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M I C R O S T A T S M A N U A L M I C R O S T A T S M A N U A L
=================================
Author : Mike Hart, (c) 1987 [Public domain version,1991]
Thurnby Computing
698 Uppingham Road
Thurnby
Leicestershire
LE7 9RN
Public Domain version restricted to 30 rows x 45 columns Public Domain version restricted to 30 rows x 45 columns
Table of Contents
=================
1. INTRODUCTION AND TUTORIAL
1.1 What is MICROSTATS ? . . . . 1
1.2 Who uses MICROSTATS ? . . . . 1
1.3 How does MICROSTATS work ? . . . . 2
1.4 Getting Started . . . . 3
1.5 Going Further . . . . 5
2. LIBRARY OF MICROSTATS COMMANDS
2.1 Entering Data . . . . 9
2.2 Column Statistics . . . . 10
2.3 Printing out Data . . . . 10
2.4 Arithmetic on Columns . . . . 11
2.5 Manipulations upon Columns . . . . 11
2.6 Plots . . . . 12
2.7 Histograms . . . . 13
2.8 Bi-variate Statistics
Correlation . . . . 14
Regression . . . . 15
Contingency Tables . . . . 17
Chi-Square . . . . 18
2.9 Data Generation . . . . 21
2.10 Edit Commands . . . . 23
2.11 Row Commands . . . . 23
2.12 Statistical Tests
Twosample 't' . . . . 24
Pooled 't' . . . . 25
Mann-Whitney non-parametric . . . . 25
2.13 Save and Retrieve Files . . . . 26
2.14 General Commands . . . . 28
2.15 Avoiding Crashes ! . . . . 29
3.0 ALPHABETIC INDEX OF COMMANDS . . . . 30
4.0 GENERAL INDEX . . . . 32
- 1 -
1. INTRODUCTION AND TUTORIAL1. INTRODUCTION AND TUTORIAL
=========================
1.1 WHAT IS MICROSTATS ?1.1 WHAT IS MICROSTATS ?
------------------
MICROSTATS has been designed as a general purpose, easy-to-
use statistics package for use on IBM and IBM compatible
machines with at least 256K of memory. It closely resembles
the MINITAB statistical package written by the University of
Pennsylvania which is to be found on many main-frame
installations. ( For the technically minded, the program
is written in Turbo PASCAL which means that the execution
speed is very fast )
1.2 WHO USES MICROSTATS ?1.2 WHO USES MICROSTATS ?
-------------------
Many students in their training for commerce, industry and
the professions are exposed to a course in statistics which
they may well learn and then just as quickly forget! This
package is designed so that students may use a microcomputer
to provide a range of the most common statistical techniques
when analysing a set of data. At the same time, it is also
a tool for those who require statistical analysis but do not
require the power or sophistication of a large-scale
statistical package like SPSS.
The following categories of individuals, amongst others,
will find MICROSTATS useful to them :
- students - students at school or college undertaking a statistics
course either in its own right or as part of a wider
course.
- researchers - researchers e.g. in market research who require an
analysis of data that has been collected in
quantitative form (typically questionnaire data)
- professionals - professionals in industry, commerce or the professions
who are not 'professional statisticians' but who
would like to use a personal computer rather than a
programmable calculator to give them a fairly swift
analysis of their data. (For example, a teacher could
see if there was a statistical correlation between the
marks that were being awarded during the year and
those obtained in the end of year examination)
- 2 -
1.3 HOW DOES MICROSTATS WORK ? HOW DOES MICROSTATS WORK ?
------------------------
MICROSTATS assumes that the user has several columns of data
organised in a worksheet and is organised to reproduce ( and
eventually manipulate) that worksheet. The fundamental
concept, which it is important to appreciate early on, is
that MICROSTATS primarily manipulates COLUMNS of data and
expects that data will be numerical, and not textual, data.
In this, it differs from a spreadsheet which typically
handles text and formulae in addition to numbers.
Columns of data, which are the basic unit of analysis used
in the MICROSTATS system, may be called by a number which
MUST be preceded by the letter C ( or c) and no intervening
space e.g. C1. Columns of data may also be allocated a name
up to eight characters long and this name may also be used
to refer to data provided that the name is always enclosed
in (the same!) single quotation marks e.g. 'MARKS'
(On most IBM keyboards, this is the quotation mark found on
the same key as the @ symbol or the shifted 7)
MICROSTATS works by scanning a line which is input to the
computer after the user has responded to the MICROSTATS
prompt (Command ? )with a command and then pressed ENTER. It
picks out the keyword that it needs ( or more specifically,
the first four letters of a keyword) and then checks it
against an internal dictionary of some 70+ words. If the
keyword is found, then MICROSTATS will activate the sub-
routine which performs the analysis suggested by the user.
Notice that MICROSTATS does not necessarily display anything
on the screen after certain commands - if you get the
Command ? prompt back and no error message, you can
generally assume that the system has obeyed your request !
If you want to see the results of your commands, you can
generally PRINT out the relevant columns to satisfy yourself
that your command has worked!
If you use a command that MICROSTATS does not recognise
(e.g. you may have mis-typed it) then you will get a
warning beep and a 'Word not found' message.
MICROSTATS will take the whole of an input line and 'parse'
it which means that the relevant information will be
extracted and the rest of the line will be ignored.
Generally, everything that is not a keyword, a column number
(preceded by C) or a value will be ignored. This means that
you can address MICROSTATS with English-type commands,
provided that your first word is always a keyword. Later
on, as your familiarity with the system increases, you can
progress to the shortened form of the command if you prefer.
- 3 -
E.g. assume that we have a column of data in C1 and we
wish to multiply this column by 10 and put the results
into C2.
We can use the 'long form' of the command :
MULTiply data in C1 by 10 and put the result into C2
Or we could use the 'short' form of the command (which is
what MICROSTATS actually acts upon) :
MULT C1 10 C2
Notice in the long form of the command that only the first
four letters are significant. If we made a mistake in any
of the letters from the fifth onwards then MICROSTATS would
ignore this.
If you forget any of the words in the MICROSTATS dictionary,
then you can either press ? or type the keyword HELP which
will give you access to one of five HELP screens where the
commands are displayed in logical groups.
1.4 GETTING STARTED GETTING STARTED
---------------
Assuming that you have the computer 'booted up' under MS-
DOS, then merely insert the MICROSTATS disk and type the
filename which is MS. MICROSTATS should load, give a title
page and invite you to press RETURN to start. You will be
presented with information of the size of your worksheet and
the Command ? prompt. Try the following, not forgetting to
press RETURN (or ENTER) at the end of each line:
SET data into C1
At this stage, a new prompt will appear as follows :
DATA >
and you should then type in some data at least one space as
a separator before each number:
DATA > 3 4 5 6 7 8 9
As soon as you press RETURN, the DATA > prompt will reappear
and you could enter as many more lines of data as you wish.
