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Here are the topics for each of the modules
in Statistics Explained:
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- (1) On the need for algebra
This introductory module provides an example of the surprising
insights that are possible when we apply some basic algebra to problems that at
first glance seem to have obvious answers.
- (2) Techniques for visualizing data
Statisticians
need to be able to visualize collections of data. This module describes three
of the techniques that have proven especially useful:
- (3) Measures of Central Tendency
Statisticians
need a way to indicate where, along a scale of measures, a given distribution
is centered. This module describes three measures of central tendency.
- (4) Measures of Variability
Statisticians have developed ways to indicate the degree
to which the items in a distribution are separated from each other. The measures
of variability described in this module are:
- (5) The effects of uniform transformations of X
This module shows what happens to the mean and variance of a distribution when
every item in the distribution is transformed by either adding a given amount
to it or by multiplying it by a given amount.
- (6) The z score conversion
When the items
in a distribution have been converted to z scores, it is possible to compare the
relative standings of items from distributions that initially had different means
and variances. This module shows how and why this happens.
- (7) Introduction to the Normal Curve
When one measures a very large number of items and then graphically represents
the data using very fine class intervals, the histogram that results is often
somewhat bell shaped.
- (8) The details
of the equation for the Normal Curve
Re-examining
the equation that specifies a normal curve.
- (9) Specifying a given normal curve
Gives
an example of normal curve with the same area, mean and standard deviation as
a histogram of 100 items.
- (10) Normal
curves versus curves that are not normal
This module illustrates several kinds of curves that are not normal, including
two bell shaped curves.
- (11) Some
examples of Normal Distributions
Because
of the nature of our universe, many kinds of measures are distributed normally,
and hence we can use the equation for a normal curve to completely describe their
distribution.
- (12) The arrangement
of areas in a normal distribution
We saw
in the previous module that the ordinates in a normal curve are distributed in
a special way. In this module we learn that the areas under a normal curve are
also distributed in a special way.
- (13) The z score conversion of a normal curve
It would be tremendously burdensome, indeed impossible, to specify the areas under
various portions of all possible normal curves, because N, i, and can each have
any of an infinite number of possible
- (14) The Concept of a Random Sample
Any
set of items with a common observable characteristic can be regarded as a population
(or universe). Once we have specified a given population we regard any set of
items drawn from that population as a sample.
- (15) The Sampling Distribution of the Mean and The Central Limit Theorem
Introduces the concept of a sampling distribution
and it indicates how The Central Limit Theorem can describe the characteristics
of a particular Sampling Distribution of the Mean.
- (16) The rationale for The Central Limit Theorem
If we are to understand why the Central Limit Theorm is true, we will have to
look very closely at what happens when we take random samples.
- (17) The z score conversion of The Sampling Distribution
of the Mean
Explains how to assess the
probability of obtaining various values of a sample mean when we take a random
sample from a normal population with a known mean and variance.
- (18) A test of a hypothesis about a population's mean when
the population's variance is known
Introduces
the z test of a statistical hypothesis about the mean of a population from which
a random sample has been drawn.
- (19)
A Review and Summary of the z test
Reviews
and summarizes the ideas presented in the previous module.
- (20) An example of the z test
A practical example to help clarify the strategy described in the previous module.
- (21) An introduction to the t test
Introduces a test of a hypothesis about the mean of
a population that can be used when the variance of the population is unknown and
must be estimated from the information in the sample.
- (22) The rational for using S-squared x to estimate Sigma-squared x
This is because the expression sigma(X-Mu
x) squared is algebriacally equal to sigma(X-Xbar+Xbar-Mu
x) squared.
- (23) Some notes on the use of n-1
and the concept of degrees of freedom
This
module summarizes the concepts introduced in the previous two modules.
- (24) The Sampling Distribution of s
squared
Describes
the sampling distribution of the statistic s squared and shows how its form is determined by the statistic's degrees of freedom.
