1. Action Research Review
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Glenn Booker
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2. Why do we do this?
Measurements are needed to understand a system, and predict its future behavior
Statistical techniques provide a commonly accepted means of analyzing measurements
Statistics is based on recognizing that measurements tend to fall over a range of values, not just one precise number
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3. Types of Research Historical (what happened?)
Descriptive (what is happening?)
Developmental (over time)
Case and Field (study an organization)
Correlational (does A affect B?)
Causal Comparative (what caused it)
True Experimental (single / double blind)
Quasi-Experimental
Action Research
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4. Data Analysis Raw data, such as one survey result
Refined data, such as the distribution of ages of Philadelphia residents
Derived data, such as comparing the age distribution of Philadelphia residents to that of the country
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5. Population vs. Sample
Often the subject of interest (population) is so big it isn’t feasible to measure it all
Then a sample of measurements can be made, and we want to relate the sample measurement to the population
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6. Sampling
Sampling can be done using probabilistic techniques (e.g. various random samples)
Simple or stratified random,
Cluster (geographic), or
Systematic (every Nth) samples
Or using non-probabilistic methods (whoever’s convenient, specific groups, or experts)
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7. Customer Satisfaction Surveys
A special case of sampling, customer satisfaction surveys are often done using:
In person interview
Telephone interview
Questionnaire by mail
Sample sizes are based on the allowable error, population size, and the result obtained
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8. Measurement Scales
Measurements can use four major types of scales; the types of analysis possible depend strongly on the type of measurements used
Nominal (named buckets, without sequence)
Ordinal (ordered buckets)
Interval (intervals mean something, can +-)
Ratio (you can form ratios, can +-*/ )
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9. Discrete versus Continuous
Discrete (nonparametric) measurements use nominal or ordinal scales; only specific values are allowed
Car make = Chevy, or cost = High
Continuous (parametric) measurements use interval or ratio scales, and generally have integer or real number values
Temperature = 98.6 deg F, Height = cm
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10. Descriptive Statistics
Many common statistics can describe the central tendency of a set of measurements
Average (arithmetic mean)
Minimum, Maximum, Range
Median (middle value)
Mode (most common value)
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11. Normal Distribution
Many measurements can be described by a “normal” distribution, which is summarized by an average value and a standard deviation, s or s
We can predict how likely any range of values is to occur for a normal distribution (how often is X between 5 and 8?)
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12. Z Score
Z scores measure how far from the mean a single measurement is z = (Xi - m) / s
Same formula used for finding “t” too
Does not only apply to a normal distribution, but if it does, then we can predict the probability of that value or higher/lower occurring
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13. Standard Error
A sample of N measurements will have a standard error SEx = s / sqrt(N)
The standard error allows us to define the confidence interval, CI CI = mean +/- crit*SEx where “crit” is the critical z score for a large sample, or the critical t score for a small sample
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14. Critical z and t
The critical z score is only a function of the desired confidence level of the results (zc = 1.96 for 95% confidence level)
Critical t score is a function of the sample size (degrees of freedom, df = n-1) and the desired confidence level
As df gets very large, critical t  critical z
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15. Confidence Level
We have to accept some level of uncertainty in a statistical analysis – our conclusion might be wrong!
Generally, a 95% level of confidence is used, unless life is on the line - then a 99% level of confidence is required
Use 95% typically, hence critical significance is 0.050
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16. Confidence Level
The level of confidence of your results, plus the critical significance, always equals exactly one
For practically every statistical test, having the Significance of the result less than the critical value means to reject the null hypothesis
If Sig actual < Sig crit, reject null hypothesis
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17. Frequency and Percentage
Frequency graphs and crosstabs can provide a lot of information just from counts of a nominal or ordinal measurement occurring, possibly given with the percentages of each event’s occurrence
Histograms can provide similar charts for ratio or interval scaled data
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18. Scatterplots
Scatter plots or diagrams show the relationship between two or more measures
The horizontal axis is generally the independent variable (X), sometimes also called a factor or grouping variable
The vertical axis is generally the dependent variable (Y), which is the measure you’re trying to understand
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19. Hypothesis Testing
Some statistics are used in the context of testing a hypothesis - a statement whose truth you wish to determine
Are Philadelphians more likely to be Nobel Prize winners?