You should NOT attempt to enter more than a one full line of
data (i.e. 80 characters of data) at a time - it is always
possible to enter additional data on subsequent lines.
- 4 -
To finish entering your data, make sure that you have typed
ENTER at the end of the previous line and the DATA > prompt
is still in front of you. Now type END to signify that you
have finished and the data you have just entered will be
displayed to you on the screen with an indication of the
number of items that you have input.
The whole transaction will look like this :
Command ? SET data into C1
DATA > 3 4 5 6 7
DATA > END
1. 3.000
2. 4.000
3. 5.000
4. 6.000
5. 7.000
5 data items entered in C1
Command ?
When the Command ? prompt re-appears, you may now issue
another command. Print out the data that you have just
entered with the following command :
Command ? PRINT data in C1
C1
(n= 5)
1. 3.000
2. 4.000
3. 5.000
4. 6.000
5. 7.000
and the data that you have just entered will appear on the
screen.
Now that you have some data in the computer, try the command :
Command ? DESCRIBE C1
and you should see the following output :
Count of C1 = 5
Minimum of C1 = 3.000
Maximum of C1 = 7.000
Sum of C1 = 25.000
Mean of C1 = 5.000
Median of C1 = 5.000
Standard dev-n [pop] of C1 = 1.414
Standard dev-n [samp] of C1 = 1.581
- 5 -
You may now like to try some of the other commands that will
perform statistics upon a single column of data. For
example, try giving the command :
Command ? SUM data in C1
and then try extending the command even further by trying :
Command ? SUM data in C1 - put data into C2
In this case, the sum of the data in C1 will be put into as
many rows of C2 as there are rows in C1.
To give names to your two data columns then use the NAME
command as follows :
Command ? NAME C1 'DATA' and C2 'SUM'
Now type INFO or the full command INFOrmation and you will
see the status of your worksheet displayed as follows :
C1 'DATA' n=5
C2 'SUM' n=5
58 columns of length 150 still available for use
You should get used to using the following two commands
often as they help you keep track of your worksheet :
INFO ( checks on the number and length of your columns)
PRINT Cx ( where x is a column number e.g. C1 or a range
e.g. PRINT C1-C5)
1.5 GOING FURTHER GOING FURTHER
-------------
Let us assume that you still have the numbers
3 4 5 6 7
contained in C1. You can PRINt C1 to check whether this is
still the case. Now we are going to create a second column
of data to explore some of the other possibilities offered
by MICROSTATS. Try the following :
Command ? MULTiply C1 by C1 - put result into C2
If you prefer, you may use the short form of the command
which merely gives MICROSTATS the bare minimum that it
requires i.e. the first four letters of the keyword and the
columns that it must use to operate upon. The short form of
the command is :
Command ? MULT C1 C1 C2
- 6 -
MICROSTATS will respond equally well to either the short or
the long form of the command - feel free to use whichever
seems more natural to you.
If the Command ? prompt returns and there has been no error
message, we can assume that the command has been
successfully performed. To check this out, we can ask
MICROSTATS to perform the following :
Command ? PRINt C1 and C2
and we should find that C2 contains the square of C1 i.e.
each number in C1 has been multiplied by itself and then put
into C2.
In the print-out on the screen, you will notice that
MICROSTATS has printed out each number to three places of
decimals. It does this 'by default' but it is possible to
alter this by using a command such as
Command ? DISPlay C1 to 2 dp
Command ? DISPlay C2 to 2 dp
If you then PRINt C1 and C2 once again, you will see that
have output to two places of decimals. You can experiment
with this if you like.
Now we are going to do some more serious work. If a
statistician wishes to see how much of one variable is
associated with another (s)he generally wishes to generate a
statistical measure known as a CORRELATION COEFFICIENT. If
you have ever done this long-hand, you will know what a lot
of effort is involved. MICROSTATS will do this for you
quickly and easily by using the command :
Command ? CORRelate C1 and C2
You should obtain a result that informs you that the
correlation of C1 and C2 is equal to 0.9931, as well as
other statistical information giving you the probability
that you could have obtained this result by chance alone
( in this case, about 3 in every 10,000 cases ! )
This indicates a very high positive correlation. The
correlation coefficient can take a value of +1 for a perfect
positive correlation, -1 for a perfect negative correlation
or any value in between. A value very close to +1 indicates
a very high degree of association between the two sets of
values, which is not surprising considering that one is the
square of the other.
- 7 -
If we wish to derive a mathematical equation to link
together the two sets of observations, we use what
statisticians call a REGRESSION EQUATION. This is an
equation that informs you on the basis of the data that the
computer has what would be a predicted value for one
variable once we are a given a specific value for the other.
Try the command :
Command ? REGRESS C2 upon C1
and your output should inform you that the regression
equation of C2 upon C1 is equal to :
y= -23.0000 + 10.0000 * X
The asterisk is a standard method in computing of saying
'multiplied by' and this regression equation is telling us
that y will take a value of 10.000 times a particular value
of X minus 23.0
Now give names to the two variables created by using the
commands :
Command ? NAME C1 'Number' and C2 'Square'
Command ? INFO
Finally, we are going to do something which is visually more
exciting and also shows that you can use names in commands
as well as column numbers. Once we have allocated names to
columns we are quite free to use them instead of column
numbers. We must ensure that the names are exactly those
which have been allocated. For example the names 'NUMBER'
and 'NUMBER ' look exactly alike to a human reader but are
regarded as quite different names by the computer, where the
space would be regarded as an additional character.
If we now use the PLOT command as follows :
Command ? PLOT 'Number' v 'Square'
then we will find a plot of the two variables in which the
maximum and minimum of each variable are displayed, as well
as the names of the variables and the correlation
coefficient. If you perform the Plot in reverse order i.e.
PLOT 'Square' v 'Number' you will find the axes are now
reversed.
To clear the system of data, we can now type
Command ? ERASE C1 C2 and the two columns of data will be
'rubbed out' You can check that this is so by using INFO
which will inform you that all of the columns are empty and
un-named and give you the total available on the system.
- 8 -
This concludes the tutorial on the use of the MICROSTATS
system. The following sections will describe how to use the
more sophisticated features contained in the package. This
also assumes that you have a certain degree of statistical
knowledge so that you can understand what type of analysis
is being performed, and also that you are aware of the
underlying assumptions.