- (25) The Sampling Distribution of the statistic
(s squared x/sigma-squared x)
Introduces the statistic and describes some of the situations
where it can be used.
- (26) Testing
a Statistical Hypothesis about the Variance of a Population
Describes an example that summarizes the procedure to test a statistical hypothesis
about the value of sigma-squared x.
- (27) The Sampling Distribution of
F
Provides a description of the statistic
F which was named after the great statistician Ronald Fisher. F is the ratio of
two s squared xs.
- (28) The table of F values
Introduces the
table of F values and describes the basic features of its entries.
- (29) Two-Tailed tests with the F Table
Provides further insight into the F table by describing
how to use it to carry out a Two-Tailed test.
- (30) Two-Tailed tests with the F Table -- A summary
Provides a short account of the sequence of steps required to carry out a Two-Tailed
test using the F tables.
- (31) The
t-statistic
Provides an introduction to
the t test.
- (32) More on the t test
Provides additional details on the differences between
t and z.
- (33) The Sampling Distribution
of t
Compares the t distribution to a normal
distribution.
- (34) The effect on t
of the s square in its denominator
Provides an account of how and why the
t distribution differs from a normal distribution.
- (35) An example of a t test
The use of
t as a test of a hypothesis about the mean when the variance of the population
is not known.
- (36) The relationship
between t and F
Indicates the nature of
the relationship between t and F.
- (37) More on the relationship between t and F
Indicates the nature of the relationship between t and F.
- (38) Uncovering the value of t within an F table
Provides a quantitative example of how values of t
are contained within the F table.
- (39) Tests of hypotheses about the means of two populations
Introduces the Z test of a hypothesis about the means of two populations when
the population variances are known.
- (40) More on the Z test of the difference between two means
Examines the details of the Z test of the difference between two means.
- (41) An example of a test of a hypothesis
about the means from two populations
Summarizes
the Z test of a difference between two sample means with known variances.
- (42) The t test of the difference between
two sample means
Introduces the t test
of a hypotheses about the means of two populations when the variances are not
known.
- (43) Some constraints on the
use of the t test
Describes some of the
assumptions underlying the t test and discusses how to use t to best advantage.
- (44) Some computing formulae for the
t test
Introduces several of the computational
formulae for the t test and it shows their derivations.
- (45) More on the assumptions that underlie the t test
Discusses the use of t when its assumptions may not have been met.
- (46) More on the relationship between t and F
Explains the relationship between t and F when dealing
with the t test of the difference between two sample means.
- (47) An example of a t test of the difference between two
means when the population variances are not known
Provides a summary of the steps in performing a t test of the observed difference
between the means of two samples.
- (48) The concept of beta and the power of a test
This somewhat complex module provides an explanation
of the important concepts of alpha, beta and Power. Because of the central role of these concepts
to all of statistical inference, this module is worthy of especially careful study.
- (49) Correlation
Introduces the concept of a correlation between two measures on each of a set
of items.
- (50) Quantifying the relationship
(the coefficient of correlation)
This module
shows how to quantify a measure of correlation (r).
- (51) A Computational Formula for r
There
are times when it is convenient to express r in a form that is designed to make
its numerical calculation as convenient as possible. This module shows the derivation
of one of the various ways that r can be conveniently calculated.
- (52) The concept of regression
Introduces the concept of regression and relates it to the concept of correlation.
- (53) Specifying a regression line
Shows the derivation of the regression equations that
are used to specify the parameters of the line that specifies the best estimate
of the value of Y given a value of X.
- (54) The z score transformation in the context of correlation
This module provides a graphic illustration of the relationship
between regression and correlation by showing what happens when z score transformations
are applied to a set of related measures.
- (55) The regression equations after z score transformations
This module explains why, after the Z score conversion, the value of b is equal
to r and the value of a is equal to zero.
