Practical Statistics for Educators - User contributions [en]

Analysis of Variance

2022-05-05T18:05:08Z

Jblueteach1:

ANOVA is a statistical method for making simultaneous comparisons between two or more means. This statistical method can yield values that can be tested to determine whether a significant relation exists between variables. ''contributed by Jennifer Blue''

While the t-test is limited to the comparison of two means,The ANOVA produces an F statistic that is a ratio of the variability of the means being compared to the variability of the observations within each set of data on which the means are based. The ANOVA can be used to compare two or more means (in theory the number of means is limitless) taken from different groups (resulting from a between-subjects design) or to compare means taken from the same group of subjects under varying conditions (resulting from a within-subjects or repeated-measures design) – the repeated measures ANOVA. The one-way ANOVA examines the effect of a single independent variable on a dependent variable. The two-way ANOVA examines the effects of two separate independent variables, as well as their interaction, on a dependent variable.

Consider an experimental investigation in which the effects of work environment on productivity are examined using a between-subjects design. A large corporation randomly assigns its workers to one of three different work environments; single closed office, single open cubby where the worker works alone but can see and hear other workers, or shared open cubby where the worker shares the work space with another worker and can see and hear other workers. They then measure each worker’s productivity on a standardized productivity schedule. The independent variable is the type of work environment and it has three levels rather than two as in the previous studying example. The dependent variable is productivity, and the null hypothesis is that work environment has no effect on productivity or that the productivity of all workers will be the same regardless of work environment. These data would be most appropriately analyzed with an ANOVA.

== Identify the independent variable, dependent variable, and the null hypothesis from the following scenario: ==

A researcher would like to know if highlighting a textbook helps students to score better on exams. She randomly selects one-half of the students in an introductory class and instructs them to highlight their textbooks as they read. The other students are instructed to do NO highlighting as they read.

If the experimental manipulation has no effect, the experimental and control groups in the over-learning study would not differ significantly in their performance on the exam and the workers in the different work environments would all be equally productive. In those cases, we would fail to reject the null hypothesis. If, in the over-learning study, the experimental manipulation has an effect, the two groups would differ significantly in their performance on the exam. In that case, we would reject the null hypothesis. This would indirectly support the research hypothesis, which would predict that over-learning affects exam performance. But how large must a difference be between groups for it to be significant? How much more productive must one group of workers be than another for us to conclude that work environment affects productivity? To determine whether the difference between groups is large enough to minimize chance variation as an alternative explanation of the results, we must determine the statistical significance of the difference between them.

== Critical Values ==

To determine whether a t statistic or F statistic has less than a .05 (or .01) probability of occurring by chance, the observed (i.e., calculated) statistic is compared to a critical value taken from a probability table. The exact critical value used for any one comparison depends on the level of significance chosen, the number of observations in each group, the number of groups being compared, and whether the researcher has a directional or non-directional hypothesis. Using information about the above factors a researcher obtains a critical value and compares the observed value to it. If the observed value fails to exceed the critical value, the null hypothesis is retained and the results of the study are said to be inconclusive. If the observed value is more extreme than the critical value, the null hypothesis is rejected, the results are said to be statistically significant, and the research hypothesis is said to have received support from the study.

Note that statistical significance is a statement of probability. We can never be certain that what is true of our samples is also true of the populations they represent. This is one of the reasons why all scientific findings are tentative. Moreover, statistical significance does not indicate practical significance. A statistically significant effect may be too small or be produced at too great a cost of time or money to be useful. What if those who practice over-learning must study two extra hours each day to improve their exam performance by a statistically significant, yet relatively small, 3 points? Knowing this, students might choose to spend their time in another way. As the American statesman Henry Clay (1777-1852) noted, in determining the importance of research findings, by themselves "statistics are no substitute for judgment."

== Power ==

The difference between the means of groups will more likely be statistically significant under the following conditions:

1. When the samples are large.

2. When the difference between the means is large.

3. When the variability within the groups is small.

These are all factors involved in the power of a study. Power is the probability of your experiment allowing you to detect an effect that really exists in the world. The difference between the means of your groups is a measure of effect size – how big of an impact your independent variable has on your dependent variable. Larger samples are apt to be more representative of the population in question and, as sample size increases within groups variance typically decreases. Since one rarely has precise control over the difference between means, a good method for improving power is to increase the number of participants in a study.

