一些统计学基本概念

What are variables. Variables are
things that we measure, control, or manipulate in research. They differ
in many respects, most notably in the role they are given in our
research and in the type of measures that can be applied to them.

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Correlational vs. experimental research.
Most empirical research belongs clearly to one of those two general
categories. In correlational research we do not (or at least try not
to) influence any variables but only measure them and look for
relations (correlations) between some set of variables, such as blood
pressure and cholesterol level. In experimental research, we manipulate
some variables and then measure the effects of this manipulation on
other variables; for example, a researcher might artificially increase
blood pressure and then record cholesterol level. Data analysis in
experimental research also comes down to calculating "correlations"
between variables, specifically, those manipulated and those affected
by the manipulation. However, experimental data may potentially provide
qualitatively better information: Only experimental data can
conclusively demonstrate causal relations between variables. For
example, if we found that whenever we change variable A then variable B
changes, then we can conclude that "A influences B." Data from
correlational research can only be "interpreted" in causal terms based
on some theories that we have, but correlational data cannot
conclusively prove causality.

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Dependent vs. independent variables.
Independent variables are those that are manipulated whereas dependent
variables are only measured or registered. This distinction appears
terminologically confusing to many because, as some students say, "all
variables depend on something." However, once you get used to this
distinction, it becomes indispensable. The terms dependent and
independent variable apply mostly to experimental research where some
variables are manipulated, and in this sense they are "independent"
from the initial reaction patterns, features, intentions, etc. of the
subjects. Some other variables are expected to be "dependent" on the
manipulation or experimental conditions. That is to say, they depend on
"what the subject will do" in response. Somewhat contrary to the nature
of this distinction, these terms are also used in studies where we do
not literally manipulate independent variables, but only assign
subjects to "experimental groups" based on some pre-existing properties
of the subjects. For example, if in an experiment, males are compared
with females regarding their white cell count (WCC), Gender could be
called the independent variable and WCC the dependent variable.

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Measurement scales. Variables
differ in "how well" they can be measured, i.e., in how much measurable
information their measurement scale can provide. There is obviously
some measurement error involved in every measurement, which determines
the "amount of information" that we can obtain. Another factor that
determines the amount of information that can be provided by a variable
is its "type of measurement scale." Specifically variables are
classified as (a) nominal, (b) ordinal, (c) interval or (d) ratio.

Nominal variables allow for only qualitative classification. That
is, they can be measured only in terms of whether the individual items
belong to some distinctively different categories, but we cannot
quantify or even rank order those categories. For example, all we can
say is that 2 individuals are different in terms of variable A (e.g.,
they are of different race), but we cannot say which one "has more" of
the quality represented by the variable. Typical examples of nominal
variables are gender, race, color, city, etc.

Ordinal variables allow us to rank order the items we measure
in terms of which has less and which has more of the quality
represented by the variable, but still they do not allow us to say "how
much more." A typical example of an ordinal variable is the
socioeconomic status of families. For example, we know that
upper-middle is higher than middle but we cannot say that it is, for
example, 18% higher. Also this very distinction between nominal,
ordinal, and interval scales itself represents a good example of an
ordinal variable. For example, we can say that nominal measurement
provides less information than ordinal measurement, but we cannot say
"how much less" or how this difference compares to the difference
between ordinal and interval scales.

Interval variables allow us not only to rank order the items
that are measured, but also to quantify and compare the sizes of
differences between them. For example, temperature, as measured in
degrees Fahrenheit or Celsius, constitutes an interval scale. We can
say that a temperature of 40 degrees is higher than a temperature of 30
degrees, and that an increase from 20 to 40 degrees is twice as much as
an increase from 30 to 40 degrees.

Ratio variables are very similar to interval variables; in
addition to all the properties of interval variables, they feature an
identifiable absolute zero point, thus they allow for statements such
as x is two times more than y. Typical examples of ratio scales are
measures of time or space. For example, as the Kelvin temperature scale
is a ratio scale, not only can we say that a temperature of 200 degrees
is higher than one of 100 degrees, we can correctly state that it is
twice as high. Interval scales do not have the ratio property. Most
statistical data analysis procedures do not distinguish between the
interval and ratio properties of the measurement scales.

