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| Appendix 2: Statistical Distributions
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|
|
t |
t2
|
|
1 |
6.31 |
39.87 |
|
2 |
2.92 |
8.53 |
|
3 |
2.35 |
5.54 |
|
4 |
2.13 |
4.54 |
|
5 |
2.02 |
4.06 |
|
6 |
1.94 |
3.78 |
|
7 |
1.90 |
3.59 |
|
8 |
1.86 |
3.46 |
|
9 |
1.83 |
3.36 |
|
10 |
1.81 |
3.28 |
|
15 |
1.75 |
3.07 |
|
20 |
1.72 |
2.97 |
|
30 |
1.70 |
2.88 |
|
40 |
1.68 |
2.84 |
|
60 |
1.67 |
2.79 |
|
120 |
1.66 |
2.75 |
|
|
1.64 |
2.71 |
Values of the t and t2 corresponding to the five percent area of the t-distribution for selected degrees of freedom (one-tailed test). The degrees of freedom equal to n - 2. The t2 equals F for 1 degree of freedom. For infinitely large degrees of freedom, t equals z.
Using the t distribution for estimation of probability associated with the strength of a relationship in lieu of the normal distribution increases the threshold of the significance criterion and thus makes results less likely to be significant when a small number of subjects is used for analysis. For groups of subjects larger than 60, the z-test and t-tests can be used interchangeably.
In
the above equation
(
)
= (
- 1) ! For its both degrees of freedom equal to 10, the above
equation was written for Microsoft Excel as
=630 * a1^4 * (1 + a1) ^ -10
The constant 630 within the above expression was computed as (9! / 4! 4!). This F(10,10) distribution is shown in the figure below.

The ease with which Microsoft Excel permits to visualize higher transcendental functions removes much of the mythology and obfuscation from the statistical data analysis.
Values
of F for selected degrees of freedom at the five percent
level of significance (one-tailed test) are shown in the
table below.
|
|
1 |
2 |
3 |
|
|
1 |
39.87 |
49.5 |
53.6 |
63.3 |
|
2 |
8.53 |
9.00 |
9.16 |
9.49 |
|
3 |
5.54 |
5.46 |
5.39 |
5.13 |
|
4 |
4.54 |
4.32 |
4.19 |
3.76 |
|
5 |
4.06 |
3.78 |
3.62 |
3.11 |
|
6 |
3.78 |
3.46 |
3.29 |
2.72 |
|
7 |
3.59 |
3.26 |
3.07 |
2.47 |
|
8 |
3.46 |
3.11 |
2.92 |
2.29 |
|
9 |
3.36 |
3.01 |
2.81 |
2.16 |
|
10 |
3.28 |
2.92 |
2.73 |
2.06 |
|
15 |
3.07 |
2.70 |
2.49 |
1.76 |
|
20 |
2.97 |
2.59 |
2.38 |
1.61 |
|
30 |
2.88 |
2.49 |
2.28 |
1.49 |
|
40 |
2.84 |
2.44 |
2.23 |
1.38 |
|
60 |
2.79 |
2.39 |
2.18 |
1.29 |
|
120 |
2.75 |
2.35 |
2.13 |
1.19 |
|
|
2.71 |
2.30 |
2.08 |
1.00 |
For one degree of freedom, F equals t2.
The equation for the chi square distribution is
The above equation conforms to the general form of the Euler's gamma function
The constant a equals
the constant b equals
the constant c equals .5 and the constant d equals 2.
For example, for 10 degrees of freedom
b equals 4, c equals .5 and d equals 2.
As shown in the table (p = .05) below,
|
|
|
|
1 |
3.841 |
|
3 |
7.815 |
|
5 |
11.07 |
|
10 |
18.307 |
|
20 |
31.410 |
|
30 |
43.773 |
|
40 |
55.759 |
|
50 |
67.505 |
|
|
|
for one degree of freedom, chi-square equals z-square. For infinite number of degrees of freedom, chi-square, divided by the degrees of freedom, equals F with both of its degrees of freedom equal to infinity.
The above figure was plotted by using Microsoft Excel, using the equation (a1^4*2.71828^ (-a1/2))/768.
Within the statistical computer programs, the probabilities associated with the z, t F, and Chi Square ratia may be calculated by a single subroutine. This subroutine normalizes the F distribution as
After the normalization, this subroutine uses the polynomial approximations to find areas under the normal distribution, corresponding to standard z scores, as
where c1 = .196854, c2 = .115194, c3 = .000344, and c4 = .019527.
For the example of the F(10,10) distribution, the conversion equation was written for Microsoft Excel as
=(0.9778*A2^(1/3)-0.9778) / (0.9778*A2^(2/3)+0.9778)^(1/2))*10
and the distribution was standardized as

The slight irregularity in the left tail of the standardized distribution is, in the course or real life computer implementations, removed by Kelley's correction.
Since F equals z-Square with (1, infinity) degrees of freedom, the probability associated with the z-Square ratio can be obtained as p = fSig (1, 1000, z-Square). The infinity is represented by a large number, usually equal to 1,000.
Since F equals t-Square with (1, df) degrees of freedom, the probability associated with the t-Square ratio can be obtained as p = fSig (1, df, t-Square).
This probability is obtained by calling the fSig subroutine as p = fSig(df1,df2,F).
Since F equals Chi Square with (df, infinity) degrees of freedom, the probability associated with the chi square ratio can be obtained as p = fsig(df, 1000, Chi-Square / df).