''contributed by Karen Burke, EdD''

==ANOVA: Significance and Eta Squared==

When looking at the significance column when analyzing the results from an ANOVA in SPSS, look at the column labeled "Sig." This column indicates the level of significance (the likelihood the result is due to chance) the lower the significance, the less likely the differences between the groups are due to chance and the more likely they are due to the independent variable. For example, a significance level or probability of less than .01 means there's a fewer than 1 possibility in 100 that the results are due to chance.

The next column is labeled "Eta Squared," which is the measure of effect size. It is the percentage of the dependent variable explained by the independent variable. The higher the percentage (the closer to 1), the more important the effect of the independent variable. For example, an Eta Squared of .75 means that 75% of the independent variable is explained by the independent variable.

''contributed by Sheri Prendergast''

==Why an ANOVA instead of multiple t-tests==

An ANOVA is a comparison of means. A t-test can also be used to compare means, but for each T-test that’s run, the Type I error rate increases. If there are five groups being studied, we could run ten separate t-tests to compare all possible pairs of means…but if significance is set at .05 for each test, the Type I error rate is computed as follows: 1- (1-.05)10 = .40. This means that the probability of making at least one Type I error across the comparison of means for five groups is .40 (Hinkle, Wiersma, & Jurs, 2003, pp. 331-332).

Hinkle, D.E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral sciences (5th edition). Boston, M.A.: Houghton Mifflin Company.

''contributed by Emily Kilbourn''

Analysis of Variance

2022-05-05T18:04:42Z

Jblueteach1:

ANOVA is a statistical method for making simultaneous comparisons between two or more means. This statistical method can yield values that can be tested to determine whether a significant relation exists between variables. ''contributed by Jennifer Blue''
While the t-test is limited to the comparison of two means,The ANOVA produces an F statistic that is a ratio of the variability of the means being compared to the variability of the observations within each set of data on which the means are based. The ANOVA can be used to compare two or more means (in theory the number of means is limitless) taken from different groups (resulting from a between-subjects design) or to compare means taken from the same group of subjects under varying conditions (resulting from a within-subjects or repeated-measures design) – the repeated measures ANOVA. The one-way ANOVA examines the effect of a single independent variable on a dependent variable. The two-way ANOVA examines the effects of two separate independent variables, as well as their interaction, on a dependent variable.

Consider an experimental investigation in which the effects of work environment on productivity are examined using a between-subjects design. A large corporation randomly assigns its workers to one of three different work environments; single closed office, single open cubby where the worker works alone but can see and hear other workers, or shared open cubby where the worker shares the work space with another worker and can see and hear other workers. They then measure each worker’s productivity on a standardized productivity schedule. The independent variable is the type of work environment and it has three levels rather than two as in the previous studying example. The dependent variable is productivity, and the null hypothesis is that work environment has no effect on productivity or that the productivity of all workers will be the same regardless of work environment. These data would be most appropriately analyzed with an ANOVA.

== Identify the independent variable, dependent variable, and the null hypothesis from the following scenario: ==

A researcher would like to know if highlighting a textbook helps students to score better on exams. She randomly selects one-half of the students in an introductory class and instructs them to highlight their textbooks as they read. The other students are instructed to do NO highlighting as they read.

If the experimental manipulation has no effect, the experimental and control groups in the over-learning study would not differ significantly in their performance on the exam and the workers in the different work environments would all be equally productive. In those cases, we would fail to reject the null hypothesis. If, in the over-learning study, the experimental manipulation has an effect, the two groups would differ significantly in their performance on the exam. In that case, we would reject the null hypothesis. This would indirectly support the research hypothesis, which would predict that over-learning affects exam performance. But how large must a difference be between groups for it to be significant? How much more productive must one group of workers be than another for us to conclude that work environment affects productivity? To determine whether the difference between groups is large enough to minimize chance variation as an alternative explanation of the results, we must determine the statistical significance of the difference between them.

== Critical Values ==

To determine whether a t statistic or F statistic has less than a .05 (or .01) probability of occurring by chance, the observed (i.e., calculated) statistic is compared to a critical value taken from a probability table. The exact critical value used for any one comparison depends on the level of significance chosen, the number of observations in each group, the number of groups being compared, and whether the researcher has a directional or non-directional hypothesis. Using information about the above factors a researcher obtains a critical value and compares the observed value to it. If the observed value fails to exceed the critical value, the null hypothesis is retained and the results of the study are said to be inconclusive. If the observed value is more extreme than the critical value, the null hypothesis is rejected, the results are said to be statistically significant, and the research hypothesis is said to have received support from the study.