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Relations between variables. Regardless
of their type, two or more variables are related if in a sample of
observations, the values of those variables are distributed in a
consistent manner. In other words, variables are related if their
values systematically correspond to each other for these observations.
For example, Gender and WCC would be considered to be related if most
males had high WCC and most females low WCC, or vice versa; Height is
related to Weight because typically tall individuals are heavier than
short ones; IQ is related to the Number of Errors in a test, if people
with higher IQ's make fewer errors.

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Why relations between variables are important. Generally
speaking, the ultimate goal of every research or scientific analysis is
finding relations between variables. The philosophy of science teaches
us that there is no other way of representing "meaning" except in terms
of relations between some quantities or qualities; either way involves
relations between variables. Thus, the advancement of science must
always involve finding new relations between variables. Correlational
research involves measuring such relations in the most straightforward
manner. However, experimental research is not any different in this
respect. For example, the above mentioned experiment comparing WCC in
males and females can be described as looking for a correlation between
two variables: Gender and WCC. Statistics does nothing else but help us
evaluate relations between variables. Actually, all of the hundreds of
procedures that are described in this manual can be interpreted in
terms of evaluating various kinds of inter-variable relations.

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Two basic features of every relation between variables.
The two most elementary formal properties of every relation between
variables are the relation's (a) magnitude (or "size") and (b) its
reliability (or "truthfulness").

Magnitude (or "size"). The magnitude is much easier to understand
and measure than reliability. For example, if every male in our sample
was found to have a higher WCC than any female in the sample, we could
say that the magnitude of the relation between the two variables
(Gender and WCC) is very high in our sample. In other words, we could
predict one based on the other (at least among the members of our
sample).

Reliability (or "truthfulness"). The reliability of a
relation is a much less intuitive concept, but still extremely
important. It pertains to the "representativeness" of the result found
in our specific sample for the entire population. In other words, it
says how probable it is that a similar relation would be found if the
experiment was replicated with other samples drawn from the same
population. Remember that we are almost never "ultimately" interested
only in what is going on in our sample; we are interested in the sample
only to the extent it can provide information about the population. If
our study meets some specific criteria (to be mentioned later), then
the reliability of a relation between variables observed in our sample
can be quantitatively estimated and represented using a standard
measure (technically called p-value or statistical significance level,
see the next paragraph).

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What is "statistical significance" (p-value). The
statistical significance of a result is the probability that the
observed relationship (e.g., between variables) or a difference (e.g.,
between means) in a sample occurred by pure chance ("luck of the
draw"), and that in the population from which the sample was drawn, no
such relationship or differences exist. Using less technical terms, one
could say that the statistical significance of a result tells us
something about the degree to which the result is "true" (in the sense
of being "representative of the population"). More technically, the
value of the p-value represents a decreasing index of the reliability
of a result (see Brownlee, 1960).
The higher the p-value, the less we can believe that the observed
relation between variables in the sample is a reliable indicator of the
relation between the respective variables in the population.
Specifically, the p-value represents the probability of error that is
involved in accepting our observed result as valid, that is, as
"representative of the population." For example, a p-value of .05
(i.e.,1/20) indicates that there is a 5% probability that the relation
between the variables found in our sample is a "fluke." In other words,
assuming that in the population there was no relation between those
variables whatsoever, and we were repeating experiments like ours one
after another, we could expect that approximately in every 20
replications of the experiment there would be one in which the relation
between the variables in question would be equal or stronger than in
ours. (Note that this is not the same as saying that, given that there
IS a relationship between the variables, we can expect to replicate the
results 5% of the time or 95% of the time; when there is a relationship
between the variables in the population, the probability of replicating
the study and finding that relationship is related to the statistical power of the design. See also, Power Analysis). In many areas of research, the p-value of .05 is customarily treated as a "border-line acceptable" error level.

