
 Save as type: Choose a file type from the list.
o XML files (*.xml)
o DAT files (*.dat)
Export to Excel
This tool will export to Excel only those columns that has a variable defined. Empty columns will not be
exported.
Exporting to Excel has some limitations. As worksheet limit size in Excel 2010 is 65,536 rows, ISSTATS
will divide the data collection in different sheets of 65530 rows everyone. If you are going to export 1’5
million values, the application will create 23 different Excel worksheets with 65530 rows each one. As
columns limit in Excel 2010 is 256 columns, ISSTATS will not export the data of more than 256
variables.
4.2. – Edit Menu
Undo & Redo

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To undo an action, click Undo Button on the Toolbar.
To redo an action that you undid, click Redo Button on the Toolbar.
Cut Cells
The selection will be copied to the clipboard buffer. The selection will be moved to a new location as
soon as you choose Paste command as described below.
Please note that the selection will remain in its current location until you paste it. If you want to delete a
selection without pasting it to a new location, use Delete command instead.
Copy Cells
The selection will be copied to the clipboard buffer. The selection will be copied to a new location as
soon as you choose Paste command as described below.
You can copy all the cells of a column or a row selecting the rows or columns and using the COPY
CELLS button. To avoid copying the variable names, when you select some columns and click the COPY
CELLS button, the name cells of the column is deselected.
Paste Cells
The contents of the clipboard will be pasted in the spreadsheet so that the selected cell becomes the top
left corner of the selection.
When you paste cells, the blank cells that you cut/copied will not modify any existing value and will not
add any new value to the column. The missing values that you copied will add new missing values or will
replace the existing ones with missing values.
If you copy a discontinuous group of cells, by selecting cells or block of cells while holding down the Ctrl
key, ISSTATS will paste it as a continuous selection where non selected cells are converted in empty
strings.
Insert Cells
You can insert copied cells above of the active cell on your worksheet, shifting other cells in the same
column down.
After you insert cells, the blank cells that you previously copied will add missing values. Inserting cells
below the last value of the column will do as pasting cells: blank cells will not add data.
Clear Cells
Erases the contents of the selected cells, without moving other cells. This menu command is available
when at least one cell is selected in the Data window. To delete selected cells, see Delete Cells.
In a numeric column, ISSTATS inserts system missing values (*) in a cleared cell.
Delete Cells

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If you are deleting cells, other cells automatically shift up. If you click a cell and then press DELETE,
Excel deletes the cell contents.
Select All Cells
Click this button to select all cells on the worksheet.
Copy Columns
When you copy various columns, alternated selected columns will be joined together in a continuous
selection once you paste or insert them (as Excel does).
Paste Columns
On pasting, empty selected columns will not modify any existing variables. Not empty selected columns
will replace any existing variables.
Insert Columns
You can insert columns that previously copied in the worksheet.
To insert a single column, select the column in the column immediately to the right of where you want to
insert the new column. To insert multiple columns, select the columns immediately to the right of where
you want to insert columns.
If you insert columns, other columns automatically shift to the right.
On inserting, empty columns (no variable defined) will add new empty columns (no variable defined).
Clear Columns Content
Erases the contents of the selected column, without moving other columns. This menu command is
available when at least one column is selected in the Data window. To delete selected columns, see Delete
Columns.
The cleared cells remain as blank cells on the worksheet.
Delete Columns
If you are deleting columns, other columns automatically shift to the left.
Select All Columns
The Select All Columns command will create a new selection that includes every column with data.
As rows have not any structural meaning in the table, there is no copy/cut/paste/insert options for rows. If
you want to copy all cells of a row just select the rows and press COPY CELLS button.
4.3. – Options Menu

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Table Decimal Places
For numbers that are already entered on a worksheet, you can increase or decrease the number of places
that are displayed after the decimal point by using the Table Decimal Places buttons. By default,
ISSTATS displays 3 decimal places when you apply a built-in number format to the cells.
Table Columns
You can increase or decrease the number of columns that are available in the worksheet by using the
Table Columns buttons. By default, ISSTATS displays 10 columns.
To avoid memory problems, limit the number of table columns when operating with huge data. If you
want to handle variables with millions of values, set the number of table columns to minimum: if you are
using only 4 variables, set only 4 columns.
Table Rows
You can increase or decrease the number of rows that are available in the worksheet by using the Table
Rows buttons. By default, ISSTATS displays 100 rows.