Note that statistical significance is a statement of probability. We can never be certain that what is true of our samples is also true of the populations they represent. This is one of the reasons why all scientific findings are tentative. Moreover, statistical significance does not indicate practical significance. A statistically significant effect may be too small or be produced at too great a cost of time or money to be useful. What if those who practice over-learning must study two extra hours each day to improve their exam performance by a statistically significant, yet relatively small, 3 points? Knowing this, students might choose to spend their time in another way. As the American statesman Henry Clay (1777-1852) noted, in determining the importance of research findings, by themselves "statistics are no substitute for judgment."

== Power ==

The difference between the means of groups will more likely be statistically significant under the following conditions:

1. When the samples are large.

2. When the difference between the means is large.

3. When the variability within the groups is small.

These are all factors involved in the power of a study. Power is the probability of your experiment allowing you to detect an effect that really exists in the world. The difference between the means of your groups is a measure of effect size – how big of an impact your independent variable has on your dependent variable. Larger samples are apt to be more representative of the population in question and, as sample size increases within groups variance typically decreases. Since one rarely has precise control over the difference between means, a good method for improving power is to increase the number of participants in a study.

''contributed by Karen Burke, EdD''

==ANOVA: Significance and Eta Squared==

When looking at the significance column when analyzing the results from an ANOVA in SPSS, look at the column labeled "Sig." This column indicates the level of significance (the likelihood the result is due to chance) the lower the significance, the less likely the differences between the groups are due to chance and the more likely they are due to the independent variable. For example, a significance level or probability of less than .01 means there's a fewer than 1 possibility in 100 that the results are due to chance.

The next column is labeled "Eta Squared," which is the measure of effect size. It is the percentage of the dependent variable explained by the independent variable. The higher the percentage (the closer to 1), the more important the effect of the independent variable. For example, an Eta Squared of .75 means that 75% of the independent variable is explained by the independent variable.

''contributed by Sheri Prendergast''

==Why an ANOVA instead of multiple t-tests==

An ANOVA is a comparison of means. A t-test can also be used to compare means, but for each T-test that’s run, the Type I error rate increases. If there are five groups being studied, we could run ten separate t-tests to compare all possible pairs of means…but if significance is set at .05 for each test, the Type I error rate is computed as follows: 1- (1-.05)10 = .40. This means that the probability of making at least one Type I error across the comparison of means for five groups is .40 (Hinkle, Wiersma, & Jurs, 2003, pp. 331-332).

Hinkle, D.E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral sciences (5th edition). Boston, M.A.: Houghton Mifflin Company.

''contributed by Emily Kilbourn''

Inferential Statistics Definition

2022-05-05T17:44:30Z

Jblueteach1:

'''Inferential statistics''' is defined as the branch of statistics that is used to make inferences about the characteristics of a populations based on sample data.

• The goal is to go beyond the data at hand and make inferences about population parameters.

• In order to use inferential statistics, it is assumed that either random selection or random assignment was carried out (i.e., some form of randomization must is assumed).

''contributed by Karen Burke, EdD''

Descriptive statistics describes data and inferential statistics allows one to make predictions about that data. In inferential statistics take data from samples and make generalizations about it. ''contributed by Jennifer Blue''

== Parametric versus non-parametric tests ==

Many statistical tests are based upon the assumption that the data are sampled from a normal distribution. These tests are referred to as parametric tests. Parametric tests are generally used on interval or ratio data.

Tests that do not make assumptions about the population distribution are referred to as nonparametric tests. Non-parametric tests are used when data ranks the outcome variable from low to high (ordinal level data) or if data groups are nominal.

''contributed by Frank LaBanca, EdD''

[[Image:Chosingstat.gif]]

''Example:''

Inferential statistics is data based upon a sample of a specified population. The objective of inferential statistics is to gain insights about a population. For example, if the researcher wanted to know if more customers would go to the Shelton Starbuck's on a double reward points day (usually Wednesday or Thursday) in contrast to a non-double rewards days between the hours of 6 A.M. - 8 A.M. The researcher would tally the number of patrons on the observed days and a one sample T-test to record and analyze the data. From this, the researcher will learn if double reward days increases business.

