Practical Statistics for Educators - User contributions [en]

File:Winter .jpg

2019-12-03T20:50:43Z

Kilbourn004:

Pearson r

2019-12-03T20:50:00Z

Kilbourn004: /* A "real life example" of using correlations to gauge winter weather */

Also known as Pearson's product-moment correlation. This technique is used to correlate the raw scores of two variables.

Also visit http://psych.csufresno.edu/psy144/Content/Statistics/relationship_strength.html for more information on Pearson r.

''contributed by Kara Kunst''

Also referred to as the Pearson Correlation Coefficient Squared, it is the proportion of variance in the criterion variable that can be accounted for by the predictor variable. (from Dr. Nancy Heilbronner)

''contributed by Mary Fernand''

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Note: Pearson r scores cannot exceed 1.00 or -1.00 (range is between -1.00 and 1.00).

The Pearson r score (say for example .80) is the number where the distribution will peak, and the remaining distribution will spread out around the number.

''contributed by Mykal Kuslis, WCSU Cohort 8''

Reference:

Meyers, S., Gamst, G., & Guarino, A.J. (2017). Applied multivariate research: Design and interpretation. Thousand Oaks, CA: Sage Publications. (P. 21)

==A "real life example" of using correlations to gauge winter weather==

Every teacher in New England has a vested interest in understanding how winter weather may impact the school calendar. I recently heard an interview with Judah Cohen on NPR, and sourced an older article from the Washington Post which includes a graph, which illuminates how this meteorologist who works for the firm, Atmospheric and Environmental Research, uses correlations to forecast East Coast weather. Specifically, Cohen evaluates the Siberian snow cover in October to predict winter weather in New England (Samenow, 2013).

Because we’ve learned about correlational statistics, specifically what’s implied by the correlation coefficient or r-value, we can look beyond the narrative offered in the Washington Post article, which describes the statistical correlation as “striking.” In fact, we can look at the r =.810 in the graph below, and determine that because this number is close to 1, the Snow Advance Index (which relates to the Siberian snow cover) and the Arctic Oscillation (which produces the winter weather patterns in the Northeast) are strongly positively correlated (Hinkle, Wiersma, & Jurs, 2003, pp.98-99).

[[File:winter.jpg]]

Given the strong positive correlation, teachers in New England might pay a little more attention to what’s happening in Siberia in October to determine how much hot chocolate to buy in advance of snow days and how far those snow days will cause us to overshoot our districts’ June calendars.

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

Samenow, J. (2013). Judah Cohen’s winter outlook: A downer for East Coast winter weather lovers. The Washington Post. Retrieved from:

Pearson r

2019-12-03T19:19:25Z

Kilbourn004:

Also known as Pearson's product-moment correlation. This technique is used to correlate the raw scores of two variables.

Also visit http://psych.csufresno.edu/psy144/Content/Statistics/relationship_strength.html for more information on Pearson r.

''contributed by Kara Kunst''

Also referred to as the Pearson Correlation Coefficient Squared, it is the proportion of variance in the criterion variable that can be accounted for by the predictor variable. (from Dr. Nancy Heilbronner)

''contributed by Mary Fernand''

----
Note: Pearson r scores cannot exceed 1.00 or -1.00 (range is between -1.00 and 1.00).

The Pearson r score (say for example .80) is the number where the distribution will peak, and the remaining distribution will spread out around the number.

''contributed by Mykal Kuslis, WCSU Cohort 8''

Reference:

Meyers, S., Gamst, G., & Guarino, A.J. (2017). Applied multivariate research: Design and interpretation. Thousand Oaks, CA: Sage Publications. (P. 21)

==A "real life example" of using correlations to gauge winter weather==

Every teacher in New England has a vested interest in understanding how winter weather may impact the school calendar. I recently heard an interview with Judah Cohen on NPR, and sourced an older article from the Washington Post which includes a graph, which illuminates how this meteorologist who works for the firm, Atmospheric and Environmental Research, uses correlations to forecast East Coast weather. Specifically, Cohen evaluates the Siberian snow cover in October to predict winter weather in New England (Samenow, 2013).

Because we’ve learned about correlational statistics, specifically what’s implied by the correlation coefficient or r-value, we can look beyond the narrative offered in the Washington Post article, which describes the statistical correlation as “striking.” In fact, we can look at the r =.810 in the graph below, and determine that because this number is close to 1, the Snow Advance Index (which relates to the Siberian snow cover) and the Arctic Oscillation (which produces the winter weather patterns in the Northeast) are strongly positively correlated (Hinkle, Wiersma, & Jurs, 2003, pp.98-99).

[Note: I would like to include the graph/image]

Given the strong positive correlation, teachers in New England might pay a little more attention to what’s happening in Siberia in October to determine how much hot chocolate to buy in advance of snow days and how far those snow days will cause us to overshoot our districts’ June calendars.

