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Chapter 6
Statistical Concepts
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Statistics
• Statistics is simply an objective means of interpreting a collection of
observations.
• Various statistical techniques are necessary to describe the
characteristics of data, test relationships between sets of data, and test
the differences among sets of data.
Types of Statistics
• Descriptive
• example: Mean - A statistical measure of central tendency that is
the average score of a group of scores.
• Associations
• example: Pearson product moment coefficient of correlation—
The most commonly used method of computing correlation
between two variables; also called interclass correlation, simple
correlation, or Pearson r.
• Differences
• example: t test—A statistical technique to assess
differences between two groups.
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Selecting Research Samples
• The sample is the group of participants, treatments, and situations on
which the study is conducted.
Random Selection
The sample of participants might be randomly selected from some larger
group, or a population (the larger group from which a sample is taken.)
Potential numbered subject pools could be selected form a list using
random numbers table (A table in which numbers are arranged in two-digit (or greater)
sets so that any combination of rows or columns is unrelated - see Table 1 in Appendix)
Stratified Random Sampling
In stratified random sampling, the population is divided (stratified) on some
characteristic before random selection of the sample.
• example : the selection of 200 students from a population of 10,000; with 30%
freshmen, 30% sophomores, 20% juniors, and 20% seniors. Stratify on class before
random selection to make sure that the sample was exact in terms of class
representation. Here, you would randomly select 60 students from the 3,000
freshmen, 60 from the 3,000 sophomores, 40 from the 2,000 juniors, and 40 from the
2,000 seniors. This procedure still yields a total sample of 200.
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Selecting Research Samples
Systematic Sampling
If the population from which the sample is to be selected is very large,
assigning a numeric ID to each potential participant is time consuming.
Example : Suppose that you want to sample a town with a population of 50,000
concerning the need for new sport facilities. One approach would be to use
systematic sampling from the telephone book. You might decide to call a sample of
500 people. To do so, you would select every 100th name in the phone book
(50,000/500 = 100)
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Random Assignment
In experimental research, groups are formed within the sample. The issue
here is not how the sample is selected but how the groups are formed
within the sample.
All true experimental designs require that the groups within the sample
be randomly assigned or randomized. (this will be discussed in greater detail in
Chapter 18) A good sample leads to a generalization statement that states it
is plausible the findings apply to a broader population.
This process allows the researcher to assume that the groups are
equivalent at the beginning of the experiment, which is one of several
important features of good experimental design that is intended to
establish cause and effect.
Post Hoc Explanations
Frequently, the sample for research is not randomly selected; rather, the
researcher attempts a post hoc justification that the sample represents
some larger group. A post hoc attempt at generalization may be better
than nothing, but it is not the equivalent of random selection, which allows
the assumption that the sample does not differ from the population on the
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characteristics measured.
Unit of Analysis
Unit of analysis — The concept, related to sampling and statistical
analysis, that refers to what is considered the most basic unit from which
data can be produced. This concept refers to what can be considered the
most basic unit from which data can be produced.
Example : Fitness Categories 1) High Fitness, 2) Moderate Fitness, 3) Low
Fitness Groups (See also example in book, p 104-105)
Any of the following could be a unit of analysis in a study:
• individuals
• groups
• artifacts (books, photos, newspapers)
• geographical units (town, census tract, state)
• social interactions (dyadic relations, divorces, arrests)
The Unit of Analysis can significantly affect the sample size.
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Measures of Central Tendency and Variability
Central tendency (measure of) — A single score that best
represents all the scores.
When you have a group of scores, one number may be used to
represent the group. That number is generally the mean, median, or
mode. These terms are ways of expressing central tendency.
Within the group of scores, each individual score differs to some
degree from the central tendency score. The degree of difference
is the variability of the score.
Thus, you can have variability within a group of scores (within group variance) ,
and/or variability between groups (between group variance)
Between group variability and within group variability are both components of
the total variability in the combined distributions. What we are doing when we
compute between and within variability is to partition the total variability into
the between and within components. So: Between variability + within
variability = total variability
Variability can be estimated with the standard deviation (see later slide)
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Mean: The mean is the average score in a group of scores.
Example: (M), or average: M = ΣX/N (sum of scores divided by the number of
scores)
Median — A statistical measure of central tendency that is the middle score
in a group. The median is defined as the value in the middle; the middle value
is the value that occurs in the place (N + 1)/2 when the values are put in
order.
The median score may be used because the mean may not be the most
representative or characteristic score, especially if there is an nonrepresentative (outlier) score in the distribution of scores.