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How to determine that a result is "really" significant. There
is no way to avoid arbitrariness in the final decision as to what level
of significance will be treated as really "significant." That is, the
selection of some level of significance, up to which the results will
be rejected as invalid, is arbitrary. In practice, the final decision
usually depends on whether the outcome was predicted a priori or only
found post hoc in the course of many analyses and comparisons performed
on the data set, on the total amount of consistent supportive evidence
in the entire data set, and on "traditions" existing in the particular
area of research. Typically, in many sciences, results that yield p

.05
are considered borderline statistically significant but remember that
this level of significance still involves a pretty high probability of
error (5%). Results that are significant at the p

.01 level are commonly considered statistically significant, and p

.005 or p

.001
levels are often called "highly" significant. But remember that those
classifications represent nothing else but arbitrary conventions that
are only informally based on general research experience.

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Statistical significance and the number of analyses performed. Needless
to say, the more analyses you perform on a data set, the more results
will meet "by chance" the conventional significance level. For example,
if you calculate correlations between ten variables (i.e., 45 different
correlation coefficients), then you should expect to find by chance
that about two (i.e., one in every 20) correlation coefficients are
significant at the p

.05
level, even if the values of the variables were totally random and
those variables do not correlate in the population. Some statistical
methods that involve many comparisons, and thus a good chance for such
errors, include some "correction" or adjustment for the total number of
comparisons. However, many statistical methods (especially simple
exploratory data analyses) do not offer any straightforward remedies to
this problem. Therefore, it is up to the researcher to carefully
evaluate the reliability of unexpected findings. Many examples in this
manual offer specific advice on how to do this; relevant information
can also be found in most research methods textbooks.

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Strength vs. reliability of a relation between variables. We
said before that strength and reliability are two different features of
relationships between variables. However, they are not totally
independent. In general, in a sample of a particular size, the larger
the magnitude of the relation between variables, the more reliable the
relation (see the next paragraph).

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Why stronger relations between variables are more significant. Assuming
that there is no relation between the respective variables in the
population, the most likely outcome would be also finding no relation
between those variables in the research sample. Thus, the stronger the
relation found in the sample, the less likely it is that there is no
corresponding relation in the population. As you see, the magnitude and
significance of a relation appear to be closely related, and we could
calculate the significance from the magnitude and vice-versa; however,
this is true only if the sample size is kept constant, because the
relation of a given strength could be either highly significant or not
significant at all, depending on the sample size (see the next
paragraph).

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Why significance of a relation between variables depends on the size of the sample. If
there are very few observations, then there are also respectively few
possible combinations of the values of the variables, and thus the
probability of obtaining by chance a combination of those values
indicative of a strong relation is relatively high. Consider the
following illustration. If we are interested in two variables (Gender:
male/female and WCC: high/low) and there are only four subjects in our
sample (two males and two females), then the probability that we will
find, purely by chance, a 100% relation between the two variables can
be as high as one-eighth. Specifically, there is a one-in-eight chance
that both males will have a high WCC and both females a low WCC, or
vice versa. Now consider the probability of obtaining such a perfect
match by chance if our sample consisted of 100 subjects; the
probability of obtaining such an outcome by chance would be practically
zero. Let's look at a more general example. Imagine a theoretical
population in which the average value of WCC in males and females is
exactly the same. Needless to say, if we start replicating a simple
experiment by drawing pairs of samples (of males and females) of a
particular size from this population and calculating the difference
between the average WCC in each pair of samples, most of the
experiments will yield results close to 0. However, from time to time,
a pair of samples will be drawn where the difference between males and
females will be quite different from 0. How often will it happen? The
smaller the sample size in each experiment, the more likely it is that
we will obtain such erroneous results, which in this case would be
results indicative of the existence of a relation between gender and
WCC obtained from a population in which such a relation does not exist.