If your statistical knowledge is limited, then you can try
to use some of the other simple commands that are listed in
the complete list of MICROSTATS commands and experiment to
discover what they do.
- 9 -
2. LIBRARY OF MICROSTATS COMMANDS2. LIBRARY OF MICROSTATS COMMANDS
==============================
2.1 ENTERING DATA2.1 ENTERING DATA
-------------
SET data into C1
DATA > 3 4 5 6 7 8 9
DATA > 10 11 12 13 14
DATA > END
Use for entering data one column at a time. A long column
of data may be split over several MICROSTATS input lines but
your input line should not exceed about 100 characters. As
a rule of thumb then whenever the cursor goes onto the
second 'screen line' then you should press ENTER to register
that line of data.
The SET command is actually also an command that will append
data. This means that if you SET data into a column which
already has some data in it, then the new data will be
appended to the bottom of the existing data. Make sure that
you ERASE the column if you want to ensure that the new data
is the only data in the column.
Remember to exit with END which should be on its own DATA >
line and not put at the end of a series of numbers.
READ data into C1-C3
DATA Row 1 > 1 4 7
DATA Row 2 > 2 5 8 3
Too many data items - re-enter line
Data Row 2 > 2 5
Too few data items - re-enter line
Data Row 2 > 2 5 8
Data Row 3 > 3 6 9
Data Row 4 > END
3 rows of data entered into C1-C3
Use for entering data into several columns simultaneously.
This is best used when you are more experienced in data
entry. Note that the number of columns indicated and the
number of data items per line must tally exactly. Do not
leave spaces on either side of the hyphen mark.
In the example, exactly three data items per line are
expected. If you supply more or less then the system will
warn you and prompt you to re-enter the correct number of
data items.
NAME data in C1 'name1' and C2 'name2'
Names the data - check that you use the correct apostrophe !
- 10 -
2.2 COLUMN STATISTICS2.2 COLUMN STATISTICS
-----------------
SUM data in C1
SUM data in C1 put into C2
If a second column is specified, then the value generated by
the command is put into as many rows of the second column
specified as there rows in the first column specified.
Commands which work in an identical fashion are :
COUNt Number of data items in the column
MAXImum The MAXIMUM value in a column
MINImum The MINIMUM value in a column
AVERage or ) Both commands give the arithmetic
MEAN ) mean
MEDIan The MEDIAN measure of central
tendency i.e. the value which
occupies the central position
once the data has been sorted
into ascending order.
STANdard dev-n Standard deviation calculated
with a divisor of N (population)
STDE Standard deviation calculated
with a divisor of N-1 (sample)
DESCribe This command will give summary
statistics of any one column :
Count Sum
Minimum Mean
Maximum Median
Standard deviation [population]
Standard deviation [sample]
2.3 PRINTING OUT DATA2.3 PRINTING OUT DATA
-----------------
PRINT C1
PRINT C1-C7
You may print out any one column or up to seven adjacent
columns of data. Any columns in excess of seven will be
ignored. Do not leave a space on either side of the hyphen.
After a screenfull of data [ 20 items ] you will be prompted
whether to view more data or exit the printing of the data.
- 11 -
2.4 ARITHMETIC ON COLUMNS2.4 ARITHMETIC ON COLUMNS
---------------------
You may perform simple arithmetic on your columns using
either another column or a number. The results may be put
into another column or even stored straight back into the
original column, in which case they overwrite the previous
contents. Examples include :
ADD C1 to C2 put into C3
SUBT C2 from C1 put into C3
MULTiply C1 by 10 put into C3
DIVIde C4 by 10 put into C4
RAISe C1 to the power of 3 put into C5
NB. Once you have performed these manipulations, the
MICROSTATS Command ? prompt will return. Print out
the relevant columns if you want to satisfy yourself
that the results are as you intended them to be.
2.5 MANIPULATIONS UPON COLUMNS2.5 MANIPULATIONS UPON COLUMNS
--------------------------
These manipulations allow you to transform the data in some
way ( e.g. by taking a log or a square root) - you may then
put the transformed data into another or even back into the
same column. The manipulations include the following :
SQRT of C1 put into C2 ( SQRT = SQUARE ROOT )
ABSolute of C1 put into C2 ( ABS = ABSOLUTE )
ROUNd C1 to x decimal ( ROUND to specified
decimal places no. of decimal places )
LOGE C1 put into C2 ( LOGE = Natural LOG )
EXPOnent C2 put into C3 ( EXPO = EXPONENT )
SORT C1 put into C2 ( SORT into ascending order )
( To sort data in C1 into a descending order, use the
following sequence :
MULT C1 -1 put into C2
SORT C2 put back into C2
ABS C2 put back into C2 )
- 12 -
RANK data in C1,put into C2 ( RANK gives a ranking number )
COPY data in C1 to C3 ( copies data from C1 into C3 )
COPY data from C1-C5 to block starting at C11
( Copies the BLOCK of data from C1-C5 into a new block
starting at C10.
NB no spaces on either side of the hyphen )
2.6 PLOTS2.6 PLOTS
-----
MICROSTATS will perform a SCATTER PLOT or SCATTERGRAM of
data on two matching variables.
The typical plotting command is :
PLOT C1 C2
but before you issue the command, you should take some
elementary precautions to prevent a program 'crash' or
'abort'. You should first :
- ensure that the columns are of equal length
- not try to plot data in which all of the value of one
or other variable are identical
If you observe numbers rather than asterisks ( * ) in your
plot, this is because MICROSTATS is informing you that two
data points are 'mapped' onto the same data co-ordinates.
Notice that the plots will be scaled and the minimum, mid-
point and maximum of each given as well as the column names
( if any )
The first column that you specify will be the vertical axis
and this is usually the 'independent' variable. The second
column specified will be on the horizontal axis and this is
usually termed the 'dependent' variable ( as it 'depends' in
causal terms upon the first variable) For example,if we had
two variables in which one was a student grant ('GRANT')
whilst the other was level of parental income ('INCOME').
then 'INCOME' would be the independent variable and would be
entered first whilst 'GRANT' would be the dependent variable
and be entered second, thus :
PLOT 'INCOME' v. 'GRANT'
Notice also that the correlation coefficient is calculated
and displayed in the top right hand corner of the graph.
- 13 -
Plots are useful to see if there is a tendency for the data
to cluster and form one of the following patterns :
- a 'line' sloping forwards from bottom left to top right.
The more closely the data clusters around such a line,the
more it suggests a 'positive correlation' in which as one
variable increases, so does the other.