The above section provides insight to the apparent inconsistency in the conceptualization if the t-square ratio
and the chi-square ratio where
The t-square ratio is characteristic of the Fisherian conceptualization of statistical inference with the degrees of freedom used throughout all computations leading to the t-square ratio. The chi-square ratio is characteristics of the Pearsonian conceptualization of statistical inference where the degrees of freedom are introduced only during the last phase of the computation of probability associated with the chi-square.
A seminal idea in statistical mechanics is that of Maxwell's demon. Named after the Scottish physicist James Clerk Maxwell, Maxwell's demon is a hypothetical homunculus that is considered to admit or block passage of individual molecules between adjacent compartments. If provided with information about the speed of individual molecules, Maxwell's demon would be able to violate the second law of thermodynamics.
The notion of Maxwell's demon can be adapted for use within the classical theory of statistics and its associated theories of probability distributions and scaling. Within this context, let us assume that a group of Maxwell's demons operate within an environment of gates and compartments, provided by Galton's Quincunx. Let us further assume that each demon occupies a single decision node in the Quincunx and acts in accordance with the principles of formal logic, as defined by functions of propositional calculus. The experimenter can select functions, determining the demons' behavior, for each experimental run of the Quincunx. In this paper we describe the results of three trial runs of the above-defined Quincunx using different logical functions for each trial.
Aside from its characteristic honeycomb lattice of decision points connected with bottom compartments characteristic of Galton's Quincunx, the Quincunx of Maxwell's demons also contains an incipient data matrix of all possible responses to a set of binary scored questions. This data matrix is called plenum and is defined as a truth table of formal logic. A plenum of possible responses to four binary variables p, q, r, and s is shown on the left side of the following diagrams. On the right side of these figures can be observed a matrix of outcomes containing response patterns congruent with the logical function sent to Maxwell's demons. This matrix of outcomes defines the behavior of a ball moving through the grid of Quincunx's decision points. The elements of the data matrix of the outcomes containing ones (corresponding to true values of the logical truth tables) signify a path leading toward the right side of the Quincunx. The elements containing zeroes (corresponding to false values of the logical truth tables) signify a path leading toward the left side. The trajectory of the ball, traveling through the Quincunx is controlled by a group of Maxwell's demons operating the gate mechanism of the Quincunx according to principles of Boolean algebra.
This simulation corresponds to Galton's original model. Let us submit to the demons a tautological function f = taut (p, q, r, s). The computerized version of Maxwell's demons operates in this case as follows.
The plenum of responses to a set of binary scored questions is solved as if it would be a truth table of formal logic. The solution is tautological, shown as a column of true (1) values in the second column of the diagram. The response patterns corresponding to the true values of the tautology function are replicated in the matrix of outcomes, as shown in the third column of the diagram below. The values of the data matrix of outcomes are associated with a display of the hexagonal layers of gates and bottom bins that comprise the Quincunx. This matrix of outcomes controls the movement of the balls, traveling through layers of the hexagonal lattices. Zeroes move the ball toward the left, ones toward the right. The balls are stacked inside of the compartments located underneath the hexagonal lattices. This binomial distribution is the same as that corresponding to the frequency counts in the last column of the diagram. The outcome of the simulation is a binomial distribution, approximating the normal distribution. Both the binomial and normal distributions reflect the influence of causal determinants on the outcome of events. According to the binomial model, phenomenon that has one determinant has two possible outcomes, phenomenon with two determinants has four outcomes, etc. Within this context, the logical model may help to understand the ubiquity of the binomial and normal distributions as reflections of determinants of such diverse phenomena as biological characteristics, physical and mental traits, and societal events. For example, allowing Maxwell's demons to realize possible combinations of determinants within the phylogenetic and ontogenetic repertoire of organisms reflect the optimum strategy of species survival. The logic here is that anything that is possible may and will be tried. Analogous outcomes can also be observed for individual behavior and behavior of societies. Laws of society and its ethical precepts may curtail manifestations of some outcomes, however, the magnitude of the environmental urgency is typically matched by the degree to which the outcome is usual or unusual, expected or unexpected, moderate or extreme.
If we submit to Maxwell's demons a logical function f = (p -> q) & (q -> r) & (r -> s), the outcome results in a rectangular distribution.
The solution to the conjunction of implication functions is shown in the second column of the diagram and the table of outcomes is shown toward the right side of the diagram. The implication (->) returns a false (0) value only in the case of the (1,0) response pattern. The conjunction (&) returns a true (1) value only in the case of the (1,1) response pattern. The arguments within the parentheses are solved first, the conjunctions of implications next.
On a trial run, the balls moving through the lattice of decision points will form a rectangular distribution that corresponds to an idealized version of a perfect Guttman scale, also called an implicational scale. From the standpoint of formal logic, the definition of the Guttman scales as conjunctions of implication functions indicates that implicational scales are renderings of Aristotelian syllogisms.
To simulate a test of statistical significance, the behavior of Maxwell's demons has to be determined by two logical functions, f1 = .not. p & taut(q, r, s), and f2= p & taut(q, r, s). The algorithm for this simulation is shown in the diagram below. The solution for the first function is presented in the second column and the solution for the second function in the third column. The outcomes are shown on the right side of the diagram.
On a trial run, the balls moving through the lattice of the Quincunx decision points will form two binomial distributions, shifted to the extent the determining attribute of a particular outcome is present. From the standpoint of formal logic, a test of statistical significance suggests that a particular determinant has significant influence on the outcome of an event. Within the context of an experiment, the question of whether a treatment determines a yield, a factor an outcome, or the independent variable the dependent variable, reflect the same type of logical reasoning.
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