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5. – Working in the Output window
New Document
Creating a new output document closes the current document. If your current document has been changed
since it was last saved, ISSTATS will prompt you to save the document before closing it.
Open Document
Opening a new document closes the current document. If your current document has been changed since
it was last saved, ISSTATS will prompt you to save the document before closing it.
The file format of the opened document must be ISSTAT XAML package (a XAML package file
produced by ISSTAT).
Save Document
Saves the current document in a ISSTAT XAML package file (a XAML package file produced by
ISSTAT). If the document has been saved before, Save Document saves the file with the current file
name without displaying a dialog box. If the document has not been saved before, Save Document opens
the Save Document As dialog box.
Use Windows Clipboard to copy and paste the document content in other applications, such as MS Word,
MS Excel, Notepad…
Save Document As
Saves the document in a file. Use this command if you want to rename your document or save it to a new
location.
Print Document
Prints the document.
Cut, Copy & Paste
The way to copy or move a selection is to use cut, copy, and paste operations.
Use this commands to copy a selection from Output window to other applications, such as MS Word, MS
Excel, Notepad…
Undo & Redo
To undo an action, click Undo Button on the Toolbar.
To redo an action that you undid, click Redo Button on the Toolbar.
Bold, Italic, Underline, Grow Font, Shrink Font, Bullets, Numbering, Align Left, Align Center,
Align Right, Align Justify, Increase Indent & Decrease Indent

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These are the formatting buttons, familiar to anyone who has used a word processor. These formats are
applied by selecting some text in the window and clicking the button in the editor toolbar.
Select All
The Select All command will create a new selection that includes everything on the document.

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6. – Descriptive Statistics
Overview
Produces descriptive statistics for each column. The data columns must be numeric and contain at least
one numeric value. The dialog box allows to choose the statistics that you wish to display.
Available Variables list shows numeric data columns containing at least one no-missing value.
Dialog box items
Mean: Choose to display the arithmetic mean.
Sample Variance: Choose to display the unbiased variance of the data. Estimates population variance
based on a sample. If your data represents the entire population, then compute the variance by using Total
Variance.
Sample Standard Deviation: Choose to display the standard deviation of the data. Estimates population
standard deviation based on a sample. If your data represents the entire population, then compute the Std.
Deviation by using Total Std. Deviation.
Sample Coefficient of variation: Choose to display the coefficient of variation.
Sample Skewness: Choose to display the skewness value. Estimates population skewness based on a
sample. If your data represents the entire population, then compute the skewness by using Total
Skewness.
Sample Kurtosis: Choose to display the kurtosis value. Estimates population kurtosis based on a sample.
If your data represents the entire population, then compute the kurtosis by using Total Kurtosis.

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Total Variance: Choose to display the variance of the data. Calculates variance based on the entire
population.
Total Standard deviation: Choose to display the standard deviation of the data. Calculates Std. Deviation
based on the entire population.
Total Coefficient of variation: Choose to display the coefficient of variation. Calculates coefficient of
variation based on the entire population.
Total Skewness: Choose to display the skewness value. Calculates skewness based on the entire
population.
Total Kurtosis: Choose to display the kurtosis value. Calculates kurtosis based on the entire population.
SEM: Choose to display the standard error of the mean.
Sum: Choose to display the data sum.
Minimum: Choose to display the data minimum.
Maximum: Choose to display the data maximum.
Range: Choose to display the data range. Data Range is the difference between the maximum and
minimum.
Quartiles: Choose to display the first quartile, the median and the third quartile.
Interquartile range: Choose to display the difference between the first and third quartiles.
Deciles: Choose to display the nine values that divide the sorted data into ten equal parts.
Percentiles: To request a percentile:
 Enter the desired value in the box placed in Percentiles group box. For example, if you wanted the
7th
percentile, you would enter a 7 in the box.
 Click the add button to add the percentile to the list of requested percentiles.
 Repeat Step 1 and 2 to add additional percentiles as desired.
 If you need to delete a percentile, select it in the list and click the remove button.
Mode: Choose to display the mode and the number of times it occurs. If multiple modes exist, Minitab
displays the smallest modes, up to a total of four, along with their frequency.
Sum of squares: Choose to display the sum of the squared data values. This is the uncorrected sums of
squares, without first subtracting the mean.
MSSD: Choose to display half the Mean of Successive Squared Differences.
N nonmissing: Choose to display the number of nonmissing column entries.
N missing: Choose to display the number of missing column entries.
N total: Choose to display the total (nonmissing and missing) number of column entries.