- a 'line' sloping backwards from bottom right to top left.
The more closely the data clusters around such a line, the
more it suggests a 'negative correlation' i.e. one
variable increases as the other decreases.
- No apparent pattern at all. This suggests the absence of
association which would be no correlation ( or only a very
small one)
2.7 HISTOGRAMS2.7 HISTOGRAMS
----------
The HISTogram command requests a plot of a single variable
so that you can examine its shape. A typical histogram of
random numbers from 1-100 would show the following :
Command ? HISTogram of data in C1
Choose first midpoint,interval (y/n?) y
First mid-point ? 5.5 Interval ? 10
Middle of Number of
Interval Observations
5.5 10 **********
15.5 8 ********
25.5 13 *************
35.5 8 ********
45.5 8 ********
55.5 12 ************
65.5 6 ******
75.5 9 *********
85.5 9 *********
95.5 17 *****************
There are several things to note about the HISTogram command :
- You are given the option to choose the first mid-point
and the interval. If you press 'n' or just RETURN then
the command will choose what appears to be sensible
defaults depending upon the shape of the data but which
may appear strange to you. If you choose to select the
midpoint and the interval, then you should have at least
a rough idea of what the data 'looks like' before you
start.
- 14 -
- Notice that statistically the mid-points may not be just
where you expect them to be. For example, in the example
given above, then all of the data lying in the range 0.5
upwards to 10.49999 downwards would be regarded as lying
within the first block. The mid-position of a range
which extends from 0.5 - 10.4999 is (10.4999 - 0.5) + 5
which is 5.5 and not 5.0 !
- The HISTogram command cannot deal very sensibly with very
small or very large numbers. Under such circumstances,
it is probably sensible to scale them up ( or down)
yourself and put the data into a new column and then try
the effects of HIST on the scaled column. For example, a
range of 100 numbers in the range 0-1 are best scaled up
to 0-10 or even 0-100. After all, the HISTogram analysis
is only intended to give you a visual representation of
the actual 'shape' of the data rather than a precise
mathematical result and therefore such scaling up or down
is quite legitimate.
2.8 BI-VARIATE STATISTICS2.8 BI-VARIATE STATISTICS
---------------------
Bi-variate statistics is the name given to the statistical
analysis of pairs of data, such as that dealt with already
in the PLOT command. The following bivariate statistics are
provided :
CORRelate C1 and C2
In this case, the 'Pearson product-moment correlation
coefficient' between the two stated variables will be
performed. Before you issue the command, then check the
following two points :
- via INFO make sure that you have equal numbers of data in
each column
- via PRINT make sure that all of the values of one or other
column are not identical. If so, the command will fail and
computer may well abort.
In order to INTERPRET the value of the correlation
coefficient, then read the entry under PLOT on pp.12-13. It
is particularly important to remember the following two
points :
- a high positive ( or negative ) correlation correlation
coefficient cannot be taken to imply CAUSATION
- a high ( or low ) correlation may be significant in purely
statistical terms but not be significant in social
scientific terms. For example, a high correlation between
heights and weights of a general sample of the population
is not surprising, as taller people are generally heavier.
- 15 -
- Conversely, the absence of a correlation may not achieve
statistical significance but may be highly significant in
terms of a social scientific model. The absence of a
relationship where we might be led to expect one ( for
example between unemployment and mental illness) might
prove to be highly significant in terms of social
scientific theory, even though the result does not achieve
a degree of statistical significance.
REGRess C1 upon C2
REGRess C1 upon C2 with intercept in C3, slope in C4,
value of x, put predicted y in C5
e.g. REGR C1 C2 C3 C4 10 C5
A regression line, sometimes known also as a 'least squares'
line is a line that best fits a series of data pairs and
which can be used to predict one variable once we know :
- the regression equation itself
- the value of the independent variable.
The general form of a regression equation is :
y= a + b * (x)
where y = dependent variable ( that we wish to discover )
x = independent variable ( which may be given )
a = intercept
b = slope
WORKED EXAMPLE
Put the following data pairs into C1 and C2 where :
C1 = Salary
C2 = Years of Education since age 15
C1 C2
5000 2
3000 4
6000 5
4000 6
7000 7
6000 8
9000 9
- 16 -
Then get the regression equation, as follows :
Command ? REGRess C1 upon C2
Regression of C1 on c2 =
y=2401.6393+ 565.5738 * X
Now try the longer form of the command, but this time we
wish to know what salary that can be expected from an
individual with 3 years years of 15+ education.
Command ? REGRess c1 on c2,a in C3,b in C4,x=3,result in C5
y=2401.6393+ 565.5738 * X
Command ? PRINt C3-C5
C3 C4 C5
(n= 1) (n= 1) (n= 1)
1. 2401.6393 565.574 4098.361
Here the critical result is in C4 that tells us that with 3
years of post 15+ education ( x=3) the predicted level of
salary will be :
y = 2401.6393 + (565.574 * 3 )
= 4098.361
Whereas in correlation the result does not depend upon which
variable is C1 and C2, the same is NOT true of regression.
In regression, the DEPENDENT variable is regressed onto the
INDEPENDENT variable. In terms of our example, then SALARY
( the dependent variable) will be regressed upon EDUCATION
( the independent variable) If you experiment by trying to
regress C2 on C1 then you will see a very different result,
so it is important that the order of variables is understood
before you use this command.
MICROSTATS only supports simple regression. If you require
multiple regression, then you will need a more sophisticated
package. If you suspect that the data is curvilinear ( e.g.
the kind of relationship that is met when one number is the
square or higher power of the other) then try a logarithmic
transformation of the data before you regress.
- 17 -
CONTingency table of data in C1 and C2
The CONTINGENCY TABLE command is designed to 'table' those
cases where we have integer numbers in two columns which
represent 'coding' numbers e.g. in C1 we might have the
numbers 1-2 which represent female and male whilst in C2 we
might have numbers 1-5 representing five categories of
political identification. Such data is often known as
'categorical' data. If we wish to see how many of one
category are represented in the other ( e.g. how many female
Conservatives) then we would use the CONTingency table
command.
Worked Example :
Use the following commands which put 100 random cases of
either 1,2 in C1 and either 1,2,3,4,5 in C2.