- (56) Partitioning the variance in the context of correlation
Shows why the variance of the Y values in the context of correlation can be partitioned
into two variances.
- (57) More on the
partitioning of variance of Y
An algebraic
proof of the equation describing the partitioning of the variance of Y in the
context of correlation.
- (58) Some
graphic examples of the key ideas behind correlation and regression
Provides a graphic illustration of the deviations
that make up:
- (59) The second regression
line
Introduces the second regression line,
namely the regression of X on Y.
- (60)
How the regression lines change with changes in strength of a relationship
Provides a graphic illustration of the scatterplot's
characteristic of both weak and strong positive and negative correlations.
- (61) Regression toward the mean
Provides a detailed account of the concept of regression
towards the mean.
- (62) Some final
considerations with respect to correlation and regression
Discusses some of the factors that must be considered to properly interpret the
implications of an obtained correlation coefficient.
- (63) A test of a hypothesis about the means of a set of related measures
Introduces the formula for a z test of a set of related
measures.
- (64) Some algebraic relationships
in the z test of related measures
Introduces
a second formula for the z test of related measures and shows how it is derived.
- (65) Tests involving independent measures
versus tests involving related measures
Highlights some of the special features of tests of related measures.
- (66) Test of hypotheses about population means
using related measures when the population variances are unknown
Introduces the t test for related measures.
- (67) Comparing t and z tests
Illustrates the differences between the z and t tests for related versus independent
measures.
- (68) An example of a t test
of related measures
Summarizes the steps
in performing a t test of related measures.
- (69) Chi-square: An enumeration statistic
Each of the statistical procedures that we have discussed thus far can be conceptualized
as being based on the circumstances that arise when we have a set of n or more
items where each item yields a measure of a given magnitude.
- (70) The logic of the Chi-square test
We can gain some appreciation of the logic of Chi Sq. if we consider an "real
life" example.
- (71)
Chi-square: A test for independence
There
are times when each of two aspects of an item are categorized and we wish to determine
if the categorizations across items are independent.
- (72) An introduction to the analysis of variance
Up to now we have examined a variety of ways to decide whether or not an observed
difference between two means is statistically reliable. At this point, we will
examine a technique for deciding whether or not the observed differences among
a set of K means is statistically reliable.
- (73) The analysis of variance and the F distribution
Shows how, by using the F distribution, the analysis of variance can be used to
test a statistical hypothesis about the means of a set of samples.
- (74) A preferred format for reporting the results of an
analysis of variance
Shows both the conceptual
and computing formula for an analysis of variance arranged in a preferred format.
- (75) Derivation of the computing formula
for the between sum of squares
Demonstrates
the algebraic equivalence of the conceptual and computing formula for the between
sum of squares.
- (76) Derivation of
the computing formula for the within sum of squares
Demonstrates the algebraic equivalence of the conceptual and computing formula
for the within sum of squares.
- (77)
Derivation of the computing formula for the total sum of squares
Demonstrates the algebraic equivalence of the conceptual
and computing formula for the total sum of squares.
- (78) An example of a problem requiring a One-Way Analysis of Variance
Describes an experiment for which a One-Way Analysis
of Variance is needed to determine if the results are statistically significant.
- (79) The calculations for the example
of a One-Way Analysis of Variance
Illustrates
both the conceptual and computing formula approach to the calculations for the
example of a One-Way Analysis of Variance.
- (80) An additional comment on the analysis of variance
Describes an experiment for which a One-Way Analysis of Variance is needed to
determine if the results are statistically significant.
- (81) An introduction to the Two-Way Analysis of Variance
Provides an example of a Two-Way Analysis of Variance and shows how to calculate
the grand mean.
- (82) The calculation
of the cell means
This module shows how
to calculate the cell means.
- (83)
The calculation of the row and column means
This module shows the data used to calculate the means of the several (R) rows
and (C) columns.
- (84) The calculation
of row and column effects
Introduces a
procedure for assessing the effects of the row and the column treatments.