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Example. "Baby boys to baby girls ratio." Consider
the following example from research on statistical reasoning (Nisbett,
et al., 1987). There are two hospitals: in the first one, 120 babies
are born every day, in the other, only 12. On average, the ratio of
baby boys to baby girls born every day in each hospital is 50/50.
However, one day, in one of those hospitals twice as many baby girls
were born as baby boys. In which hospital was it more likely to happen?
The answer is obvious for a statistician, but as research shows, not so
obvious for a lay person: It is much more likely to happen in the small
hospital. The reason for this is that technically speaking, the
probability of a random deviation of a particular size (from the
population mean), decreases with the increase in the sample size.

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Why small relations can be proven significant only in large samples. The
examples in the previous paragraphs indicate that if a relationship
between variables in question is "objectively" (i.e., in the
population) small, then there is no way to identify such a relation in
a study unless the research sample is correspondingly large. Even if
our sample is in fact "perfectly representative" the effect will not be
statistically significant if the sample is small. Analogously, if a
relation in question is "objectively" very large (i.e., in the
population), then it can be found to be highly significant even in a
study based on a very small sample. Consider the following additional
illustration. If a coin is slightly asymmetrical, and when tossed is
somewhat more likely to produce heads than tails (e.g., 60% vs. 40%),
then ten tosses would not be sufficient to convince anyone that the
coin is asymmetrical, even if the outcome obtained (six heads and four
tails) was perfectly representative of the bias of the coin. However,
is it so that 10 tosses is not enough to prove anything? No, if the
effect in question were large enough, then ten tosses could be quite
enough. For instance, imagine now that the coin is so asymmetrical that
no matter how you toss it, the outcome will be heads. If you tossed
such a coin ten times and each toss produced heads, most people would
consider it sufficient evidence that something is "wrong" with the
coin. In other words, it would be considered convincing evidence that
in the theoretical population of an infinite number of tosses of this
coin there would be more heads than tails. Thus, if a relation is
large, then it can be found to be significant even in a small sample.

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Can "no relation" be a significant result? The
smaller the relation between variables, the larger the sample size that
is necessary to prove it significant. For example, imagine how many
tosses would be necessary to prove that a coin is asymmetrical if its
bias were only .000001%! Thus, the necessary minimum sample size
increases as the magnitude of the effect to be demonstrated decreases.
When the magnitude of the effect approaches 0, the necessary sample
size to conclusively prove it approaches infinity. That is to say, if
there is almost no relation between two variables, then the sample size
must be almost equal to the population size, which is assumed to be
infinitely large. Statistical significance represents the probability
that a similar outcome would be obtained if we tested the entire
population. Thus, everything that would be found after testing the
entire population would be, by definition, significant at the highest
possible level, and this also includes all "no relation" results.

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How to measure the magnitude (strength) of relations between variables. There
are very many measures of the magnitude of relationships between
variables which have been developed by statisticians; the choice of a
specific measure in given circumstances depends on the number of
variables involved, measurement scales used, nature of the relations,
etc. Almost all of them, however, follow one general principle: they
attempt to somehow evaluate the observed relation by comparing it to
the "maximum imaginable relation" between those specific variables.
Technically speaking, a common way to perform such evaluations is to
look at how differentiated are the values of the variables, and then
calculate what part of this "overall available differentiation" is
accounted for by instances when that differentiation is "common" in the
two (or more) variables in question. Speaking less technically, we
compare "what is common in those variables" to "what potentially could
have been common if the variables were perfectly related." Let us
consider a simple illustration. Let us say that in our sample, the
average index of WCC is 100 in males and 102 in females. Thus, we could
say that on average, the deviation of each individual score from the
grand mean (101) contains a component due to the gender of the subject;
the size of this component is 1. That value, in a sense, represents
some measure of relation between Gender and WCC. However, this value is
a very poor measure, because it does not tell us how relatively large
this component is, given the "overall differentiation" of WCC scores.
Consider two extreme possibilities:

If all WCC scores of males were equal exactly to 100, and those of
females equal to 102, then all deviations from the grand mean in our
sample would be entirely accounted for by gender. We would say that in
our sample, gender is perfectly correlated with WCC, that is, 100% of
the observed differences between subjects regarding their WCC is
accounted for by their gender.