Command ? IRAN 100 cases from 1 2 in C1
Command ? IRAN 100 cases from 1 5 in C2
Now table the result :
Command ? CONTingency C1 with C2
You should get a result similar in appearance to the
following - it will probably not be identical because the
random number generator may well have produced a different
pattern of data to fill C1 and C2 :
C2 > 1 2 3 4 5
-------------------------------
C1 1 ! 8 ! 12 ! 5 ! 13 ! 13 ! 51
-------------------------------
2 ! 11 ! 9 ! 10 ! 9 ! 10 ! 49
-------------------------------
19 21 15 22 23 100
There are two points to note about this command :
- Only try to table consecutive integers up to a maximum of
10 in each direction of the table
- If you wish to put the cell results into another column
for later analysis by chi-square (p.17) then specify a
third column as the starting point :
e.g. CONT C1 with C2, results at C10
- 18 -
In this case, MICROSTATS responds with a message :
Data fed into C10-C14
and you can confirm this result by PRINting out the
relevant columns of data.
Note that you can put 5 or less cell contents into the
new block. If you attempt to put more than five, then
the command will be ignored. (This is because the CHI-
SQUARE command which uses these results will only accept
a table 5 cells wide.)
CHISquare of data in C3-C5
The CHISquare command will accept a block of up to five
rows/columns and perform a chisquare test upon the data.
The underlying statistical assumption is that data is
measured at the nominal or categorical level ( e.g. code
numbers representing a sex or a political party.) To
demonstrate CHISquare. then put in the following data using
the random number generator. We are going to generate a sex
coding (1,2) in C1 and a political party coding in C2 (1,2
or 3) :
Command ? IRAN 100 nos from 1 to 2 put into C1
Command ? IRAN 100 nos from 1 to 3 put into C2
Command ? CONTingency C1 and C2 put cells into C3
C2 > 1 2 3
------------------
C1 1 ! 18 ! 14 ! 14 ! 46
------------------
C2 2 ! 19 ! 19 ! 16 ! 54
------------------
37 33 30 100
Data fed into C3-C5
- 19 -
CHISquare of data in C3-C5
Expected frequencies are printed below observed frequencies
I I I I
I C3 I C4 I C5 I Totals
------------------------------------------------------
1 I 18 I 14 I 14 I 46
I 17.02 I 15.18 I 13.80 I
------------------------------------------------------
2 I 19 I 19 I 16 I 54
I 19.98 I 17.82 I 16.20 I
------------------------------------------------------
Totals I 37 I 33 I 30 I 100
0.06 + 0.09 + 0.00 +
0.05 + 0.08 + 0.00 +
Total chi-square = 0.280 df = 2 p=0.8695
(You will not get get exactly these results because the
random number generator will have generated a different
pattern of initial data but it should not be too dissimilar)
CHISquare takes the initial sets of data in each cell ( the
'observed') data and then work out the 'expected' data in
each cell on the assumption that one variable is exactly
proportionately represented within the other.
In this case, we are trying to see if there are sex
differences in the way in which people vote. For each cell,
the 'expected' differences are worked out according to the
formula :
2
(Observed-Expected)
-------------------
Expected
and this is the chi-square for that cell. Finally all of
the chi-squares are summed, the degrees of freedom (df)
calculated according to the rule (rows-1)*(columns-1) and
the probability worked out. Any probability which is
greater than 0.05 means that there is not a statistically
significant difference in the proportions of C2 represented
in C1 ( in terms of this example, a sex difference in
voting behaviour)
- 20 -
Points to note about chisquare are :
- The data should be measured at the nominal level i.e.
categories such as male/female
- Cells with an expected frequency of less than 5 can
generate chi-squares that give a misleading result. If
this is the case, then a warning message will be given.
It is generally best to combine categories to make the
numbers in each cell so much larger.
- Zero cells will abort the analysis, with a division by
zero error! Make sure that you have no zero cells in the
analysis before you start.
- 21 -
2.9 DATA GENERATION2.9 DATA GENERATION
---------------
There are times when it is useful to generate displays of
data for demonstration purposes. The following commands are
provided in MICROSTATS :
GENErate values from 1 to 100 in C1
This will generate data from the first value to the second
value stated in the relevant column. It could be used to
provide an index number for a series of data.
DEFIne the value of 10 into the first 5 rows of C1
This allows a constant to be put in as many rows of the
column as you desire.
IRANdom 100 random integers from 1 to 100, put into C1
IRANdom 100 random integers from 1 to 100, out into C1-C5
This is an integer random number generator. You should
remember to state the numbers of integers required,
followed by the lower limit, the upper limit and the
destination column.
URANdom 100 random numbers and put into C2
URANdom 100 random numbers and put into C2-C5
The URANdom random number generator generates floating point
numbers in the range 0 to 1 and puts the required number in
the destination column. You can multiply it up if required.
2.10 EDIT COMMANDS2.10 EDIT COMMANDS
-------------
It is often necessary to edit data because it may have to be
manipulated or sifted to meet particular needs. The
following editing commands are supplied :
PICK the rows from 1 to 2 in C1 and put into C2
This command enables the user to 'top' or 'tail' a column to
ensure it is generally correct. If you had entered one too
many items in a column in error, then the PICK command could
be used to put the correct number of items back into the
same or a different column.
RECOde the values from 3 to 5 in C1 to 1 and put into C2
This command enables the user to 'degrade' the data. For
example, if there were several political parties coded 1 to
5 then they could be reduced to 2 groups by recoding all the
values from 3-5 to either a 1 or a 2.
- 22 -
CHOOse values 1 to 5 in C1 (and corresponding C2 ) and put
into C3 and C4
This is one of the most powerful editing commands, as it
enables us to make sub-groups for further analysis. For
example, if males/females were coded as 1,2 in C1 then the
'male' data could be separated from the 'female' data.
Worked example :
SET C1
DATA > 1 2 1 2 1 2 1 2 1 2
DATA > END
SET C2
DATA > 3 2 7 4 2 6 4 2 1 4
DATA > END
CHOOSE 1 in C1 (corr C2) and put into C3 and C4
PRINT C3 C4
C3 C4
(n= 5) (n= 5)
1. 1.000 3.000
2. 1.000 7.000
3. 1.000 2.000
4. 1.000 4.000
5. 1.000 1.000
As you can see, the coding number in C1 i.e. 1 and the
corresponding data from C2 have been sifted out and put into
C3 and C4.
OMIT data from 5 to 7 in C3 put into C4
This data is the obverse of the CHOOse command. Whereas
CHOOse will select the data that you request and transfer
that data over to the destination columns, the OMIT command
will transfer over all of the data except that which you
wish to omit.
JOIN the data in C2 to the end of C1 and put back into C1
Notice here that the data you wish to join to the end of the
other column is specified first - you have the opportunity
to put the newly augmented column in a new column or back
into one of the original ones.
