This presentation is about introduction to statistics and data analysis using statistical methods. Population, sampling and all the methods to test them are included.
2. Definition of Statistics
◉ The term statistics refers to a set of mathematical procedures
for organizing, summarizing, and interpreting information.
◉ Statistical procedures help ensure that the information or
observations are presented and interpreted in an accurate and
informative way. In somewhat grandiose terms, statistics help
researchers bring order out of chaos. In addition, statistics
provide researchers with a set of standardized techniques that
are recognized and understood throughout the scientific
community.
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3. Population and Sample
◉ A population is the set of all the
individuals of interest in a
particular study.
◉ As you can well imagine, a
population can be quite large,
for example, the entire set of
women on the planet Earth. A
researcher might be more
specific, limiting the population
for study to women who are
registered voters in the United
States.
◉ A sample is a set of individuals
selected from a population, usually
intended to represent the
population in a research study.
◉ Just as we saw with populations,
samples can vary in size. For
example, one study might examine
a sample of only 10 students in a
graduate program and another
study might use a sample of more
than 10,000 people who take a
specific cholesterol medication.
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5. Variable and Data
◉ A variable is a characteristic or
condition that changes or has
different values for different
individuals.
◉ Once again, variables can be
characteristics that differ from one
individual to another, such as
height, weight, gender, or
personality. Also, variables can be
environmental conditions that
change such as temperature, time of
day, or the size of the room in which
the research is being conducted.
◉ Data (plural) are measurements or
observations.
◉ A data set is a collection of
measurements or observations.
◉ A datum (singular) is a single
measurement or observation and is
commonly called a score or raw
score.
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6. Parameters and Statistics
◉ A parameter is a value, usually a
numerical value, that describes a
population.
◉ A parameter is usually derived
from measurements of the
individuals in the population.
◉ For example, we want to
know the average length of a
butterfly. This is a parameter
because it is states something
about the entire population of
butterflies.
◉ A statistic is a value, usually a
numerical value, that describes a
sample.
◉ A statistic is usually derived from
measurements of the individuals in
the sample.
◉ For example, the parameter may
be the average height of 25-year-
old men in North America. The
height of the members of a sample
of 100 such men are measured; the
average of those 100 numbers is a
statistic.
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7. Descriptive and Inferential Statistical Methods
◉ Descriptive statistics are statistical
procedures used to summarize,
organize, and simplify data.
◉ Descriptive statistics are techniques
that take raw scores and organize
or summarize them in a form that is
more manageable. Often the scores
are organized in a table or a graph
so that it is possible to see the
entire set of scores. Another
common technique is to summarize
a set of scores by computing an
average.
◉ Inferential statistics consist of
techniques that allow us to study
samples and then make
generalizations about the
populations from which they were
selected.
◉ Because populations are typically
very large, it usually is not possible
to measure everyone in the
population. Therefore, a sample is
selected to represent the
population.
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8. ◉ Sampling error is the naturally occurring discrepancy, or error, that
exists between a sample statistic and the corresponding population
parameter.
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9. Constructs and Operational Definitions
◉ Constructs are internal attributes
or characteristics that cannot be
directly observed but are useful
for describing and explaining
behaviour.
◉ Constructs exist at a higher
level of abstraction than
concepts. Justice, Beauty,
Happiness, and Health are all
constructs.
◉ An operational definition identifies
a measurement procedure (a set of
operations) for measuring an
external behaviour and uses the
resulting measurements as a
definition and a measurement of a
hypothetical construct.
◉ Note that an operational definition
has two components. First, it
describes a set of operations for
measuring a construct. Second, it
defines the construct in terms of the
resulting measurements.
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10. Discrete and Continuous Variable
◉ A discrete variable consists of separate,
indivisible categories. No values can
exist between two neighbouring
categories.
◉ Discrete variables are commonly
restricted to whole, countable
numbers—for example, the number of
children in a family or the number of
students attending class. A discrete
variable may also consist of
observations that differ qualitatively. For
example, people can be classified by
gender (male or female), by occupation
(nurse, teacher, lawyer, etc.)
◉ For a continuous variable, there are an
infinite number of possible values that
fall between any two observed values. A
continuous variable is divisible into an
infinite number of fractional parts.