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

Samenow, J. (2013). Judah Cohen’s winter outlook: A downer for East Coast winter weather lovers. The Washington Post. Retrieved from:

Z-scores

2019-12-03T19:15:17Z

Kilbourn004:

'''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 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''

Type I and Type II Errors

2019-12-03T19:04:24Z

Kilbourn004:

'''Type One Error:''' An incorrect rejection of the null hypothesis. For example, the researcher falsely states that there is a statistically significant difference between the control group and the experimental group based on their intervention program. This can also apply to correlational tests.

'''Type Two Error:''' An incorrect acceptance of the null hypothesis. For example, the researcher does not report a significance between the control and experimental group based on an intervention when, in fact, there is. In other words, an effect truly exists, but the research reports that there is none.

Contribution by: Tom Fox, WCSU Cohort 8

Lawrence, S., Meyer, G, & Guarino, A.J. (2017). ''Applied multivariate research: Design and interpretation''. Thousand Oaks, CA: Sage Publications

==A cross-disciplinary comparison of Type I and Type II errors and setting significance levels==

I found the following comparison illustrative both of the definition of Type I and Type II errors in practical application, and for understanding that there are different impacts of manipulating significance levels – as the tests we’ve run in ED826 have mostly involved setting significance levels at .05 – that have discipline-specific consequences.

Significance is under the direct control of the researcher, and is established relative to the consequences of making a Type I error. While conventionally, significance is set at .05 or .01, lower significance levels are set for example, in medical research where the implications of committing a Type I error are potentially grave. Therefore, if the potential harm to patients if a drug-in-testing is said to be more effective than an existing drug when in fact it is not (rejecting the null hypothesis when it is true), significance levels in medical research are set very low (.001) to minimize the potential of committing a Type I error (Hinkle, Wiersma, & Jurs, 2003, p. 300).

By comparison, the example given in the Hinkle, Wiersma, & Jurs (2003) text is an education researcher who is investigating two different programs types and their respective effects on student achievement. If the cost of the program and impact on teacher time are the same/similar, the consequence of making a Type I error (implementing a program that is not better than the other) is minimal. By comparison, behavioral science researchers can use higher significance levels (.10) to avoid making Type II errors (e.g. not implementing a superior program) (p.300).

Thus, setting the “right” significance level is really a matter of mitigating risk and in some fields – like medicine -- that risk is greater than in others.

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"

MANOVA

2019-12-03T19:01:40Z

Kilbourn004:

Multivariate variate analysis of variance (MANOVA) is the statistical procedure of comparing the means of several groups rather than a single group as you would find in an ANOVA. It is appropriate to use a MANOVA if the IV has 2+ levels and there are 2+ DV. Assumptions for use of MANOVA include: normal distribution. linearity, homogeneity of variances and homogeneity of variances and covariances. [http://userwww.sfsu.edu/efc/classes/biol710/manova/MANOVAnewest.pdf].

''contributed by Raymond Manka''

For example, we may conduct a study where we look at math achievement based on standardized test scores and ELA acheivement based on standardized test scores between students in different socioeconomic groups (low, moderate, high). The two dependent variables would be math achievement vased on standardized test scores and ELA achievement based on standardized test scores. The independent variable is the socioeconomic status with three levels: low, moderate, and high. (Based on a case scenario provided by Dr. Frank Labanca)

Instead of a univariate ''F'' value, we would use a multivariate ''F'' value Wilk's λ.

''contributed by Mary Fernand''

Prior calculating a MANOVA (or other statistics) in SPSS, it may be necessary to "clean the data" in order to obtain a good sample of numbers.
Here is a video that discusses how to clean the data in SPSS. While it may be easier to do this prior to importing to SPSS, this video to be a bit long, but helpful in understanding the process.
https://www.youtube.com/watch?v=Ik4Dyn8e8vA

''contributed by Sheri Prendergast''

==Why MANOVAs are a good test for dissertations==
Using Dr. Labanca’s notes as a guide, a rationale for using a MANOVA as the statistical test for our dissertations includes the following: 1. A MANOVA is a test that allows us to measure multiple dependent variables together, which is likely given the scope and scale of the quantitative data collection for dissertations; 2. Passing the Box’s M test for significance .05 (Meyers) or .01 (Huberty & Olenjnik), mitigates risk that we’ve committed a Type I error (Labanca, 2019, slides 5 & 12).

References:
Labanca, F. (2019). Multivariate Analysis of Variance (MANOVA) [PowerPoint slides]. Retrieved from http://moodle.labanca.net/course/view.php?id=4.

Analysis of Variance

2019-12-03T18:54:04Z

Kilbourn004:

While the t-test is limited to the comparison of two means, the ANOVA is an inferential statistic that can be used to compare two or more 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"