SUBStitute the value of 10 into row 9 of C2
This command would be used if you had made an error ( e.g.
in data input) that you wish to correct after having entered
the data. Remember that the value that you wish to
substitute is specified first, and the row of the column for
which it is destined is specified second.
- 23 -
DELETE row 2 from C1
DELETE row 2 from C1-C5
Be careful with any delete command, as once deleted the data
cannot be recovered. This command will remove a row of data
from either a single column or a block of columns.
ERASE DATA IN ROWS c3-c5
This command erase single columns or blocks of columns.
COPY C1 into C2
COPY the block from C3-C6 into a new block starting at C13
The simple version of copy performs a straight copy of one
column into another. The more advanced version will copy a
block of data but the user should specify the source columns
using a hyphen (no spaces!) and the start of the new block.
DISPlay the data in C3 to 1 decimal place.
DISPlay the data in unchanged cols to 2 decimal places.
This command alters the DISPLAYED value to the required
number of decimal places but not the value which MICROSTATS
holds internally which will be about 10 places of decimals.
You may either change one column specifically or the rest of
the unchanged columns by specifying no column number. To
display NO decimal places then use the command DISPlay 0
(rather than the command DISPlay with no parameters)
2.11 ROW Commands2.11 ROW Commands
------------
The ROW commands exactly parallel the Column Statistics on
p.10 except that they operate upon rows across columns
rather than individual columns which is the usual method of
analysis. When any of the ROW commands are issued, a table
is given from which users may select the value(s) in which
they are interested. The ROW commands are :
RDEScribe the data in row 1 of C1-C5
RSUM of data in row 1 of C1-C5, put results into C6
RMEAN of data in row 1 of C1-C5, put results into C6
RMEDian of data in row 1 of C1-C5, put results into C6
RMINimum of data in row 1 of C1-C5, put results into C6
RMAXimum of data in row 1 of C1-C5, put results into C6
RSTAndard dev-n [pop'n] data in row 1 of C1-C5, results in C6
RSTD dev-n [sample] data in row 1 of C1-C5, results in C6
- 24 -
2.12 STATISTICAL TESTING2.12 STATISTICAL TESTING
-------------------
To perform a statistical test of data which has been
measured with a ratio or interval level of measurement, we
can use either of the TWOSample or the POOLed commands.
The TWOSample command assumes that we wish to test the
hypothesis that the means of two samples differ
statistically from each other. The underlying assumption is
that the population variances need not be approximately
equal.
WORKED EXAMPLE
IRAN 100 values from 1 100, put into C1 and C2
TWOS C1 and C2
Twosample t C1 vs. C2
n mean stdev se mean
C1 100 52.9200 29.9504 2.9950
C2 100 48.4400 27.2528 2.7253
95.00 PCT C.I. for mu C1 - C2 : (-3.508, 12.468 )
ttest mu C1 = mu C2 (vs. n.e.) :
T= 1.106 p=0.2699 approx. d.f.= 196
The output requires some interpretation. For information,
the means, standard deviations, standard errors and
95% confidence intervals (= C.I.) of the mean are displayed.
The critical results come on the last line of the display
where the critical values are those for T and p. As a rough
rule of thumb, we would expect a significant result for T to
be anything in excess of the value of 2.00. The value of p
gives us the probability that the means differ by amount
that they do under the influence of chance factors alone. A
significant result is achieved when p is equal or less to
the value of 0.05 ( i.e. there is a 5% chance or less that
the observed differences could have occurred by chance
alone.) The inference is, therefore, that non-chance
factors are operating in which case we reject the null
hypothesis that the population means (mu) are equal and
accept the alternative hypothesis that the populations means
(mu) are not, in fact, equal.
The d.f. (degrees of freedom figure is used internally by
MICROSTATS to calculate the values for T and p.)
- 25 -
POOLed test for C1 vs. C2
The output and interpretation of the POOLed test is almost
identical to that of the TWOSample command. However, this
command may be used if the user is confident that the two
populations have approximately equal variances. If in
doubt, the user should generally use the TWOSample test.
MANN-Whitney test.
This is a NON-parametric test which is generally regarded as
almost as powerful as its parametric analogue ( TWOSample
test or 't' test) The data may be measured at the ordinal
level ( as internally, the calculations are performed upon
ranked data) For a full interpretation of 'u', the user
should consult a statistical source. The Mann-Whitney test
is regarded as closely related to the Wilcoxon rank-sum
test. Strictly speaking, the test is used to evaluate the
difference between population distributions, not population
means but when the distributions of the groups are similar
the test does in fact measure differences in central
tendency.
As with the TWOSample and POOLed test, the critical value
are those for T and the probability. As a rule of thumb,
one is looking for a T value of approx. 2.00 or greater and
a probability of equal or less than 0.05 (2-tailed test) in
order to achieve evidence that the distributions differ from
each other significantly.
TTEST data in C1 against a value of 50
This test is used to test a sample mean against a known
value or population mean. The critical value is to observe
whether or not the value is equal to or less than p= 0.05 in
which case we conclude that there is a statistically
significant difference between the sample mean and the
value.
TDIST 1.9603 at 2500 df
This calculation will give the user the proportion of a
distribution ( one and two-tailed) that correspond to the
value for the degrees of freedom specified. It may be
thought of as an alternative to look-up tables.
TINT for data in C1 at 95%
This command gives the user the confidence intervals for the
data in the specified column at the confidence level
requested.
- 26 -
CHID for value 3.84 at 1 df
This command is another alternative to a look-up table. It
provides the user with the probability of achieving the
specified value at specific degrees of freedom. In the
above example, we would get the response
Probability = 0.0500
which informs us that with a normal chi-square table of 2x2
i.e. two rows and two columns which is 1 d.f. then a value
of 3.84 would be achieved only 0.05 (5% of the time) by
chance factors alone.
2.13 SAVE AND RETRIEVE FILES2.13 SAVE AND RETRIEVE FILES
-----------------------
SAVE ( and then follow instructions )
SAVE a:myfile
This command will SAVE the workfile for the user. No
extension should be used as MICROSTATS actually saves two
files, one of which is in specially coded numerical format
(for fast access and compact disk storage) and the other of
which is a text file in which names, if allocated, are
stored. The user does need to be concerned with such
details but it does explain why two files will appear on the
disk for every worksheet saved, one with a .MCS extension
and the other with a .NAM extension.
RETRieve ( and then follow instructions )
RETRieve a:myfile
This command RETRieves files that have been previously saved
under MICROSTATS. It will NOT retrieve other files which
might be accessed with FREAD (q.v.)
