Practical Statistics for Educators - User contributions [en]

Central Tendency

2019-11-07T21:26:55Z

Sheri.prendergast:

Central Tendency is commonly referred to as the "measure of central tendency". A measure of central tendency is used to describe a data set by identifying the central position within that set of data. In statistics, the three most commonly used measures of central tendency are mean, median, and mode.

Mean: The mean is the average of all the numbers within a data set. To find the mean value, one would add up all the values in a data set and divide that sum by the total number of data points within the data set.
Example - 3+4+5+6+7 = 25. There are 5 values in this data set. 25/5 = 5. In this scenario the mean, or average, is 5.

Median: The median is the middle point in a sorted set of data. The median is identified by organizing the data set into order of magnitude (starting with the smallest number).
Once sorted the median is identified as the number directly in the middle of that sorted data.
Example - 6, 9, 23, 15, 2. If we put this data set in order by magnitude it is displayed as: 2, 6, 9, 15, 23. In this data set 9 is the median.

Mode: The mode is the number that occurs most frequently in a data set.
Example - 2, 3, 15, 3, 5, 7, 8, 3, 2, 1, 10, 9.
In this data set, 3, is the number that occurs most frequently and would be identified as the mode.

''contributed by Scott Trungadi''

''Having finally mastered the skills necessary to report my data, and thanks to the help from fellow students in my doctoral program, I was ready to write up a description of what all the numbers meant. I was excited to have reached this point in my central tendency assignment, as there is one thing I love doing, and that is write. Finally, something I might be good at! However, this also meant that I needed to understand and be able to explain what all the numbers meant.''

In October of 2007 during the kindergarten year, the Letter Naming Fluency
( LNF) portion of the Dynamic Indicators of Basic Early Literacy Skills (Dibels) was administered to a class of 18 kindergarten students, 10 males and 8 females.
The SPSS helped to process the following data regarding the testing administration. The mean score for the eighteen students for the Beginning LNF portion of Dibels was 32.33 with the median or midpoint being 35.0(Table 1). By gender, the boys in the class scored slightly lower with the mean score of 32.30 and the midpoint or median score of 34.50 as compared to the girls’ mean score of 32.38 and a median of 37.00(Table 5).

The standard deviation is based on all the scores in the group and is determined by how much each score deviates from the mean, or in other words, it is an estimate of what the range of scores probably was. The standard deviation tells me that in the Beginning and Ending LNF administration, all students tested, in a similar range-16.01 and 16.87(Tables 1 and 2).
However, in looking at the Beginning LNF scores analyzed by gender, there is a large discrepancy between how well the boys did when compared to the girls. There was a higher standard of deviation for the boys than the girls, respectively 16.34 for the boys and 6.71 for the girls. In trying to understand the possible reasons for this, one must consider birthdates. Although birthdates were not considered in this collection of data, it is important to note that there was a higher incidence of younger birth dates for the boys than the girls which might account for this wide range in the boys’ scores.

The Z scores in this statistical analysis refer to how many standard deviations a particular raw score lies above or below the group means. Table 6 indicates the range of Z scores for the students who had taken the Ending LNF portion of the Dibels test. The score range from 1.44, or 1.44 standard deviations above the group mean of 64.67 to -1.82, or 1.82 below the group mean of 64.67.

''I actually felt that I began to really understand what everything meant as I was scripting my report. It was very helpful. Hope this helps someone!''

''contributed by Debbie Mumford''

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Skewness and Central Tendency
[[File:Skeweness.jpg]]

Identifying where the mean, median, and mode are in relation to each other can help to determine the skew of a curve.

In a normal distribution, the mean, median, and mode will all be very close (mean = median = mode)

In a distribution that is skewed ''left'', otherwise known as a negatively skewed distribution, it is outliers on the lower end of the number line that is impacting the shape of the curve. In a curve that is skewed left the mean (which is the mathematical average) will be furthest to the left on the number line, the median remains at the mid point of the distribution on the number line, and the mode will be the farthest point to the right.

In a distribution that is skewed ''right'', otherwise known as a positively skewed distribution, it is outliers on the higher end of the number line that is impacting the shape of the curve. In a curve that is skewed right the mean (which is the mathematical average) will be furthest to the right on the number line, the median remains at the mid point of the distribution on the number line, and the mode will be the farthest point to the left.

