Statistical Analysis of Data

1
Statistical Analysis of Data

• Graziano and Raulin
• Research Methods Chapter 5

2
Individual Differences

• A fact of life
• People differ from one another
• People differ from one occasion to another
• Most psychological variables have effects that
  are small compared to individual differences
• Statistics give us a way to detect such subtle
  effects in a sea of individual differences

3
Descriptive Statistics

• Are used to describe the data
• Many types of descriptive statistics
• Frequency distributions
• Summary measures such as measures of central
  tendency, variability, and relationship
• Graphical representations of the data
• A way to visualize the data
• The first step in any statistical analysis

4
Frequency Distributions

• First step in organization of data
• Can see how the scores are distributed
• Used with all types of data
• Illustrate relationships between variables in a
  cross-tabulation
• Simplify distributions with a large range by
  using a grouped frequency distribution

5
Histograms

• A bar graph, as shown at the right
• Can be used to graph either
• Data representing discrete categories
• Data representing scores from a continuous
  variable

6
Histograms (2 distributions)

• Possible to graph two or more distributions on
  the same histogram to see how they compare
• Note that one of the two groups in this histogram
  was the same group graphed previously

7
Frequency Polygon (1 group)

• Like a histogram except that, instead of a bar
  representing the mean score or frequency, a dot
  is used, with the dots connected as shown
• Data could be either discrete or continuous

8
Frequency Polygon (2 groups)

• Can compare two of more frequency polygons on the
  same scale as shown
• Easier to compare frequency polygons because the
  graph appears less cluttered than multiple
  histograms

9
Shapes of Distributions

• Many variables in psychology are distributed
  normally
• The distribution is skewed if scores bunch up at
  one end
• Illustrate on the left are symmetric and skewed
  distributions

10
Measures of Central Tendency

• Mode the most frequently occurring score
• Easy to compute from frequency distribution
• Median the middle score in a distribution
• Less affected than the mean by a few deviant
  scores
• Mean the arithmetic average
• Most commonly used central tendency measure
• Used in later inferential statistics

11
Computing the Mean

• Compute the mean of 3, 4, 2, 5, 7, 5
• Sum the numbers
• 26
• Count the numbers
• 6
• Plug these values into the equation at the right

12
Measuring Variability

• Range lowest to highest score
• Average Deviation average distance from the
  mean
• Variance average squared distance from the mean
• Used in later inferential statistics
• Standard Deviation square root of variance
• expressed on the same scale as the mean

13
Measures of Relationship

• Pearson product-moment correlation
• Used with interval or ratio data
• Spearman rank-order correlation
• Used when one variable is ordinal and the second
  is at least ordinal
• Scatter plots
• Visual representation of a correlation
• Helps to identify nonlinear relationships

14
Regression

• Using a correlation (relationship between
  variables) to predict one variable from knowing
  the score on the other variable
• Usually a linear regression (finding the best
  fitting straight line for the data)
• Best illustrated in a scatter plot with the
  regression line also plotted (see Figure 5.6)

15
Reliability Indices

• Test-retest reliability and interrater
  reliability are indexed with a Pearson
  product-moment correlation
• Internal consistency reliability is indexed with
  coefficient alpha
• Details on these computations are included on the
  CD supplement

16
Standard Scores (Z-scores)

• A way to put scores on a common scale
• Computed by subtracting the mean from the score
  and dividing by the standard deviation
• Interpreting the Z-score
• Positive Z-scores are above the mean negative
  Z-scores are below the mean
• The larger the absolute value of the Z-score, the
  further the score is from the mean

17
Inferential Statistics

• Used to draw inferences about populations on the
  basis of samples from the populations
• The statistical tests that we perform on our
  data are inferential statistics
• Provide an objective way of quantifying the
  strength of the evidence for our hypothesis

18
Populations and Samples

• Population the larger groups of all
  participants of interest to the researcher
• Sample a subset of the population
• Samples almost never represent populations
  perfectly (termed sampling error)
• Not really an error just the natural variability
  that you can expect from one sample to another

19
The Null Hypothesis

• States that there is NO difference between the
  population means
• Compare sample means to test the null hypothesis
• Population parameters sample statistics
• Population parameter is a descriptive statistic
  computed from everyone in the population
• Sample statistics is a descriptive statistic
  computed from everyone in your sample

20
Statistical Decisions

• We can either Reject or Fail to Reject the null
  hypothesis
• Rejecting the null hypothesis suggests that there
  is a difference in the populations sampled
• Failing to reject suggests that no difference
  exists
• Decision is based on probability (reject if it is
  unlikely that the null hypothesis is true)
• Alpha the statistical decision criteria used
• Traditionally alpha is set to small values (.05
  or .01)
• Always a chance for error in our decision

21
Statistical Decision Process
22
Testing for Mean Differences

• t-test for independent groups tests mean
  difference of two independent groups
• Correlated t-test tests mean difference of two
  correlated groups
• Analysis of Variance tests mean differences in
  two or more groups
• Groups may or may not be independent
• Also capable of evaluating factorial designs

23
Power of a Statistical Test

• Sensitivity of the procedure to detect real
  differences between the populations
• Not just a function of the statistical test, but
  also a function of the precision of the research
  design and execution
• Increasing the sample size increases the power
  because larger samples estimate the population
  parameters more precisely

24
Statistical versus Practical Significance

• Statistical significance means that the observed
  mean differences are not likely due to sampling
  error
• Can get statistical significance, even with very
  small population differences, if the sample size
  is large enough
• Practical significance looks at whether the
  difference is large enough to be of value in a
  practical sense

25
Effect Size

• Gives an indication of the size of the difference
  between groups
• Unlike the statistical test, the effect size is
  NOT affected by the size of the sample
• More details on effect size
• In Chapter 15
• On the CD supplement

26
Meta-Analysis

• A relatively new statistical technique
• Allows researchers to statistically combine the
  results of several studies of the same phenomenon
  to get a sense of how powerful the effect is
• Discussed in more detail in Chapter 15

27
Summary

• Statistics allow us to detect and evaluate group
  differences that are small compared to individual
  differences
• Descriptive versus inferential statistics
• Descriptive statistics describe the data
• Inferential statistics are used to draw
  inferences about population parameters on the
  basis of sample statistics
• Statistics objectify evaluations, but do not
  guarantee correct decisions every time