◉ For example, two people who both
claim to weigh 150 pounds are probably
not exactly the same weight. However,
they are both around 150 pounds. One
person may actually weigh 149.6 and
the other 150.3. Thus, a score of 150 is
not a specific point on the scale but
instead is an interval
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12. Nominal Scale and Ordinal Scale
◉ A nominal scale consists of a set
of categories that have different
names.
◉ Measurements on a nominal scale
label and categorize
observations, but do not make
any quantitative distinctions
between observations. The rooms
or offices in a building may be
identified by numbers.
◉ An ordinal scale consists of a set of
categories that are organized in an
ordered sequence. Measurements
on an ordinal scale rank
observations in terms of size or
magnitude.
◉ Often, an ordinal scale consists of a
series of ranks (first, second, third,
and so on) like the order of finish in
a horse race. Occasionally, the
categories are identified by verbal
labels like small, medium, and large
drink sizes at a fast-food restaurant.
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13. Interval Scale and Ratio Scale
◉ An interval scale consists of
ordered categories that are all
intervals of exactly the same size.
◉ Equal differences between
numbers on scale reflect equal
differences in magnitude.
However, the zero point on an
interval scale is arbitrary and
does not indicate a zero amount
of the variable being measured.
◉ A ratio scale is an interval scale with
the additional feature of an absolute
zero point. With a ratio scale, ratios
of numbers do reflect ratios of
magnitude.
◉ For example, you know that a
measurement of 80° Fahrenheit is
higher than a measure of 60°, and
you know that it is exactly 20°
higher.
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14. Shape of Frequency Distribution
◉ In a symmetrical distribution, it is possible to draw a vertical line
through the middle so that one side of the distribution is a mirror image
of the other.
◉ In a skewed distribution, the scores tend to pile up toward one end of
the scale and taper off gradually at the other end.
◉ The section where the scores taper off toward one end of a distribution is
called the tail of the distribution.
◉ A skewed distribution with the tail on the right-hand side is positively
skewed because the tail points toward the positive (above-zero) end of
the X-axis. If the tail points to the left, the distribution is negatively
skewed.
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17. Introduction to Measures of Central Tendency
Mean
The mean for a
distribution is the sum of
the scores divided by the
number of scores
Median
If the scores in a distribution
are listed in order from
smallest to largest, the
median is the midpoint of
the list. More specifically, the
median is the point on the
measurement scale below
which 50% of the scores in
the distribution are located.
Mode
If the scores in a distribution
are listed in order from
smallest to largest, the median
is the midpoint of the list. More
specifically, the median is the
point on the measurement
scale below which 50% of the
scores in the distribution are
located.
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19. Introduction to Variability
◉ Variability provides a quantitative measure of the differences between
scores in a distribution and describes the degree to which the scores are
spread out or clustered together.
◉ The range, is the distance covered by the scores in a distribution, from
the smallest score to the largest score.
◉ Deviation is distance from the mean:
Deviation score = X - μ
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20. ◉ SS, or sum of squares, is the sum of the squared deviation scores.
◉ Variance equals the mean of the squared deviations. Variance is the
average.
◉ Standard deviation is the square root of the variance and provides a
measure of the standard, or average distance from the mean.
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23. Introduction to z-score
◉ The z-score definition is adequate for transforming back and forth from X
values to z-scores as long as the arithmetic is easy to do in your head.
◉ Z-scores are often used in academic settings to analyze how well a
student's score compares to the mean score on a given exam. For example,
suppose the scores on a certain college entrance exam are roughly normally
distributed with a mean of 82 and a standard deviation of 5.
◉ For more complicated values, it is best to have an equation to help structure
the calculations. Fortunately, the relationship between X values and z-scores is
easily expressed in a formula. The formula for transforming scores into z-
scores is
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24. ◉ The numerator of the equation, X – μ, is a deviation score.
◉ It measures the distance in points between X and μ and
indicates whether X is located above or below the mean.
◉ The deviation score is then divided by σ because we want the z-
score to measure distance in terms of standard deviation units.
◉ The formula performs exactly the same arithmetic that is used
with the z-score definition, and it provides a structured equation
to organize the calculations when the numbers are more
difficult.
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27. Introduction to Hypothesis Testing
◉ A hypothesis test is a statistical method that uses sample data to
evaluate a hypothesis about a population.