Mode — A statistical measure of central tendency that is the most frequently
occurring score of the group. Mode scores are helpful if you are looking for
the most frequented response or score in a distribution of scores.
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Variability Scores
Variability — The degree of difference between each individual score and
the central tendency score.
An estimate of the variability, or spread, of the scores can be calculated
as the standard deviation. ( see Table 6.1, p106, for an example of how to
calculate the standard deviation) The square of the standard deviation is
called the variance, or s2
Table 6.1, contains the scores, deviation scores, and squared deviation
scores.
The mean and standard deviation together are good descriptions of a
set of scores. If the standard deviation is large, the mean may not be
a good representation. Roughly 68% of a set of scores fall within ±1s,
about 95% of the scores fall within ±2s, and about 99% of the scores
fall within ±3s. This distribution of scores is called a normal
distribution
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Range of Scores
Sometimes the range of scores (highest and lowest) may also be reported,
particularly when the median is used rather than the mean. The RANGE is
the difference between the lowest and highest values. ( high score – low
score)
Confidence intervals (CI)
CI’s should be used because statistics vary in how well they represent target
populations.
• A CI provides an expected upper and lower limit for a statistic at a
specified probability level, usually either 95% or 99%.
• CI’s are based on the fact that any statistic possesses sampling error.
This error relates to how well the statistic represents the target
population.
• When we compute a mean for a sample, we are making an estimate
of the mean of the target population. A CI provides a band within
which the estimate of the population mean is likely to fall instead of a
single point (review example on p 107-108, text)
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Frequency Distribution and the Stem-and-Leaf Display
(see figure 6.2, p 108)
• A common technique for summarizing data is to produce a picture
(called a histogram) of the distribution of scores by means of a frequency
distribution.
• Frequency distribution - A distribution of scores including the
frequency with which they occur.
• If there is a wide range of values, a grouped frequency distribution is used
in which scores are grouped into small ranges called frequency intervals.
• Frequency intervals - Small ranges of scores within a frequency
distribution into which scores are grouped.
• One drawback to a grouped frequency distribution is that information is lost; that is, a
reader does not know the exact score of each individual within a given interval. Thus, Stem
and leaf designs are helpful in understanding both the shape of the distribution and the
exact scores.
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Frequency Distribution and the Stem-and-Leaf Display
• Stem-and-leaf display — A method of organizing raw scores by
which score intervals are shown on the left side of a vertical line and
individual scores falling into each interval are shown on the right side.
This representation is similar to a grouped frequency distribution, but no
information is lost.
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Categories of Statistical Tests
The two general categories of statistical tests are parametric and
nonparametric. Using the various tests in each category requires meeting
the assumptions for those tests. The first category, parametric statistical
tests, has three assumptions about the distribution of the data:
♦ The population from which the sample is drawn is normally
distributed on the variable of interest.
♦ The samples drawn from a population have the same variances on
the variable of interest.
♦ The observations are independent.
The second category, nonparametric statistics, is called distribution free
because the previous assumptions need not be met.
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Parametric Statistics
Whenever the assumptions are met, parametric statistics are often said to
have more power. Having more power increases the chances of rejecting
a false null hypothesis.
The assumptions can be tested by using estimates of skewness and
kurtosis. Normal distributions of scores will resemble the normal
curve.
Skewness — Description of the direction of the hump of the curve of the
data distribution and the nature of the tails of the curve.
Kurtosis — Description of the vertical characteristic of the curve
showing the data distribution, such as whether the curve is more peaked
or flatter than the normal curve.
Normal curve — Distribution of data in which the mean, median, and
mode are at the same point (center of the distribution) and in which ±1s
from the mean includes 68% of the scores, ±2s from the mean includes
95% of the scores, and ±3s includes 99% of the scores.
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Skewness and Kurtosis (Figure 6.3-6.5, p110, text)
Skewness of the distribution describes the direction of the hump of the
curve (labeled A in figure 6.4) and the nature of the tails of the curve
(labeled B and C). If the hump (A) is shifted to the left and the long tail (B)
to the right (figure 6.4a), the skewness is positive. If the shift of the hump
(A) is to the right and the long tail (C) is to the left (figure 6.4b), the
skewness is negative.
Kurtosis describes the vertical aspect of the curve, such as whether the
curve is more or less peaked than the normal curve. Figure 6.5a shows a
more peaked curve, and figure 6.5b shows a flatter curve.
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Looking Ahead…
What statistical techniques tell us….
Reliability (significance) of effect
Strength of the relationship (meaningfulness)
Types of statistical techniques…
Relationships among groups (units)
Differences among groups (units)
END OF PRESENTATION
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