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Cut–Points: Divide the data into a number of equal groups. For example, to create deciles, you would
enter 10 in the box .Enter 3 to divide the data into tertiles.
Check statistics
Check None: Choose to clear all check boxes and then individually check the statistics to display.
Check All: Choose to check all boxes. You can uncheck statistics as needed.

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7. – One Sample Test
Overview
In the box Test Type, choose the test to perform:
 Z Test
 T Test
 Variance Test
Hypothesis Test
To perform a hypothesis test, check the box Perform Hypothesis Test and choose the alternative
hypothesis of the test:
 Less than: Perform a level  test of H0:  ≥ 0 against the one-sided alternative H1:  < 0
 Not Equal: Perform a level  test of H0:  = 0 against the two-sided alternative H1:  ≠ 0
 Greater than: Perform a level  test of H0:  ≤ 0 against the one-sided alternative H1:  > 0
Significance level of the test () derive from the 1- value set in the Confidence Level text box.
If you choose a lower-tailed hypothesis test, an upper confidence bound will be constructed. If you
choose an upper-tailed hypothesis test, a lower confidence bound will be constructed.
Available Variables list shows numeric data columns containing at least one no-missing value.
7.1. – Z Test
Use 1-Sample Z to compute a confidence interval or perform a hypothesis test of the mean when the
standard deviation of the population σ is known. The samples should come from a normal population if n
is low; if however n>30 the distribution of the population does not have to be normal.

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Dialog box items
Data: Choose the origin of the data.
 Samples in columns: Choose if you have entered raw data in columns. Enter the columns
containing the sample data in the list Computing Variables. Move these variables from Available
Variables list to Computing Variables list using Add and Remove buttons. Enter the value for the
population standard deviation in the text box. Entering multiple columns, ISSTATS performs
separate one-sample analyses on each column.
 Summarized data: Choose if you have summary values for the sample size and mean.
o Sample size: Enter the value for the sample size.
o Mean: Enter the value for the sample mean.
o Population Standard deviation: Enter the value for the population standard deviation.
Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9
will result in a 90% confidence interval. The default is 0,95 = 95%.
Perform hypothesis test: Check to perform the hypothesis test.
 Hypothesized mean: Enter the test mean 0.
 Alternative hypothesis: Choose the alternative hypothesis of the test.
7.2. – T Test
Performs a one-sample t-test or t-confidence interval for the mean.
Use T Test for one sample to compute a confidence interval and perform a hypothesis test of the mean
when the population standard deviation, σ, is unknown. Use this test when samples come from a normal
population or n > 30.
Dialog box items
Variables list to Computing Variables list using Add and Remove buttons. Entering multiple
columns, ISSTATS performs separate one-sample analyses on each column
 Summarized data: Choose if you have summary values for the sample size and mean.
o Mean: Enter the value for the sample mean.
o Sample Standard deviation: Enter the value for the sample standard deviation.

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 Hypothesized mean: Enter the test mean 0.
7.3. – Variance Test
This procedure calculates confidence intervals for the variance of a population, and performs a hypothesis
test to determine whether the population variance equals a specified value. Use this test when samples
come from a normal population.
Dialog box items
Variables list to Computing Variables list using Add and Remove buttons. Entering multiple
columns, ISSTATS performs separate one-sample analyses on each column.
 Summarized data: Choose if you have summary values for the sample size and variance.
o Sample Variance: Enter the value for the sample variance.
 Hypothesized variance: Enter the test variance σ2
0.

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8. – Two-Sample Test
 2-Sample t-Test
 Paired t-Test
 2 Variances F-Test
Hypothesis Test
To perform a hypothesis test, check the box Perform Hypothesis Test and choose the Alternative
Hypothesis of the test:
 Less than: Perform a level  test of H0:  ≥ d against the one-sided alternative H1:  < d
 Not Equal: Perform a level  test of H0:  = d against the two-sided alternative H1:  ≠ d
 Greater than: Perform a level  test of H0:  ≤ d against the one-sided alternative H1:  > d
Significance level of the test () derive from the 1- value set in the Confidence Level text box.
If you choose a lower-tailed hypothesis test, an upper confidence bound will be constructed. If you
choose an upper-tailed hypothesis test, a lower confidence bound will be constructed.
Available Variables list shows numerical and text columns containing at least one no-missing value.
Text columns may be used as Subscripts.
8.1. – 2 Sample t-Test
Performs an independent 2-sample t-test and generates a confidence interval.