The Null hypothesis is the opposite of the hypothesis, and generally says there is no difference or no effect observed
Philadelphians no more likely to be Nobel Prize winners than any other group
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20. Hypothesis Testing
Can’t truly PROVE anything - only determine if the differences observed are “not likely to be due to chance”
Select one or more “Tests of Significance” to determine if there is a statistically significant difference (Yes/No); if Yes, then can
Select one or more “Measures of Association” to describe the strength of the difference, and possibly its direction
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21. One versus Two Tailed Tests
A null hypothesis which tests for “no difference” uses a two tailed test
A null hypothesis which specifically tests for “greater than” uses a one tailed test
A null hypothesis which specifically tests for “less than” uses a one tailed test
One versus two tailed changes the critical z or t score; generally makes the test easier to show significance – that’s why two-tailed tests are used
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22. Z or T Test
The z or t tests can be used to compare two distribution means, or compare one distribution mean to a fixed value (interval or ratio data)
Compare the actual z or t score to the critical z or t score
If the actual z or t score is closer to zero than the critical value, accept the null hypothesis
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23. Z or T Test (Two Tailed)
Notice this is for the x or t value, NOT the significance of that value
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24. Z or T Test (One Tailed)
(Case here is testing if the actual value is greater than the mean; for a “less than” case, use only the negative critical value.)
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25. Is My Sample Normal?
Boxplots and stem-and-leaf diagrams can help show graphically whether a sample has a fairly normal distribution
The skewness and kurtosis of a data set can help identify non-normality, if their values are more than two times their own standard errors
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26. T Tests T tests compare means for ratio or interval data
Independent t test is for two different strata within one data set
Paired t test is to compare measures of the same group before and after some event (drug test), or the samples are otherwise believed to be dependent on each other
One-sample t test compares one sample to a fixed value
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27. T Tests
Null hypothesis is that there is no difference between the means
Results (e.g. significance) may differ if variances are not equal, since df changes
The Levene test checks for equal variances
Null hypothesis for the Levene test is that the variances are equal
If the Levene significance < 0.050, variances are not equal (reject the null hypothesis)
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28. Independent T Test Evaluation
Three ways to check the results of a T test
If the T test’s significance < 0.050, reject the null hypothesis
Check the stated t value against the critical t value for this ‘df’ level; if t(actual) > t(critical) reject the null hypothesis
If the confidence interval for the difference between the means does not include zero, reject the null hypothesis
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29. Evaluating Significance
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30. Paired T Test Evaluation
Checks before and after test cases
Includes a correlation factor (like ‘r’)
Can use paired test if significance < 0.050
Larger correlation factor means stronger relationship between the variables
Test evaluation as Independent T Test
Significance, ‘t’ value, and confidence interval
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31. One-Sample T Test Compare a sample mean to a fixed value
Test shows the actual values of means, with their std deviation and std error
Same interpretation of results
Significance, ‘t’ value, and confidence interval
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32. F Test and ANOVA
Compare several means against each other using Analysis of Variance (ANOVA) and the F test
Like extending the T tests to many variables
Want data from random samples of normal populations with equal variances
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33. F Test and ANOVA Output includes the Levene test
Want significance for Levene > 0.050, so that equal variances can be assumed
Otherwise, should not use ANOVA
Evaluate F by its significance
If Sig. < 0.050, reject the null hypothesis (there is a significant difference among the means)
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34. Additional ANOVA Tests
Once the F test shows there is some difference in the means across a subset, additional ANOVA tests can help identify more specific trends and differences
Types of tests (see end of lecture 6) include
Pairwise Multiple Comparisons
Post Hoc Range Tests
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35. Pairwise Multiple Comparisons
Pairwise Multiple Comparisons check two subsets of data at a time
Bonferroni test is better for a small number of subsets
Tukey test is better for many subsets
Both assume subset variances are equal
For each pair of subset values, Sig < means the difference in means is significant
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36. Post Hoc Range Tests
Post Hoc Range Tests look for groups within each subset which all have similar variances
Tukey and Tukey’s-b tests include Post Hoc Range Tests
Each column of the output is a subset with statistically similar means
Subsets may overlap substantially
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37. Contrasts Across Means
Look across subset means to see if there is a trend, such as a linear increase or decrease across subsets
Can check for Linear, Quadratic, or Cubic relationships
(i.e. first, second, or third order polynomials)
Check Significance of F for the Unweighted version of each relationship (Linear, etc.) if Sig. < 0.050, reject the null hypothesis
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38. Determine Linearity
An option under Compare Means / Means allows checking just for linearity
This confirms the ANOVA test result for Linearity
And gives R and Eta parameters, which are Measures of Association
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39. R and Eta
Pearson’s R * measures how well the data fits the regression (-1 is a perfect negative correlation, 0 is no relationship, 1 is perfect positive correlation), and describes the amount of shared variance between them
Eta squared gives how much of the variance in one variable is caused by the changes in the other variable
* Named for English statistician Karl Pearson, (per
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40. Regression Analysis
Regression Analysis looks at two interval or ratio-scaled variables (generically X and Y) and tries to fit an equation between them
A dozen different equations are available
Linear, Power, Logarithmic, Exponential, etc.