DIREctory
DIREctory A:
DIREctory with no parameter will give the list of all of the
files on the logged drive, or the files on a specified drive
if this is requested.
Note that this command does not change the logged disk
drive. In addition to the normal directory display, a
separate list of MICROSTATS files is given. Wildcard
characters such as * or ? are not implemented in this
command.
- 27 -
LOGD
LOGD A:
LOGD with no parameter will remind you of the drive upon
which you are currently logged, and at the same time issue a
directory of files.
LOGD with a legitimate file disk-drive parameter will BOTH
log the user onto the specified drive and also issue a
directory.
FREAD
FREAD a:datafile
The FREAd command will read, or attempt to read, any file in
which data has been saved in a straight ASCII format. As
FREAD can only read in completely 'rectangular' blocks of
data, it is IMPORTANT that any data that is exported by
another package should be absolutely rectangular. For
example, to export a spreadsheet of 2 x 10 columns and 2 x 5
columns, then pad the last columns to 10 with zeros to make
a 'rectangle' that is 2 x 10. Adjustments could be made
once the data is successfully imported into MICROSTATS.
Remember that MICROSTATS will only read numerical and not
textual data.
The user will be prompted for the start column of the data,
which will then be read into consecutive columns.
If data is prepared using a text or word processor for input
into MICROSTATS as well as other packages, then any
legitimate data separator (; or , or <space> ) may be used.
FWRITE
FWRIte will write out data as a straight ASCII file with a
comma delimiting the data items. Before using this command,
remind yourself of the start and end columns by using INFO
as FWRIte will request your start and end columns.
FERAse
FERAse myfile
FERAse will erase any type of file whether saved under
MICROSTATS or not. If there are non-MICROSTATS files that
the user wishes to erase, then the full file name with drive
letter, name and extension should be given.
- 28 -
2.14 GENERAL COMMANDS2.14 GENERAL COMMANDS
----------------
HELP ( or ? )
HELP gives access to five help screens. At the bottom of
each the user may access the (N)ext Page, (L)ast Page or the
(E)xit command.
STOP
STOP completes the work-session. The user is prompted to
save the work-sheet and also asks the user to confirm exit
to ensure that an accidental exit does not occur.
INFOrmation
This command informs the user of the numbers of columns (and
their length) still available for use. The column numbers,
names allocated (if any) and number of data items in each
column will be notified.
MICROSTATS users should use this command FREQUENTLY to check
on the status of their work-sheet and to confirm that the
data that they have in their work-sheet conforms with their
expectations. Similarly, PRINT should be used in
conjunction with INFO to check on the data in columns.
PRINT C1
PRINT C1-C5
PRINT C10 C2 C5 C8
PRINT is a command which ALWAYS requires information as to
which columns of data to print. If a range of columns is
requested, then it should be specified with a hyphen but
with no spaces on either side of the hyphen. Only seven
consecutive columns may be printed if the hyphen form of the
command is used and it is not generally sensible to attempt
to print out more than seven columns if the user wishes to
preserve a 'clean' screen display.
Long columns will stop after twenty items ( a screenfull)
and prompt the user to view the next screenfull or to exit
to the next command. Names are displayed together with
column contents.
- 29 -
PRON
The PRON command stands for PRinter ON. Output normally
directed to the screen will now appear on the printer. In
some cases, this may mean performing 'blind' so the user
should have rehearsed a particular sequence of commands
first to verify their effect, taken a note of the same and
then repeated the same with the PRON switch toggled on.
PROF
The PROF command stands for Printer OFf. Output will be re-
directed back to the screen for a normal 'dialogue'.
NOTE
NOTE displays a comment for documentation purposes.
MICROSTATS will ignore any data on a note line and in this
respect it resembles REM in a BASIC program.
2.15 AVOIDING CRASHES !2.15 AVOIDING CRASHES !
----------------
Despite the warnings built in at various points, MICROSTATS
will occasionally crash or 'abort' In part, this was due to
the fact that the author was sparing in the use of too many
warnings as the more space devoted to these meant less space
devoted to additional commands. But, in addition,
MICROSTATS cannot cope with certain error conditions e.g. a
calculation which involves a division by zero.
Here are some tips and hints to minimise the occasions upon
which MICROSTATS will abort, or at least to make sure that
the consequences are not to dire !
- DO make sure that you do not enter more than a line-
full ( or a little over) of data in the SET command.
- DO save your precious data after a fair amount of
typing or column manipulation. A SAVE every 15-20
minutes only takes a few seconds and ensures that the
potential loss of time is limited to this amount.
- DO make sure that you do not attempt to plot data in
which all of the data in either columns is identical.
- DO make use of INFO, PRINT and the HELP screens in
order to keep a check on the status of the worksheet.
- DO keep a note of events that caused the system to
crash and avoid them in the future !