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When you have dependent samples, use 8.2. – Paired t-Test.
Use 2-Sample t-Test to perform a hypothesis test and compute a confidence interval of the difference
between two population means when the population standard deviations, σ's, are unknown.
Dialog box items
 Samples in one column: Choose if the sample data are in a single column, differentiated by
subscript values (group codes) in a second column.
o Computing Variables: Enter the columns containing the sample data in the list
Computing Variables. Move these variables from Available Variables list to Computing
Variables list using Add / Remove buttons.
o Grouping Variable: Enter the column containing the sample subscripts to define the
groups. It may be a numerical or text column.
o Group 1/Group 2: Enter the subscripts that define both groups.
o Automatic Grouping: Check to let the application define automatically both groups. It
will identify the first two different subscripts in the grouping variable.
 Samples in different columns: Choose if the data of the two samples are in separate columns.
o First Sample: Enter the column containing one sample from Available Variables list.
o Second Sample: Enter the column containing the other sample from Available Variables
list.
 Summarized data: Choose if you have summary values for the sample size, mean, and variance
for each sample.
o First Sample
 Sample size 1: Enter the sample size for the first sample.
 Mean 1: Enter the value for the mean of the first sample.
 Sample Variance 1: Enter the value for the variance of the first sample.
o Second Sample
 Sample size 2: Enter the sample size for the second sample.
 Mean 2: Enter the value for the mean of the second sample.
 Sample Variance 2: Enter the value for the variance of the second sample.
Assume Population equal variances: Check to assume that the populations have equal variances.
 Hypothesized mean: Enter the hypothesized difference between the two population means 1-2.
A difference of 0 suggests in the null hypothesis the equality between mean populations; H0: 1-2 = 0
against an alternative H1: 1-2 ≠ 0

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A difference equal or less than 0 suggest in the null hypothesis that the first mean is equal or less than the
second; H0: 1-2 ≤ 0 against an alternative H1: 1-2 > 0
A difference equal or greater than 0 suggest in the null hypothesis that the first mean is equal or greater
than the second; H0: 1-2 ≥ 0 against an alternative H1: 1-2 < 0
Optionally, test ratios other than 0 (equality) can be specified. A difference of 2 suggests in the null
hypothesis the first mean is the second mean plus 2.
8.2. – Paired t-Test
Performs a paired t-test. This is appropriate for testing the mean difference between paired observations
when the paired differences follow a normal distribution.
Use the Paired t command to compute a confidence interval and perform a hypothesis test of the mean
difference between paired observations in the population. A paired t-test matches responses that are
dependent or related in a pairwise manner. A typical example of the repeated measures t-test would be
where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are
tested again after treatment with a blood-pressure lowering medication. Paired samples t-tests are often
referred to as "dependent samples t-tests". When the samples are drawn independently from two
populations, use 8.1. – 2 Sample t-Test.
Dialog box items
 Sample in columns: Choose if you have entered raw data in two columns.
o First sample: Enter the column containing the first sample from Available Variables list.
o Second sample: Enter the column containing the second sample from Available Variables
list.
Pairs must have two numerical values. Pairs that have a missing data in any of the
members are ignored.
 Summarized data: Choose if you have summary values for the sample size, mean, and variance
of the difference.
o Mean of Differences: Enter the value for the mean of differences ̅.
o Variance of Differences: Enter the value for the variance of differences s2
d.
 Hypothesized mean: Enter the hypothesized population mean of the paired differences d.
8.3. – 2 Variances F-Test