''contributed by Héctor Huertas''

Z-scores

2022-04-07T16:37:01Z

Jblueteach1:

'''A Brief Explanation of Z-Scores:'''
A z-score is a standard score that is used by researchers to add focus and clarity to data. Z-scores indicate how many standard deviations a raw score is from the mean. The mean is fixed at zero and standard deviations are fixed at 1. For example, suppose the mean test score for a sample is 80 with a standard deviation of 12 and you scored a 98 on that test. Your z-score is +1.5, indicating that you scored 1.5 standard deviations above the mean. If a z-score is close to zero the corresponding raw score is close to the mean for the test. If a z-score is -2 the corresponding raw score is 2 standard deviations below the mean.

''contributed by Helen Knudsen''

==Impact of converting raw scores to z-scores==

When each raw score is converted to a z score:

1. The distribution of standard scores is similar in shape to the distribution of raw scores
2. The mean of the distribution of z scores will always equal 0, regardless of the value of the mean in the raw score distribution.
3. Both the variance of the distribution and the standard deviation of z scores always equals 1.

[I would like to insert an image here and am unsure how]

It is helpful to see what this looks like in a side-by-side distribution of raw and standard scores. Notice that the mean raw scores are 6.0; whereas the standard score is set at a mean of zero. And whereas the standard deviation from the raw scores was 3.18, when converted to standard scores, the standard deviation is 1.00 (Hinkle, Wiersma, & Jurs, 2003, p. 72).

In sum, calculating a z score for each raw score in a distribution will transform the original distribution of scores into one with identical shape but a mean of 0 and a standard deviation of 1 (Hinkle, Wiersma, & Jurs, 2003, p. 71).

References:
Hinkle, D.E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral sciences (5th edition). Boston, M.A.: Houghton Mifflin Company.

''contributed by Emily Kilbourn''

== Calculating Z-scores using SPSS ==

''I, among others were having a hard time calculating Z-scores using the SPSS program. Amy, Michelle and I brainstormed last week, but had no luck. The book is vague in terms of how to approach it. Thanks for the guidance, Frank.''

'''<big>When calculating Z-scores on SPSS, follow these directions:</big>'''

1) Once you have the data entered in SPSS, click on "Analyze", "Descriptive Statistics", "Descriptives".

2) Move the variable over that you want to analyze.

3) Click on the small box that states, "Save standardized values as variables".

4) Click on "Options" if you would like to calculate mean, median, mode, etc. in addition to Z-scores.

5) Click "OK".

6) The Z-scores will appear in a separate column in the data editor.

''contributed by Chris Longo''

Rules of thumb for interpreting the size of a correlation coefficient

2022-04-07T16:22:35Z

Jblueteach1:

Correlation is simply the relationship between two variables (x & y), varying between -1 and +1. Correlation is a descriptive measure of a central tendency and does not necessarily indicate causal relationships. When looking at graphs, an upward trend indicates a positive correlation; a downward trend indicates a negative correlation; and a scattered or messy graph indicates no correlation at all.

''contributed by Jennifer Blue''

{| class="wikitable"
|-
! Size of positive correlation
! Size of negative correlation
! Interpretation
|-
| .90 to 1.00
| -.90 to -1.00
| Very high positive (negative) correlation
|-
| .70 to .90
| -.70 to -.90
| High positive (negative) correlation
|-
| .50 to .70
| -.50 to -.70
| Moderate positive (negative) correlation
|-
| .30 to .50
| -.30 to -.50
| Low positive (negative) correlation
|-
| .00 to .30
| .00 to -.30
| Little, if any, correlation
|}

''contributed by Frank LaBanca EdD''

Interpreting the line of best fit can show outliers. Outliers can lead to different interpretations of data, and an easy method for spotting outliers is through a scatterplot.

''contributed by Lauren Moyer''

Recall that this is the r value, not the p value, when interpreting the r value. The Pearson correlation does not show causation. For example, if the r value has a high positive correlation between a teacher shortage and the deterioration of the ozone layer, it does not necessarily mean that the teacher shortage caused the deterioration of the ozone layer.

''Contributed by Lisa Daigle''

Main Page

2022-04-07T15:41:52Z

Jblueteach1: /* Philosophy */

<big>'''Practical Statistics for Educators'''</big>
edited and maintained by Frank LaBanca, EdD

----

== Philosophy ==

Quantitative statistical analyses can be intimidating for many educators pursing an advanced academic degree. The thought of computational math can sometimes trigger unwarranted fears. Quantitative research in education and other fields of inquiry is expressed in numbers and measurements. This type of research aims to find data to confirm or test a hypothesis. Quantitative study requires extensive statistical analysis, which can be difficult to perform for researchers from non- statistical backgrounds. Statistical analysis is based on scientific discipline and hence difficult for non-mathematicians to perform. But once one begins to embark on understanding all of the representations, descriptions, and analyses of particular data sets, statistics becomes an educator’s friend not foe.