Significance is checked by ANOVA F, and Sig. of the regression coefficients; association is measured with R Squared
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41. Regression Analysis
For a regression to have any significance, we must have ANOVA’s Sig. F < 0.050
Then each variable’s coefficient (b0, b1, etc.) must have significance < 0.050
Otherwise the coefficient might be zero
Then the better regression equations are ranked in order of strength by R Square, which is confirmed visually by plotting
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42. Regression Analysis
The standard error of coefficients is given, so confidence intervals can be formed
Also helps report them meaningfully, so you don’t report a value as if it has a standard error of 0.92
Depending on the accuracy of the source data, you could report that result as 5 +/- 1, or 4.9 +/- 0.9, or /- 0.92
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43. Crosstabs
Crosstabs display data sorted by two or more variables in table form
Often just counts of each category, and/or the percentage of counts
Recoding data allows interval or ratio scale data to be put into groups (e.g. age 18-25)
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44. Pearson’s Chi Square
Measures how well the actual (observed) data differs from a even (expected) distribution of data
The “expected” data can be a random distribution (same number of counts per cell), or adjusted for the actual total counts for each row and column
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45. Pearson’s Chi Square Evaluation
When chi square is larger than the critical value, reject the null hypothesis
Or if the significance of chi square is < 0.050, reject the null hypothesis
Can also generate Chi square for a single variable
Beware that Chi square is less meaningful for large matrices
Or, it’s too easy for large matrices to show significance falsely using Chi square
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46. Residuals
A residual is the difference between the Observed and Estimated values for a cell
Residuals can be plotted to look for outliers
Residuals can be standardized by dividing by their standard deviation
Cells with a standardized residual magnitude > 2 contribute a lot to Chi square
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47. Measures of Association
Measures of Association between two variables can be symmetric or directional
Dozens of measures have been developed to work with chi square test
Interpret them like ‘r’ - zero means no correlation, larger values mean a stronger correlation
Some can be > 1
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48. Measures of Association
Symmetric measures don’t care which variable is dependent (Y)
Directional measures DO care which variable is dependent (A = f(B) is not B = f(A))
Some directional measures have a “symmetric” value, the weighted average of the other two
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49. Symmetric Measures
The “Contingency Coefficient” is the main symmetric measure with a Chi Square test
Works even with nominal data
Evaluated like Pearson’s r
Phi and Cramer’s V are other symmetric measures
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50. Directional Measures Directional measures range from 0 to 1
Lambda is the recommended directional measure - tells what proportion of the dependent variable is predicted by the independent variable (like Eta)
Eta can be applied here if one variable is interval or ratio scaled
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51. Relative Risk and Odds Ratio
Use only with 2x2 tables
Are quite directional
Tells how much more likely one cell is to occur than the others
Need to be very careful when interpreting
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52. Square Tables
Tables with the same number of rows and columns (RxR), and the same variables in those rows and columns, can use kappa
Measures strength of association, like ‘r’
Check results for significance (<0.050)
Then judge the value of kappa using a fixed scale
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53. General RxC Measures
Many measures can be used with a general table of R rows and C columns
Gamma is the recommended measure (symmetric)
Spearman’s Correlation Coefficient is also widely used
Ranges from -1 to +1, based on ordered categories
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54. Yule’s Q Yule’s Q is a special case of gamma for a 2x2 table
Is judged on a fixed scale, like ‘r’
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