If WCC scores were in the range of 0-1000, the same
difference (of 2) between the average WCC of males and females found in
the study would account for such a small part of the overall
differentiation of scores that most likely it would be considered
negligible. For example, one more subject taken into account could
change, or even reverse the direction of the difference. Therefore,
every good measure of relations between variables must take into
account the overall differentiation of individual scores in the sample
and evaluate the relation in terms of (relatively) how much of this
differentiation is accounted for by the relation in question.

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Common "general format" of most statistical tests. Because
the ultimate goal of most statistical tests is to evaluate relations
between variables, most statistical tests follow the general format
that was explained in the previous paragraph. Technically speaking,
they represent a ratio of some measure of the differentiation common in
the variables in question to the overall differentiation of those
variables. For example, they represent a ratio of the part of the
overall differentiation of the WCC scores that can be accounted for by
gender to the overall differentiation of the WCC scores. This ratio is
usually called a ratio of explained variation to total variation. In
statistics, the term explained variation does not necessarily imply
that we "conceptually understand" it. It is used only to denote the
common variation in the variables in question, that is, the part of
variation in one variable that is "explained" by the specific values of
the other variable, and vice versa.

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How the "level of statistical significance" is calculated. Let
us assume that we have already calculated a measure of a relation
between two variables (as explained above). The next question is "how
significant is this relation?" For example, is 40% of the explained
variance between the two variables enough to consider the relation
significant? The answer is "it depends." Specifically, the significance
depends mostly on the sample size. As explained before, in very large
samples, even very small relations between variables will be
significant, whereas in very small samples even very large relations
cannot be considered reliable (significant). Thus, in order to
determine the level of statistical significance, we need a function
that represents the relationship between "magnitude" and "significance"
of relations between two variables, depending on the sample size. The
function we need would tell us exactly "how likely it is to obtain a
relation of a given magnitude (or larger) from a sample of a given
size, assuming that there is no such relation between those variables
in the population." In other words, that function would give us the
significance (p) level, and it would tell us the probability of error
involved in rejecting the idea that the relation in question does not
exist in the population. This "alternative" hypothesis (that there is
no relation in the population) is usually called the null hypothesis.
It would be ideal if the probability function was linear, and for
example, only had different slopes for different sample sizes.
Unfortunately, the function is more complex, and is not always exactly
the same; however, in most cases we know its shape and can use it to
determine the significance levels for our findings in samples of a
particular size. Most of those functions are related to a general type
of function which is called normal.

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Why the "Normal distribution" is important. The
"Normal distribution" is important because in most cases, it well
approximates the function that was introduced in the previous paragraph
(for a detailed illustration, see Are all test statistics normally distributed?).
The distribution of many test statistics is normal or follows some form
that can be derived from the normal distribution. In this sense,
philosophically speaking, the Normal distribution represents one of the
empirically verified elementary "truths about the general nature of
reality," and its status can be compared to the one of fundamental laws
of natural sciences. The exact shape of the normal distribution (the
characteristic "bell curve") is defined by a function which has only
two parameters: mean and standard deviation.

A characteristic property of the Normal distribution is that 68% of
all of its observations fall within a range of �tandard deviation from
the mean, and a range of �tandard deviations includes 95% of the
scores. In other words, in a Normal distribution, observations that
have a standardized value of less than -2 or more than +2 have a
relative frequency of 5% or less. (Standardized value means that a
value is expressed in terms of its difference from the mean, divided by
the standard deviation.) If you have access to STATISTICA, you
can explore the exact values of probability associated with
different values in the normal distribution using the interactive
Probability Calculator tool; for example, if you enter the Z value
(i.e., standardized value) of 4, the associated probability computed by
STATISTICA
will be less than .0001, because in the normal distribution almost all
observations (i.e., more than 99.99%) fall within the range of �tandard
deviations. The animation below shows the tail area associated with
other Z values.