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The 2 Variances F-Test procedure performs hypothesis tests and computes confidence intervals for the
ratios between two populations' variances. Use this test to determine if one treatment condition has more
variability than the other. Each population must follow the normal distribution.
Dialog box items
 Samples in one column: Choose if you have entered data into a single column with a second
column of subscripts that identify the samples.
o Computing Variables: Enter the columns containing the sample data in the list
Computing Variables. Move these variables from Available Variables list to Computing
Variables list using Add and Remove buttons. Entering multiple columns, ISSTATS
performs separate analyses on each column.
o Grouping Variable: Enter the column containing the sample subscripts to define the
groups. It may be a numerical or text column.
o Group 1/Group 2: Enter the subscripts that define both groups.
o Automatic Grouping: Check to let the application define automatically both groups. It
will identify the first two different subscripts in the grouping variable.
 Samples in different columns: Choose if you have entered the data for the two samples into
separate columns.
o First Sample: Enter the column that contains the data for the first sample from Available
Variables list.
o Second Sample: Enter the column that contains the data for the second sample from
Available Variables list.
 Summarized data: Choose if you have summary values for the sample sizes and variances.
o First Sample
 Sample size 1: Enter the sample size for the first sample.
 Sample Variance 1: Enter the variance for the first sample.
o Second Sample
 Sample size 2: Enter the sample size for the second sample.
 Sample Variance 2: Enter the variance for the second sample.
 Hypothesized ratio: Enter the hypothesized ratio between two population’s variances σ2
1/σ2
2.
A ratio of 1 suggests in the null hypothesis the equality between variance populations; H0: σ2
1/σ2
2 = 1
against an alternative H1: σ2
1/σ2
2 ≠ 1.
A ratio equal or less than 1 suggest in the null hypothesis that the first variance is equal or less than the
second; H0: σ2
1/σ2
2 ≤ 1 against an alternative H1: σ2
1/σ2
2 > 1.

25
A ratio equal or greater than 1 suggest in the null hypothesis that the first variance is equal or greater than
the second; H0: σ2
1/σ2
2 ≥ 1 against an alternative H1: σ2
1/σ2
2 < 1.
Optionally, test ratios other than 1 (equality) can be specified. A ratio of 2 suggests the first variance is
double the second variance.

26
9. – One-Way ANOVA
Performs a one-way analysis of variance. You can also perform multiple comparisons. The response
variable must be numeric. The factor level column (grouping variable) may be numeric or text.
Dialog box items
 Groups in different Columns
o Computing Variables: Enter the columns containing the response. Each column must
contain the data for one of the groups.
 Groups in 1 Column
o Computing Variables: Enter the column or columns containing the response. ISSTATS
performs a ANOVA test for each of these columns.
o Grouping Variable: Enter the column containing the factor levels.
Confidence level: Enter the confidence level.
For further details, see ANOVA Test.
Multiple Comparisons: Use to generate grouping information tables and confidence intervals for the
differences between means, by different methods. Check to obtain confidence intervals for all pairwise
differences between level means using any of these methods.
 Scheffe
 Tukey HSD: Tukey's Honestly Significant Difference test.
 Sidak

27
 Fisher LSD: Fisher's Least Significant Difference test.
 Bonferroni
For further details, see ANOVA Multiple Comparisons.
Include Brown-Forsythe Test for equality of Means
Use to test the equality of means when distribution violates the assumption of equal variances.
The Brown-Forsythe test cannot be computed if all groups have zero variance. To take part into Brown–
Forsythe test, a group must have at least two elements. In the situation that some groups have zero
standard deviations, the statistic can be computed but the approximation may not work.
For further details, see Brown–Forsythe Test for equality of means.
Include Welch’s Test for equality of Means
Use to test the equality of means when distribution violates the assumption of equal variances.
To take part into Welch’s test, a group must have non zero variance. Moreover, sample sizes of a group
have to be greater than or equal to 2.
For further details, see Welch’s Test for equality of means.
Remark: ISSTATS does not cancel a test if any of the groups does not fulfill the conditions. It simply
rejects the group. To take part into a ANOVA test, a group must have at least one element. To take part
into a Welch’s Test, a group must have at least two elements; moreover, the group must have a non-zero
variance. Thus, ISSTATS may use 4 of the groups to perform ANOVA Test but only 2 of the groups to
perform a Welch’s Test. It depends on the number of elements of each group. You should read the
Descriptive Statistics information to check the Rejected Groups and also should read the Total Groups
info of each test to check the number of groups being used in that test.
Outputs
Outputs include:
 Descriptive statistic for every level of the factor’s variable.
 Total mean.
 Degrees Of Freedom: DFTotal, DFInter and DFIntra
 The Sum Of The Squares: SSTotal, SSInter and SSIntra
 The Mean Square: MSTotal, MSInter and MSIntra
 The F ratio
 The p-value
 R-squared
 R-squared adjusted
Inter is also referred as Between Groups or Between Treatments. Intra is also referred as Within Groups
or Error Term.