Here, we approach statistics from a straightforward conceptually-based perspective. Our goal is to collaborate and provide insight for statistics that make them meaningful tools in the educational arena.

Each "module" corresponds with the topics presented each week, and will expand as the course progresses. A topical outline can be found @ [http://docs.google.com/Doc?id=dfqvtcqp_46hhzzcsgt ]

Comments and edits are welcome and encouraged! Please give yourself credit as you contribute. At the end of a section you insert please add the following in italics:
''contributed by <your name>''
If you are modifying content, add the following under the contribution line:
''modified by <your name>'' We are glad to accept as many modifications as necessary to give the most meaning to each section. As we asynchronously socially construct knowledge together, we can recognize the accomplishments and contributions of each writer.

''contributed by Frank LaBanca, EdD''
''contributed and modified by Jennifer Blue''

== Contributions ==

Our contributors [[contributions here]].

Please submit your contribution at [https://forms.gle/PBaVrxFffbk5CKgg8]

== Modules ==

1.1 [[The Greek Alphabet]] and its significance in statistics

1.2 [[An introduction to probability]] PowerPoint @[http://docs.google.com/Presentation?id=dfqvtcqp_97wcrbtsn]

1.3 [[Some Probability Formulas]]

2.1 [[Types of Data]]

2.2 [[Visualizing Data]]

2.3 Visually representing data PowerPoint @ [http://docs.google.com/Presentation?docid=dfqvtcqp_27dwth2zz2#]

2.3.1 Table 2 from LaBanca dissertation @ [http://docs.google.com/Doc?id=dfqvtcqp_25cb5pqcfw]

2.3.2 Cool graph of movie box office from NY Times [http://www.nytimes.com/interactive/2008/02/23/movies/20080223_REVENUE_GRAPHIC.html#]

2.3.4 [[Histograms]]

2.3.5 Scatterplots YouTube @ [http://youtu.be/HFuU1uxJ1tQ]

2.4 [[Shapes of distribution]]

2.5 Survey of Attitudes Toward Statistics (SATS) Data Set @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7PokF_oAoCRUg]

2.6 [[Data Screening]]

3.1 [[Central Tendency]]

3.1.1 Central Tendency and Normal Distribution PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_47dfhfr4nw ]

3.1.2 Central Tendency YouTube @ [http://youtu.be/Fn4z8RDpwDY]

3.2 [[Interquartile ranges]]

3.2.1 [[The Box Plot]]

3.2.2 Interpreting a Box Plot - video [https://www.youtube.com/watch?v=b2C9I8HuCe4]

3.3 [[Standard deviation]]

3.3.1 [[Identifying percentile ranks and scores based on standard deviation]]

3.3.1.a [[Practice Identifying percentile ranks and scores based on standard deviation]]

3.4 [[z-scores]]

4.1 [[Percentile Rank]]
4.1.1 Areas under the standard normal curve for z values @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7NstEjJ40jJOQ]

4.1.2 z scores corresponding to divisions of the area under the normal curve @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7P6SVNiBdWbEg]

4.2 Conversion of data PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_98chfzg2tf]

4.2.1 Descriptive analysis of USRT data @ [http://docs.google.com/Doc?id=dfqvtcqp_61db6smpdm ]

4.3 [[Normal Curve Equivalent scores]]

4.3 [[Standard Error of Measurement]]

4.4 [[Confidence Intervals]]

4.4 z score machine @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7N8oLJZwmr3Zw]

5.1 [[Pearson r]]

5.2 [[Rules of thumb for interpreting the size of a correlation coefficient]]

5.3 Critical values for the correlation coefficient @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7MkuRpIgceTRQ]

5.4 [[Spearman rho]]

5.5 Correlation PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_134hsxg7td7]

5.6 [[Writing samples for correlations]]

6.1 [[Inferential Statistics Definition]]

6.2 [[Sampling]]

6.3 [[Sampling distributions]]

6.4 t test PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_153gr9f3hgd]

6.4.1 t -t test video [https://www.youtube.com/watch?v=N2dYGnZ70X0]

6.5 Sample data set @ [http://wolfweb.unr.edu/homepage/liu/stat/help/help.htm]

6.6 Critical values for t @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7OZzyZeHg9MIA]

6.7 Helpful Tutorial for Running a t-Test in Excel @ [https://www.rwu.edu/sites/default/files/downloads/fcas/mns/running_a_t-test_in_excel.pdf]