- 30 -
3.0 ALPHABETICAL INDEX OF COMMANDS3.0 ALPHABETICAL INDEX OF COMMANDS
==============================
11 ABS
11 ADD
10 AVERage
26 CHID
18 CHISquare
22 CHOOse
17 CONTingency
23 COPY
14 CORRelate
10 COUNt
21 DEFIne
23 DELEte
10 DESCribe
26 DIREctory
23 DISPlay
11 DIVIde
23 ERASe
11 EXPO
27 FERAse
27 FREAd
27 FWRIte
21 GENErate
28 HELP
13 HISTogram
28 INFOrmation
17 IRANdom
22 JOIN
27 LOGD
11 LOGE
25 MANN-Whitney
10 MAXImum
10 MEAN
10 MEDIan
10 MINImum
11 MULTiply
7 NAME
29 NOTE
22 OMIT
- 31 -
21 PICK
12 PLOT
25 POOL
28 PRINt
29 PROF
29 PRON
11 RAISe
12 RANK
23 RDEScribe
9 READ
21 RECOde
15 REGRess
26 RETRieve
23 RMAXimum
23 RMEAn
23 RMEDian
23 RMINimum
11 ROUNd
23 RSTAndard_dev'n
23 RSTD_dev'n
23 RSUM
26 SAVE
9 SET
11 SORT
11 SQRT
10 STANdard_deviation
10 STDEv-n
28 STOP
22 SUBStitute
11 SUBTract
11 SUM
25 TDIStribution
25 TINTerval
25 TTESt
24 TWOSample
21 URANdom
- 32 -
4.0 GENERAL INDEX4.0 GENERAL INDEX
=============
Abort, 12, 14, 20, 29
ABS, 11, 30
ABSolute, 11
Access, 3, 26, 28
ADD, 11, 30
Addition, 2, 26, 29
Adjustments, 27
ALPHABETICAL, 30
Alternative, 24, 25, 26
Append, 9
Arithmetic, 10, 11
Ascending, 10
ASCII, 27
Assumptions, 8
Asterisks, 12
Augmented, 22
AVERage, 10, 30
Axis, 12
Block, 12, 14, 18, 23
Blocks, 27
Booted, 3
Categorical, 17, 18
Causal, 12
CAUSATION, 14
Cell, 17, 18, 19, 20
Cells, 18, 20
Central, 10, 25
Chi-square, 17, 18, 19, 26
Chi-squares, 19, 20
CHID, 26, 30
CHISquare, 18, 19, 30
CHOOse, 22
Co-ordinates, 12
Coding, 17, 18, 22
COEFFICIENT, 6, 7, 12, 14
CONT, 17
CONTingency, 17, 18, 30
COPY, 12, 23, 30
CORRelate, 6, 14, 30
Correlation, 1, 6, 7, 12, 13, 14, 15, 16
COUNt, 10
Crash, 12, 29
CRASHES, 29
Curvilinear, 16
- 33 -
Default, 6
Defaults, 13
DEFIne, 21, 30
Degrade, 21
DELETE, 23, 30
Deleted, 23
Delimiting, 27
DESCRIBE, 4, 8, 10, 30
Dev'n, 31
Dev-n, 4, 10, 23
Deviation, 10, 31
Deviations, 24
Df, 19, 25, 26
Dictionary, 2, 3
Dire, 29
DIREctory, 26, 27, 30
Disk, 3, 26
Disk-drive, 27
Display, 2, 23, 24, 26, 28, 30
Displays, 21, 29
DISPlay, 23
DIVIde, 11, 30
Division, 20, 29
Divisor, 10
Drive, 26, 27
EDIT, 21
ENTER, 2, 3, 4, 9, 29
ERASE, 7, 9, 27, 30
Error, 2, 6, 20, 21, 22, 23, 29
Errors, 24
Exit, 9, 10, 28
EXPO, 11, 30
EXPOnent, 11
Export, 27
Exported, 27
Extension, 26, 27
FERAse, 27, 30
FILES, 26, 27
Floating, 21
Format, 26, 27
Formula, 19
Formulae, 2
FREAD, 26, 27, 30
FWRITE, 27, 30
GENErate, 21
Graph, 12
- 34 -
HELP, 3, 5, 28, 29, 30
Hints, 29
HIST, 14
HISTogram, 13, 14, 30
HISTOGRAMS, 13
Horizontal, 12
Hyphen, 9, 10, 12, 23, 28
Hypothesis, 24
IBM, 1, 2
INFO, 5, 7, 14, 27, 28, 29
Insert, 3, 23
Integer, 17, 21
Integers, 17, 21
Intercept, 15
INTERPRET, 14
INTRODUCTION, 1
IRAN, 17, 18, 24
IRANdom, 21, 30
JOIN, 22, 23, 30
Keyword, 2, 3, 5
Log, 11, 27
Logarithmic, 16
LOGD, 27, 30
LOGE, 11, 30
Logged, 26, 27
MANN-Whitney, 25, 30
Maximum, 4, 7, 10, 12, 17, 30
MAXImum, 10
MCS, 26
Mean, 4, 10, 24, 25, 29, 30
Means, 1, 2, 9, 19, 24, 25
MEAN, 10
Measure, 6, 25
Measurement, 24
Median, 4, 10, 30
MEDIan, 10
Message, 2, 6, 18, 20
MINITAB, 1
MINImum, 10
MS-DOS, 3
Mu, 24
MULT, 3, 5, 11
MULTiply, 11
- 35 -
NAM, 26
Name, 2, 5, 7, 9, 14, 27, 30
NAME, 7
Names, 5, 7, 9, 12, 26, 28
Nominal, 18, 19
Non-MICROSTATS, 27
NON-parametric, 25
NOTE, 29
OMIT, 22, 30
Order, 7, 10, 11, 14, 16, 25, 29
Ordinal, 25
Output, 4, 6, 7, 24, 25, 29
Overwrite, 11
Parameters, 23
Parametric, 25
Parse, 2
PASCAL, 1
Pearson, 14
PICK, 21, 23, 31
PLOT, 7, 12, 13, 14, 29, 31
PLOTS, 12, 13
Plotting, 12
POOL, 31
POOLed, 24, 25
Precautions, 12
PRINT, 2, 4, 5, 6, 10, 11, 14, 16, 22, 28, 29, 31
Print-out, 6
PRinter, 29
PRINTING, 10, 18
Product-moment, 14
PROF, 29, 31
PRON, 29, 31
Quantitative, 1
RAISe, 11, 31
Random, 13, 17, 18, 19, 21
Range, 1, 5, 14, 21, 28
RANK, 12, 31
Rank-sum, 25
Ranked, 25
Ranking, 12
RDEScribe, 23, 31
READ, 9, 14, 27, 31
RECOde, 21, 31
Recoding, 21
REGR, 15
REGRESS, 7, 15, 16, 31
REGRESSION, 7, 15, 16
RETRIEVE, 26, 31
RETRieves, 26
RETURN, 3, 11, 13
RMAXimum, 23, 31
- 36 -
RMEAN, 23, 31
RMEDian, 23, 31
RMINimum, 23, 31
ROUNd, 6, 11, 31
RSTAndard, 23, 31
RSTD, 23, 31
RSUM, 23, 31
Sample, 10, 14, 23, 25
Samples, 24
SAVE, 26, 28, 29, 31
SCATTER, 12
SCATTERGRAM, 12
SET, 9
Slope, 15
SORT, 11, 31
Spreadsheet, 2, 27
SPSS, 1
SQRT, 11, 31
STANdard_dev'n, 10
STDE, 10
Stdev, 24
STDEv-n, 31
STOP, 28, 31
SUBStitute, 22, 31
SUBT, 11
SUBTract, 31
Sum, 4, 5, 10, 31
Summed, 19
Table, 17, 18, 23, 26
Tables, 25
TDIST, 25
TDIStribution, 31
TINT, 25
Tips, 29
Ttest, 24, 25, 31
TTESt, 25
Turbo, 1
Tutorial, 8
Two-tailed, 25
TWOS, 24
TWOSample, 24, 25, 31
URANdom, 21, 31
Users, 23, 28
Warning, 2, 20
Warnings, 29
Wilcoxon, 25
Wildcard, 26
Work-sheet, 28
Worksheet, 2, 3, 5, 26, 29