28
R-squared represents the percentage of variation in a response variable that is explained by its relationship
with one predictor variable.
R-squared adjusted is a version of r-squared that has been adjusted for the number of predictors in the
model. R-squared tends to overestimate the strength of the association, especially when there are more
than one independent variables.

29
10. – Homoscedasticity Tests
These tests are used to test if k samples are from populations with equal variances. The response variable
has to be numeric. The factor level column (grouping variable) may be numeric or text.
Dialog box items
 Levene's Test
 Brown–Forsythe Test for equality of variances
 Bartlett's Test
For further details, see Homoscedasticity Tests.
Remark: ISSTATS does not cancel a test if any of the groups does not fulfill the conditions. It simply
rejects the group. To take part into a Levene or Brown–Forsythe test, a group must have at least one
element. To take part into a Bartlett's Test, a group must have at least two elements; moreover, the group
must have a non-zero variance. Thus, ISSTATS may use 4 of the groups to perform Levene's Test but
only 2 of the groups to perform a Bartlett's Test. It depends on the number of elements of each group.
You should read the Descriptive Statistics information to check the Rejected Groups and also should read
the Total Groups info of each test to check the number of groups being used in that test.

30
11. – Methods and Formulas Help
Mean
The arithmetic mean is the sum of a collection of numbers divided by the number of numbers in the
collection.
Sample Variance
The estimator of population variance, also called the unbiased sample variance, is:
∑ ( ̅)
Source: http://en.wikipedia.org/wiki/Variance
Sample Kurtosis
The estimators of population kurtosis is:
( )
( )( )( )
∑ ( ̅) ( )
( )( )
The standard error of the sample kurtosis of a sample of size n from the normal distribution is:
√
( ) ( )
( )( )( )( )( )
Source: http://en.wikipedia.org/wiki/Kurtosis#Estimators_of_population_kurtosis
Sample Skewness
Skewness of a population sample is estimated by the adjusted Fisher–Pearson standardized moment
coefficient:
( )( )
∑ (
̅
)
where n is the sample size and s is the sample standard deviation.
The standard error of the skewness of a sample of size n from a normal distribution is:
√
( )
( )( )( )
Source: https://en.wikipedia.org/wiki/Skewness#Sample_skewness
Total Variance

31
Variance of the entire population is:
∑ ( ̅)
Source: http://en.wikipedia.org/wiki/Variance
Total Kurtosis
Kurtosis of the entire population is:
∑ ( ̅)
where n is the sample size and σ is the total standard deviation.
Source: http://en.wikipedia.org/wiki/Kurtosis
Total Skewness
Skewness of the entire population is:
∑ ( ̅)
where n is the sample size and σ is the total standard deviation.
Source: https://en.wikipedia.org/wiki/Skewness
Quantiles of a population
ISSTATS uses the same method as R–7, Excel CUARTIL.INC function, SciPy–(1,1), SPSS and Minitab.
Qp, the estimate for the kth
q–quantile, where p = k / q and h = (N–1)*p + 1, is computing by
Qp =
Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1]. When p = 1,
use xN.
Source: http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population
MSSD (Mean of the squared successive differences)
It is calculated by taking the sum of the differences between consecutive observations squared, then
taking the mean of that sum and dividing by two.
∑ ( )
( )
The MSSD has the desirable property that one half the MSSD is an unbiased estimator of true variance.

32
Confidence Bounds and One-Sided Tests
Suppose you are testing the null hypothesis H0:  ≥ 0 against the one-sided alternative H1:  < 0. Rather
than give a two-sided confidence interval for , the more appropriate procedure is to give an upper
confidence bound in this setting. This upper confidence bound has a direct relationship to the one-sided
test, namely:
1. A level  test of H0:  ≥ 0 against the one-sided alternative H1:  < 0 rejects H0 exactly when
the value 0 is above the 1–α upper confidence bound.
2. A level  test of H0:  ≤ 0 against the one-sided alternative H1:  > 0 rejects H0 exactly when
the value 0 is above the 1–α lower confidence bound.
ANOVA Test
∑ ∑( ̅)
∑ (̅ ̅)
∑ ∑( ̅ )
DFTotal = N – 1
DFInter = k – 1
DFIntra = N – k
where
 F is the result of the test
 k is the number of different groups to which the sampled cases belong
 ∑ is the total sample size
 ni is the number of cases in the i-th group
 yij is the value of the measured variable for the j-th case from the i-th group