- (85) Introduction to the concept of interaction
Introduces a procedure to test the statistical hypothesis
that the row and column treatments are producing independent effects. If we can
reject that statistical hypothesis, we will have reason to conclude that the effects
of the row and column treatments are interacting with each other.
- (86) Variance estimates in the Two-Way ANOVA when the null
hypothesis is true
Discusses the several
estimates of the population variance in the Two-Way ANOVA when the null hypothesis
is true in each case.
- (87) Variance
estimates in the Two-Way Analysis of Variance when the null hypothesis is false
Discusses the several estimates of the population
variance in the Two-Way ANOVA when the null hypothesis is false in each case.
- (88) Testing for significant effects
in the Two-Way Analysis of Variance
Describes
the tests for the statistical significance of the row, column and interaction
effects.
- (89) A preferred format for
reporting the results of a Two-Way ANOVA
Shows both the conceptual and computing formula for a two-way ANOVA, arranged
in a preferred format.
- (90) An example
of a problem requiring a two-way ANOVA
Describes an experiment for which a Two-Way ANOVA is needed to determine if the
numerical results are statistically significant.
- (91) Numerical results for our example of a Two-Way ANOVA
Shows the calculations necessary for the Two-Way ANOVA of the data presented in
the previous module. It also discusses the relationships among the various results
of those calculations.
- (92) A General
Comment about the Two-Way ANOVA
By indicating
that the observed differences among the row and column means are larger than expected
by chance, we are led to conclude that those differences are due to the treatments.
- (93) A final comment
Some notes on the proper use of statistics.
- (94) Statistics Quiz
25 Questions presented in random order.
- (95) Selected further reading
Here are
some classic as well as some obscure books for those interested in additional
concepts related to the material covered in this program.
- (96) Thanks to the following people
- (97) About Statistics
Explained
This program was originally a
book plus a diskette, written by Howard S. Hoffman.
- (98) About the authors
Howard Hoffman is
a professor emeritus at Bryn Mawr College, Bryn Mawr, Pennsylvania. He taught
psychology and statistics for nearly 50 years. Russell Hoffman, Howard's eldest
living son, is a computer programmer and animator, and owner of The Animated Software
Company. Sharon Hoffman, Russell's wife, is an editor with experience in technical
reports, newsletters and magazines. She has nearly two decades' experience in
the field of IBM AS/400 computers.
- (99) About The Animated Software Company
The Animated Software Company was founded in 1984 as "P11 Enterprises".
- (100) About Our Web Site
We began offering our software through online services in 1990. We have offered
most of our software via the Internet since then.
- (101) Why do we use P11?
We love the Internet
but don't believe its "one size fits all" approach is yet (1999) capable of the
animations or smooth user interaction we are able to achieve by using our own
development tools.
- (102) All About
Pumps
Originally released in 1995, All
About Pumps is a complete introduction to the second-most common machine on Earth,
after electric motors. YOU need to know about pumps!
- (103) The Heart: The Engine of Life
The
Heart: The Engine of Life is an animated tutorial about the human heart.
- (104) Russell's "P11" Animation Machine
Russell's "P11" Animation Machine was originally created
to produce THE HEART: THE ENGINE OF LIFE. Work on what eventually became "P11"
began in 1984. The current release is 8.68, compiled November 7th, 1994
- (105) Global Energy Network International
Global Energy Network International is a U.S. tax
exempt 501(c)(3) corporation committed to improving the quality of life for everyone
without damage to the planet. In 1995 the Animated Software Company created an
educational tutorial describing the initiative, which is available for free download
and distribution via the Internet (www.geni.org).
- (106) Assembler Language Source Code Available
This and all our products are created with a 100% Assembler Language computer
program known as "P11".
- (107) What's
next at The Animated Software Company?
We will continue to upgrade the products we already have on the market and develop
new and exciting interactive educational products along the way.
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