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Illustration of how the normal distribution is used in statistical reasoning (induction). Recall
the example discussed above, where pairs of samples of males and
females were drawn from a population in which the average value of WCC
in males and females was exactly the same. Although the most likely
outcome of such experiments (one pair of samples per experiment) was
that the difference between the average WCC in males and females in
each pair is close to zero, from time to time, a pair of samples will
be drawn where the difference between males and females is quite
different from 0. How often does it happen? If the sample size is large
enough, the results of such replications are "normally distributed"
(this important principle is explained and illustrated in the next
paragraph), and thus knowing the shape of the normal curve, we can
precisely calculate the probability of obtaining "by chance" outcomes
representing various levels of deviation from the hypothetical
population mean of 0. If such a calculated probability is so low that
it meets the previously accepted criterion of statistical significance,
then we have only one choice: conclude that our result gives a better
approximation of what is going on in the population than the "null
hypothesis" (remember that the null hypothesis was considered only for
"technical reasons" as a benchmark against which our empirical result
was evaluated). Note that this entire reasoning is based on the
assumption that the shape of the distribution of those "replications"
(technically, the "sampling distribution") is normal. This assumption
is discussed in the next paragraph.

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Are all test statistics normally distributed? Not
all, but most of them are either based on the normal distribution
directly or on distributions that are related to, and can be derived
from normal, such as t, F, or Chi-square.
Typically, those tests require that the variables analyzed are
themselves normally distributed in the population, that is, they meet
the so-called "normality assumption." Many observed variables actually
are normally distributed, which is another reason why the normal
distribution represents a "general feature" of empirical reality. The
problem may occur when one tries to use a normal distribution-based
test to analyze data from variables that are themselves not normally
distributed (see tests of normality in Nonparametrics or ANOVA/MANOVA).
In such cases we have two general choices. First, we can use some
alternative "nonparametric" test (or so-called "distribution-free test"
see, Nonparametrics); but this is often inconvenient because such tests
are typically less powerful and less flexible in terms of types of
conclusions that they can provide. Alternatively, in many cases we can
still use the normal distribution-based test if we only make sure that
the size of our samples is large enough. The latter option is based on
an extremely important principle which is largely responsible for the
popularity of tests that are based on the normal function. Namely, as
the sample size increases, the shape of the sampling distribution
(i.e., distribution of a statistic from the sample; this term was first
used by Fisher, 1928a) approaches normal shape, even if the
distribution of the variable in question is not normal. This principle
is illustrated in the following animation showing a series of sampling
distributions (created with gradually increasing sample sizes of: 2, 5,
10, 15, and 30) using a variable that is clearly non-normal in the
population, that is, the distribution of its values is clearly skewed.



However, as the sample size (of samples used to create the
sampling distribution of the mean) increases, the shape of the sampling
distribution becomes normal. Note that for n=30, the shape of that
distribution is "almost" perfectly normal (see the close match of the
fit). This principle is called the central limit theorem (this term was
first used by P髄ya, 1920; German, "Zentraler Grenzwertsatz").

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How do we know the consequences of violating the normality assumption?
Although many of the statements made in the preceding paragraphs can be
proven mathematically, some of them do not have theoretical proofs and
can be demonstrated only empirically, via so-called Monte-Carlo
experiments. In these experiments, large numbers of samples are
generated by a computer following predesigned specifications and the
results from such samples are analyzed using a variety of tests. This
way we can empirically evaluate the type and magnitude of errors or
biases to which we are exposed when certain theoretical assumptions of
the tests we are using are not met by our data. Specifically,
Monte-Carlo studies were used extensively with normal
distribution-based tests to determine how sensitive they are to
violations of the assumption of normal distribution of the analyzed
variables in the population. The general conclusion from these studies
is that the consequences of such violations are less severe than
previously thought. Although these conclusions should not entirely
discourage anyone from being concerned about the normality assumption,
they have increased the overall popularity of the
distribution-dependent statistical tests in all areas of research.