33
 ̅ is the mean of all yij
 ̅ is the mean of the yij for group i.
The test statistic has a F-distribution with DFInter and DFIntra degrees of freedom. Thus the null
hypothesis is rejected if ( )
ANOVA Multiple Comparisons
Difference of Means
̅ ̅
Standard Error of the Difference of Means Estimator
√ ( )
Scheffe’s Method
Confidence Interval for Difference of Means
( ) ̅ ̅ √ ( ) ( )
Source: http://en.wikipedia.org/wiki/Scheff%C3%A9%27s_method
Tukey's range test HSD
( ) ̅ ̅ ( ) √ ( )
Where q is the studentized range distribution.
Source: https://en.wikipedia.org/wiki/Tukey%27s_range_test
Fisher's Method LSD
If overall ANOVA test is not significant, you must not consider any results of Fisher test, significant or
not.
( ) ̅ ̅ ( ⁄ ) √ ( )
Where t is the student distribution.

34
Bonferroni's Method
The family-wise significance level (FWER) is α = 1 - Confidence Level. Thus any comparison flagged by
ISSTATS as significant is based on a Bonferroni Correction:
( )
( )
Where k is the number of groups.
( ) ̅ ̅ ( ⁄ ) √ ( )
Sidak's Method
The family-wise significance level (FWER) is α = 1 - Confidence Level. So any comparison flagged by
ISSTATS as significant is based on a Sidak Correction:
( ) ( )
( ) ( )
Where k is the number of groups.
( ) ̅ ̅ ( ⁄ ) √ ( )
Welch’s Test for equality of means
The test statistic, F*
, is defined as follows:
∑ ( ̅ ̃)
( )
∑
where

35
 F*
is the result of the test

 ∑ ∑
 ̃ ∑ ̅

( )
The test statistic has approximately a F-distribution with k-1 and ∑
degrees of freedom. Thus
the null hypothesis is rejected if ( )
Brown–Forsythe Test for equality of means
The test statistic, F*
, is defined as follows:
∑ ( ̅ ̅ )
∑ ( )
where
 F*
is the result of the test
 ni is the number of cases in the i-th group (sample size of group i)
 ̅ ∑ ̅
is the overall mean.
The test statistic has approximately a F-distribution with k-1 and df degrees of freedom. Where df is
obtained with the Satterthwaite (1941) approximation as
∑
with
( )
∑ ( )
Thus the null hypothesis is rejected if ( )
Homoscedasticity Tests
Levene's Test
The test statistic, F, is defined as follows:

36
∑ ( ̅ ̅ )
∑ ∑ ( ̅ )
where
 Yij is the value of the measured variable for the j-th case from the i-th group
 | ̅ | where ̅ is a mean of i-th group
 ̅ is the mean of all Zij
 ̅ is the mean of the Zij for group i.
The test statistic has a F-distribution with k-1 and N-k degrees of freedom. Thus the null hypothesis is
rejected if ( )
Source: http://en.wikipedia.org/wiki/Levene%27s_test
Brown–Forsythe Test for equality of variances
The test statistic, F, is defined as follows:
∑ ( ̅ ̅ )
∑ ∑ ( ̅ )
where
 Yij is the value of the measured variable for the j-th case from the i-th group
 | ̃ | where ̃ is a median of i-th group
 ̅ is the mean of all Zij
 ̅ is the mean of the Zij for group i.
The test statistic has a F-distribution with k-1 and N-k degrees of freedom. Thus the null hypothesis is
rejected if ( )
Source: http://en.wikipedia.org/wiki/Levene%27s_test
Bartlett's Test
Bartlett's test is used to test the null hypothesis, H0 that all k population variances are equal against the
alternative that at least two are different.
If there are k samples with size ni and sample variances S2
i then Bartlett's test statistic is

37
( ) ( ) ∑ ( ) ( )
( )
(∑ ( ) )
where

∑ ( )
is the pooled estimate for the variance
The test statistic has approximately a chi-squared distribution with k-1 degrees of freedom. Thus the null
hypothesis is rejected if ( ).
Source: http://en.wikipedia.org/wiki/Bartlett%27s_test

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