◉ The Four Steps of a Hypothesis Test
STEP 1
State the hypothesis. As the name implies, the process of hypothesis
testing begins by stating a hypothesis about the unknown
population. Actually, we state two opposing hypotheses. Notice that
both hypotheses are stated in terms of population parameters.
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28. ◉ The first and most important of the two hypotheses is called the
null hypothesis. The null hypothesis states that the treatment
has no effect. Thenull hypothesis is identified by the symbol H0.
The null hypothesis (H0) states that in the general population there
is no change, no difference, or no relationship. In the context of an
experiment, H0 predicts that the independent variable (treatment)
has no effect on the dependent variable (scores) for the population.
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29. ◉ The second hypothesis is simply the opposite of the null
hypothesis, and it is called the scientific, or alternative,
hypothesis (H1)
The alternative hypothesis (H1) states that there is a change, a
difference, or a relationship for the general population. In the
context of an experiment, H1 predicts that the independent
variable (treatment) does have an effect on the dependent
variable.
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34. ◉ A Type I error occurs when
a researcher rejects a null
hypothesis that is actually
true. In a typical research
situation, a Type I error
means the researcher
concludes that a treatment
does have an effect when in
fact it has no effect.
◉ A Type II error occurs when
a researcher fails to reject a
null hypothesis that is really
false. In a typical research
situation, a Type II error
means that the hypothesis
test has failed to detect a
real treatment effect.
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35. STEP 2
Set the criteria for a decision. Eventually the researcher will use the data from
the sample to evaluate the credibility of the null hypothesis. The data will either
provide support for the null hypothesis or tend to refute the null hypothesis.
The Alpha Level To find the boundaries that separate the high-probability
samples from the low-probability samples, we must define exactly what is meant
by “low” probability and “high” probability. This is accomplished by selecting a
specific probability value, which is known as the level of significance, or the alpha
level, for the hypothesis test. The alpha (α) value is a small probability that is
used to identify the low-probability samples. By convention, commonly used
alpha levels are α = .05 (5%), α = .01 (1%), and α = .001 (0.1%).
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36. ◉ The extremely unlikely values, as defined by the alpha level,
make up what is called the critical region.
◉ The alpha level, or the level of significance, is a probability
value that is used to define the concept of “very unlikely” in a
hypothesis test.
◉ The critical region is composed of the extreme sample values
that are very unlikely (as defined by the alpha level) to be
obtained if the null hypothesis is true. The boundaries for the
critical region are determined by the alpha level. If sample data
fall in the critical region, the null hypothesis is rejected.
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37. ◉ The Boundaries for the Critical Region To determine the exact
location for the boundaries that define the critical region, we use the
alpha-level probability and the unit.
◉ In most cases, the distribution of sample means is normal, and the
unit normal table provides the precise z-score location for the critical
region boundaries.
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38. ◉ Degrees of freedom describe the number of scores in a sample
that are independent and free to vary. Because the sample
mean places a restriction on the value of one score in the
sample, there are n – 1 degrees of freedom for a sample with n
scores
◉ For a sample of n scores, the degrees of freedom, or df, for the
sample variance are defined as df = n - 1. The degrees of
freedom determine the number of scores in the sample that are
independent and free to vary.
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39. The Unit Table
◉ The graph shows proportions for only a few selected z-score values. A more
complete listing of z-scores and proportions is provided in the unit normal table.
◉ This table lists proportions of the normal distribution for a full range of possible z-
score values.
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A normal distribution following a z-score
transformation
42. STEP 3
Collect data and compute sample statistics
◉ The data are as given, so all that remains is to compute the statistic.
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43. STEP 4
Make a decision
◉ The sample data are located in the critical region. By definition, a
sample value in the critical region is very unlikely to occur if the null
hypothesis is true. Therefore, we conclude that the sample is not
consistent with H0 and our decision is to reject the null hypothesis.
Remember, the null hypothesis states that there is no treatment
effect, so rejecting H0 means we are concluding that the treatment
did have an effect.
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44. Introduction to the t Statistic
◉ The t statistic is used to test hypotheses about an unknown
population mean, μ, when the value of σ is unknown. The
formula for the t statistic has the same structure as the z-
score formula, except that the t statistic uses the estimated
standard error in the denominator.