7.1 [[Effect size]]

7.1.1 Effect size calculator @ http://www.campbellcollaboration.org/resources/effect_size_input.php

7.1.2 [[Rules of thumb for interpreting effect sizes]]

7.2 Effect size PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_204f42f67dx ]

7.3 [[Hypothesis testing]]

7.4 Hypothesis testing PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_23295tm65dx]

7.5.1 Hypothesis testing template for a correlation @ [http://docs.google.com/Doc?id=dfqvtcqp_174ccchz4ds]

7.5.2 Hypothesis testing template for a t test @ [http://docs.google.com/Doc?id=dfqvtcqp_175hjcdsjff]

8.1 [[Type I and Type II Errors]]

8.2 Type I and Type II Errors PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_258wcq56rdv]

8.3 [[Levene's p versus the test statistic p]]

8.4 [[Analysis of Variance]]

8.5 ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_265ckd9j9dv]

8.6 [[ANOVA Case study]]

8.7 ANOVA video [https://www.youtube.com/watch?v=ITf4vHhyGpc]

8.8 Critical values for the F statistic @ [http://www.sussex.ac.uk/Users/grahamh/RM1web/F-ratio%20table%202005.pdf]

8.9 [[Rules of thumb for interpreting effect sizes of ANOVAs]]

9.1 Post Hoc test PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_284dbhghdc9]

9.2 [[Selecting a Post Hoc test]]

9.3 Hypothesis testing template for ANOVA @ [http://docs.google.com/Doc?id=dfqvtcqp_295ckngxdgj]

10.1 [[Chi square]]

10.1.1 Chi square video [https://www.youtube.com/watch?v=VskmMgXmkMQ]

10.2 [[Example for calculating chi square]]

10.3 Critical values for chi square @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7OhBVHQWoHTIQ]

10.4 [[Chi square analysis description/sample writing]]

10.5 Chi square PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_297dhg685g8]

11.1 [[Beyond the ANOVA]]

11.2 Beyond ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_307cmrtpxg3]

11.3 2-way ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_325hrt86ggt]

11.4 2-way ANOVA template @ [http://docs.google.com/Doc?id=dfqvtcqp_372599pr6w6]

11.5 1-way ANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1i0kIWgmXLCEIIIYCSqh2JS9Th3T_pyRC/view?usp=sharing]

11.6 2-way ANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1wX4xhQa7KGCd1Hey7VfX33Y1uT6QHdEu/view?usp=sharing]

12.1 [[MANOVA]]

12.2 [[Homogeneity vs Homoscedacity]] (Levene vs Box's M)

12.3 [[Post Hoc ANOVAs for MANOVA]] (univariate)

12.4 [[Post Hoc Discriminant Analysis]] (multivariate)

12.5 [[Covariates]]

12.6 [[MANCOVA]]

12.7 MANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1GiErYfmCdiNQlCps3anF4Bu3C_iYS7oD/view?usp=sharing]

12.8 MANCOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1TodMQy4vQ4eHStSIAevATOuJuFg9KUTD/view?usp=sharing]

13.1 [[Multiple Regression Analysis]]

13.1.1 [[Collinearity]]

13.2 [[Multiple Linear Regression]]

13.3 Reading the MLR Output: An annotated output [https://drive.google.com/file/d/141RNyYNnuDDTNvHYi4fE8EF8qWmal1aa/view?usp=sharing]

13.4 MLR Annotated SPSS Output @ [https://drive.google.com/file/d/1SsPL1YD4VYguxLtwqM_R7GFBD5RsG9H6/view?usp=sharing]

14.1 Internal Consistency Reliability[https://docs.google.com/document/d/18K16I8u9sbwpUhW9nIENF4x9wDXAy-4wFuBp6a0AljA/edit]

14.1.1 [[Internal Consistency Reliability]]

14.2 Cronbach's Alpha[https://docs.google.com/document/d/1_eyXOcFrBcDSctM27a9T2kUlx9D8TidV_YTHk-wvTu0/edit]

14.2.1 [[Cronbach's Alpha Values]]

14.2.2 Cronbach's Alpha in SPSS [https://www.youtube.com/watch?v=Kz8OdR6lV44]

== Applied Research Designs ==

15.1 [[Instrumentation]]

15.2 [[Limitations]]

15.3 [[Practice determining the stat]]

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [http://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]