''contributed by Ashley Brooksbank''

Image citation: Statistics: Basic Statistics II. (n.d.). Retrieved from ''https://guides.douglascollege.ca/c.php?g=408742&p=2970198.'''

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Descriptive Statistics

Descriptive statistics are very important because if we simply presented our raw data it would be hard to visualize what the data was showing, especially if there was a lot of it. Descriptive statistics therefore enables us to present the data in a more meaningful way, which allows simpler interpretation of the data.

Measures of central tendency: these are ways of describing the central position of a frequency distribution for a group of data. In this case, the frequency distribution is simply the distribution and pattern of marks scored by the 100 students from the lowest to the highest.

Measures of spread: these are ways of summarizing a group of data by describing how spread out the scores are. For example, the mean score of our 100 students may be 65 out of 100. However, not all students will have scored 65 marks. Rather, their scores will be spread out. Some will be lower and others higher. Measures of spread help us to summarize how spread out these scores are.

Both of these measures are important to calculate through SPSS prior to conducting other analyses as they can speak to you about what your data is saying and inform you of the next steps you should take.

"Contributed by Sheri Prendergast"

Contributions here

2019-11-07T21:15:46Z

Sheri.prendergast: /* Student Contributors */

== Editor ==
Frank LaBanca, EdD

== Faculty Contributors ==
Karen Burke, EdD

Patricia Cosentino, EdD

Deborah Hardy, EdD

Jennifer Mitchell, EdD

== Student Contributors ==
David Bozzuto

Karen Fildes

Michael Minzloff

Damien Holst

Jennifer Eraca

John Ryan

Kara Kunst

Emily Rhew

Cassandra Cosentino

Kristina Hislop

Mary Fernand

Thomas Fox

Helen Knudsen

Ashley Brooksbank

Scott Trungadi

Sheri Prendergast

Multiple Linear Regression

2019-11-07T21:15:04Z

Sheri.prendergast:

Multiple linear regression (multiple regression) is a type of correlational test in which the research is interested in finding the strength of a correlation between multiple variables. In multiple linear regression, multiple variables are used as ''predictors. '''Here, the researcher is interested in the relationship between the predicted variables (dependent) and predictor variables (also known as the independent variables).
''

Independent variables in multiple regression are usually quantitatively measured variables using summative response, interval, or ratio scales (Lawrence, Meyer, & Guarino, 2017)

Multiple Linear Regression uses the same general equation as linear regression, but accommodates for multiple IV's.

Contribution by: Thomas Fox, WCSU Cohort 8

Reference

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

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Collinearity

When conducting a multiple linear regressions, you to see if the data meets the assumption of collinearity. Therefore, you need to locate the Coefficients table in your results under the heading Collinearity Statistics, under which are two subheadings, Tolerance and VIF.

If the VIF value is greater than 10, or the Tolerance is less than 0.1, then you have concerns over multicollinearity. Otherwise, your data has met the assumption of collinearity and can be written up something like this:

Contribution by: Sheri Prendergast, WCSU Cohort 8

Dart, A., (2013). Reporting Multiple Regressions in APA format-Part One. Retrieved from: http://www.adart.myzen.co.uk/reporting-multiple-regressions-in-apa-format-part-one/

MANOVA

2019-10-26T11:10:03Z

Sheri.prendergast:

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"

Homogeneity vs Homoscedacity

2019-10-26T10:56:03Z

Sheri.prendergast:

Greek roots help us better understand hard to pronounce statistical terms such as '''Homogeneity''' and '''Homoscedacity.''' Simply stated "homo" means same and "scedacity or skedastikós" means to scatter and "geneity" means kind or stock. Homoscedacity compares the means of multivariate (two or more groups) for equal levels of variability, while homogeneity is used for the same value of the mean of univariate analyses.

==Homogeneity of Variance==
A requirement for the ANOVA test is that the variances of each comparison group are equal and you use the Levene’s Test to determine this. What you’re looking for here is a significance value that is greater than .05. You don’t want a significant result, since a significant result would suggest a real difference between variances.
For example, the significance value of the Levene statistic is .155. This is not a significant result, which means the requirement of homogeneity of variance has been met, and the ANOVA test can be considered to be robust.

"Contributed by Sheri Prendergast"

Analysis of Variance

2019-10-26T10:51:14Z

Sheri.prendergast:

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"