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45. ◉ The estimated standard error (SM) is used as an estimate of the
real standard error σM when the value of σ is unknown. It is
computed from the sample variance or sample standard
deviation and provides an estimate of the standard distance
between a sample mean M and the population mean μ.
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48. The t Test for Two Independent Samples
◉ A research design that uses a separate group of participants
for each treatment condition (or for each population) is
called an independent-measures research design or a
between-subjects design.
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49. ◉ The Estimated Standard Error In each of the t-score formulas, the standard error
in the denominator measures how accurately the sample statistic represents the
population parameter. In the single-sample t formula, the standard error measures
the amount of error expected for a sample mean and is represented by the symbol
SM. For the independent measures t formula, the standard error measures the
amount of error that is expected when you use a sample mean difference (M1 − M2)
to represent a population mean difference (μ1 − μ2). The standard error for the
sample mean difference is represented by the symbol S(M1 - M2)
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50. ◉ Pooled Variance
◉ One method for correcting the bias in the standard error is to combine the two
sample variances into a single value called the pooled variance. The pooled
variance is obtained by averaging or “pooling” the two sample variances using
a procedure that allows the bigger sample to carry more weight in determining
the final value.
◉ For the independent-measures t statistic, there are two SS values and two df
values (one from each sample). The values from the two samples are combined
to compute what is called the pooled variance.
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57. Introduction to Repeated-Measures Designs
◉ A repeated-measures design, or a within-subject design, is one in
which the dependent variable is measured two or more times for each
individual in a single sample. The same group of subjects is used in all of
the treatment conditions.
◉ In a repeated-measures design or a matched-subjects design comparing
two treatment conditions, the data consist of two sets of scores, which
are grouped into sets of two, corresponding to the two scores obtained
for each individual or each matched pair of subjects.
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61. Analysis of Variance ANOVA
◉ Analysis of variance (ANOVA) is a hypothesis-testing procedure that is used to
evaluate mean differences between two or more treatments (or populations).
◉ As with all inferential procedures, ANOVA uses sample data as the basis for drawing
general conclusions about populations.
◉ It may appear that ANOVA and t tests are simply two different ways of doing exactly
the same job: testing for mean differences. In some respects, this is true—both tests
use sample data to test hypotheses about population means.
◉ However, ANOVA has a tremendous advantage over t tests. Specifically, t tests are
limited to situations in which there are only two treatments to compare.
◉ The major advantage of ANOVA is that it can be used to compare two or more
treatments.
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62. ◉ There really are no differences between the populations (or
treatments). The observed differences between the sample means are
caused by random, unsystematic factors (sampling error) that
differentiate one sample from another.
◉ The populations (or treatments) really do have different means, and
these population mean differences are responsible for causing
systematic differences betweenthe sample means.
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63. ◉ In analysis of variance, the variable (independent or quasi-
independent) that designates the groups being compared is called a
factor.
◉ The individual conditions or values that make up a factor are called
the levels of the factor.
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68. ◉ The distribution of F-Ratios
◉ For ANOVA, we expect F near 1.00 if H0 is true. An F-ratio that is
much larger than 1.00 is an indication that H0 is not true. In the F
distribution, we need to separate those values that are
reasonably near 1.00 from the values that are significantly greater
than 1.00.
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75. The Pearson Correlation
◉ The Pearson correlation measures the degree and the direction of the linear
relationship between two variables.
◉ The Pearson correlation for a sample is identified by the letter r. The
corresponding correlation for the entire population is identified by the Greek
letter rho (ρ), which is the Greek equivalent of the letter r.
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77. ◉ The sum of products of deviations, or SP. This new value is similar to SS
(the sum of squared deviations), which is used to measure variability for a
single variable. Now, we use SP to measure the amount of co-variability
between two variables.
◉ In general, the squared correlation (r2) measures the gain in accuracy that
is obtained from using the correlation for prediction. The squared
correlation measures the proportion of variability in the data that is
explained by the relationship between X and Y. It is sometimes called the
coefficient of determination.
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78. ◉ The value r2 is called the coefficient of determination because it
measures the proportion of variability in one variable that can be
determined from the relationship with the other variable. A correlation
of r = 0.80 (or –0.80), for example, means that r2 = 0.64 (or 64%) of the
variability in the Y scores can be predicted from the relationship with X.
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