Becoming Acquainted With Statistical Concepts

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chapter 6

• Becoming Acquainted With Statistical Concepts

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Why We Need Statistics

• Statistics is an objective way of interpreting a
  collection of observations.
• Descriptions ex. Mean Score Average score of
  a group of observed score
• Mean score Sum of Scores/ Number of scores
• M ?X / N
• Associations ex. Correlations ( Pearson product
  moment coefficient of correlation, or Pearson r,
  or interclass, or simple correlation)

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Example of Correlation
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Why We Need Statistics

• Differences among groups ex. Dependent and
  independent t tests, ANOVA
• Differences in groups are either significant or
  non-significant
• Describing the data and inferring the data
  results are not the same. Statistics describe the
  sample and the impact of the independent
  variable. We then infer the results of the sample
  to the population of interest.

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Ways to Select a Sample

• Random sampling tables of random numbers
  random sampling allows for inference to the
  population
• Stratified random sampling
• Population divided by some characteristic before
  random selection occurs
• Allows sampling based upon the presence of a
  particular characteristic that naturally exists
  within the sampled population
• Results will then reflect the population sampled

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Ways to Select a Sample
Systematic sampling Logical Assignment of
Sampling that can be used for very large
populations (ex. Every 100th person listed in
phone directory) Random assignment How groups are
formed within the sample. This allows the
researcher to normalize inter-subject
variability between the experimental groups It
is the process of randomizing subjects within
the study after the experimental design is created
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Ways to Select a Sample
Justifying post hoc after the fact
explanations Does your sample reflect the
population inferred if your sample is not
randomized? How good does the sample have to
be? Can you truly have a random sample Findings
should be palusible in other participants,
treatments and situations, depending on their
similarity to the study characteristics. Good
enough for our purposes!
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Measures of Central Tendencyand Variability

• Central tendency scores using a number to
  best represent a group of numbers
• Mean (average) M ?X / N
• The mean score is a good measure of central
  tendency when there are not a small number of
  scores with a large range (ex. 1,3,2,4,2,56)
• Median (midpoint) is the middle score in the
  group of scores when the scores are place in
  chronological order
• Better choice of central tendency with larger
  range and few scores

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Measures of Central Tendencyand Variability
Mode (most frequent) The most frequently
occurring score. This is a good measure with
repeated scores, few scores and a wide range of
scores. One may also have bi- or multi-modal
distributions of scores
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Measures of Central Tendencyand Variability
Variability (variance) scores how the scores
vary in the group of scores Low variation (n5)
3,4,3,5,4 High variation (n5) 3,9,1,8,10
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Standardizing the Variation(see Table 6.1, p104,
text)

• First, decide which score best represents the
  group of scores. Calculate the mean score
• Then determine how each individual score (X)
  deviates from the mean score (M) by subtracting
  the mean score from the individual score. (X-M)
• The sum of the deviations (X-M) should equal zero
  if the mean score was the best representative
  score in the distribution of scores
• Squaring the deviation scores quantifies the
  variation of each score from the mean score

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Standardizing the Variation(see Table 6.1, p104,
text)

• Summing the squared deviation scores quantifies
  all of the variation in the group of scores
• To average the squared deviation scores, all
  scores in the distribution are used to average
  the deviation scores (with 1 degree of freedom
  (N-1 one score is held constant and others are
  free to vary ie. Point of comparison)
• The average of the squared deviations are the
  un-squared by the square root and we are left
  with the standard deviation.
• Thus the deviation scores have been standardized
  by the total variance in the group of scores
• If there is a large variance, there will be a
  large standard deviation.

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Standard Deviation

• The larger the standard deviation, the greater
  the variability in the group of scores
• The square of the standard deviation is equal to
  the total variance of the group of scores.

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Range of Scores

• Reporting the range of scores is typical when
  using the median score instead of the mean score.
• A large range of scores with a large variance
  would indicate that the median would be a better
  indicator of central tendency than the mean score

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Confidence Intervals

• A confidence interval provides an expected upper
  and lower limit for a statistic at a specified
  probability level
• A confidence interval provides a band within
  which the estimate of the population mean is
  likely to fall instead of a single point
• This pre-supposes sampling error. (The value I
  measured is likely to be represented in the
  population within this range with this degree of
  confidence (ex 95 or 99 probability)

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Standard Error

• The standard error represents the variability of
  the sampling distribution
• Thus, the standard error average variation one
  would expect from their collected sample if
  compared to the population. One could take
  multiple samples and calculate different mean
  scores. Thus if you calculated the standard
  deviation of all of these mean scores you would
  get the standard error.

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Standard Error

• Thus, the standard error is the best
  representation of the variability of the sample
  score when inferring the sample score to the
  population ( or variance in the population)
• The standard error is calculated by dividing the
  sample standard deviation by the square root of
  the sample size
• Thus one can reduce standard error by decreasing
  sample variability and/or increasing the subject
  number

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Frequency Distributions

• A frequency distribution assists the reader in
  visualizing the numbers of scores within a range
  of scores. It is a popular method in descriptive
  statistics used to develop histograms
• Example How many students scored between 90-100
  on the exam, 80-89, etc.

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Stem-and-Leaf Displays

• A major drawback to a frequency distribution is
  that there is a loss of information where the
  reader does not know how many individuals rated
  in scores within a group of scores. (See Figure
  6.2, p. 106, text)
• The Stem-and-Leaf display organizes raw scores
  where the intervals are shown on the left and the
  scores are horizontally lined to the right, from
  low to high. This is a helpful method when
  viewing the individual scores is of importance.

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Categories of Statistical Tests

• There are two categories Parametric and
  Nonparametric
• Parametric Statistical Test Assumptions
• Normal distribution the sample represents the
  population on the variable of interest
• Equal variances the sample variance is equal to
  the variance found in the population on the
  variable of interest
• Independent observations

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• Nonparametric (distribution free) the previous
  assumption of parametric statistics need not be
  met
• ex. Distribution is not normal
• Normal curve
• Mean, median, and mode are all at the same point
  at the center of the distribution
• Parametric statistics rely on the data following
  the pattern of a normal curve. The ability to
  determine if scores come from different
  distributions (curves) is the function parametric
  statistical tests.

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Normal Curve
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Normal Distribution (Curve)

• For this reason, parametric statistics can
  increase the power of the research model (ie.
  Chances of rejecting the Null hypothesis when the
  null is actually false)
• Thus, The assumptions of parametric statistics
  can be tested by estimating the degree of
  skewness or kurtosis.

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Skewness

• Skewness is the description of the direction of
  the hump or apex of the curve and the nature of
  the tails of the curve. A rightward skew of the
  TAIL of the curve is positive skew when the and
  a leftward skew is negative.

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Skewness
Negative Skew
Positive Skew
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Kurtosis

• Is a description of the vertical characteristic
  of the curve showing the data distribution. May
  be peaked or flattened but does not represent a
  normal distribution

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Kurtosis
Platykurtic
Leptokurtic
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MesokurticKurtosis 0
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Statistics

• What statistical techniques tell us about the
  data
• Reliability (significance) of effect if the
  research is repeated again, under like
  conditions, will the independent variable cause a
  significant change in the dependent variable
  scores
• Strength of the relationship (meaningfulness)
  how powerful was the introduction of the
  independent variable on the outcome of the
  dependent variable scores? What is the effect
  or magnitude of the relationship?

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Parametric Statistics - Overview

• Different Statistical Techniques

• Relationships between groups Correlation
• Cause and effect Correlation is no proof of
  causation.
• Differences between groups t Tests and ANOVA
• Differences can be significant but not meaningful