Monday, March 29, 2021

Normal Distribution or Bell Curve

 


The bell curve is also known as the normal curve or normal distribution. The bell curve has mathematical properties that allow researchers to draw conclusions about where scores (or data) are located relative to other scores (or data).

Click hyperlinks for more details.

The three measures of central tendency (mode, median, mean) are at the same middle point in a normal curve. The numbers representing the middle of the bell curve divide the distribution in half.

On the x-axis in the normal distribution, the mean is at zero and there are standard deviation units above and below the mean. 

The height of the curve indicates the percentage of scores in that area. You can see that a large percentage of the scores are between 1 and -1 standard deviations. About 68% of scores fall between +1 and -1 standard deviations from the mean. 

Look at the illustration below to see that there are about 34% of the scores in falling one standard deviation above the mean and another 34% in one standard deviation below the mean. Thus, 34% + 34% = 68%.

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Example: Intelligence test scores (IQ) appear to be distributed like the normal distribution within each age group. The mean IQ is 100 and one standard deviation is 15 points on most IQ tests. Thus, 68% of people of a similar age have IQ scores between 85 and 115.

Knowing that 100 is the average or mean IQ then we know that half of people taking the test are below average intelligence (as measured by the test) and half are above average intelligence.

Learn more about test scores in Applied Statistics: Concepts for Counselors available at  AMAZON   or   GOOGLE

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 The ends of the normal distribution are called tails. Extreme scores are in the tails. Consider the low level of the the distribution at - 2.5 or +2.5 standard deviations. At these points, the curve almost touches the x-axis; however, it never the lines of the curve never quite touch the x-axis.

Only a small percentage of scores is beyond 2.5 standard deviations in either direction. Theoretically, the tails of the curve never touch the baseline. Only a small fraction of a percent of scores is beyond 3 standard deviations.

The picture below illustrates the percentage of scores (or data) within different areas of the curve. For example, on a normally distributed test, 34.1% of scores will fall between the mean and 1 standard deviation above the mean. Because the curve is symmetrical, the same percentage will be between the mean and 1 standard deviation below the mean.


Read more about distributions in Chapter 10 of

Applied Statistics Concepts for Counselors at AMAZON    or    GOOGLE








Related posts/ pages

A-Z list of statistics

Skewed distributions


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   and see my books on   AMAZON       or  GOOGLE STORE

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You can read many published articles at no charge:

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Correlation coefficient the Pearson r in statistics

 

The term correlation can refer to a statistic and a type of research. 

Understanding correlations is an important building block of many complex ideas in statistics and research methods. My focus in this post is on the common correlation statistic, also called the Pearson r.

The Pearson r is a statistical value that tells the strength and direction of the relationship between two normally distributed variables measured on an interval or ratio scale.

Researchers examine the two sets of values and calculate a summary statistic called a correlation coefficient. The longer name for a common correlation statistic is the Pearson Product Moment Correlation Coefficient but sometimes it is referred to as the Pearson r. The symbol for correlation is a lower case and italicized r.  In behavioural research, we normally round values to two decimal points. A moderately strong positive correlation example is r = .78.

      Sometimes, the relationship between the two variables is negative. For example, the relationship between depression and self-esteem is often negative. As depression increases, self-esteem decreases. An example of a negative correlation would be written as r = -.45. The minus sign tells us that as one variable increases, the other variable decreases. The relationship is commonly described in journal articles as an inverse relationship.

An example from published research is the relationship between perceived stress and humility couples experience as they transition to parenthood. As a part of their work, Jennifer Ripley and her research team (2016) found that the correlation between a measure of perceived stress and a measure of humility ranged from -.33 to -.45, which indicates that high stress is associated with low humility.

The relationship between two variables not only varies in a positive or negative direction but it also varies in terms of strength. Large r values indicate a stronger relationship. When r = .75 or -.75, the relationship is of equal strength but in different directions. Relationships with a low number such as r = .15 or r = -.11 indicate weak relationships.

      When r values are at or near zero, we say there is no relationship between the variables. For example, we may find no relationship between scores on questionnaires about humility and depression.

Correlation is not causation

The fact that two variables have a strong relationship does not mean one variable causes the other.

Read more about correlations in Chapter 13 of 

Applied Statistics Concepts for Counselors on AMAZON or GOOGLE








Graphing the Correlations

This is an example of fictitious data illustrating a positive correlation between anxiety and depression. Anxiety and depression are different states but both may be present.



The following is an example of  fictitious data illustrating a negative correlation between self-esteem and depression. A high self-esteem score of 8 reflects low depression. Low self-esteem near 2 reflects a high level of depression at 7.


Applications

Correlations are commonly calculated in many research projects where the relationship between variables is important.

Correlations are also important to understanding the reliability of test scores and test validity.

Key concepts

Correlation coefficient

Pearson Product Moment Correlation

Inverse relationship

Positive correlation

Negative correlation

Link to A-Z list of Statistical Terms



References

Ripley, J. S., Garthe, R. C., Perkins, A., Worthington, E. J., Davis, D. E., Hook, J. N., & ... Eaves, D. (2016). Perceived partner humility predicts subjective stress during transition to parenthood. Couple and Family Psychology: Research and Practice5(3), 157-167. doi:10.1037/cfp0000063

Sutton, G. W. (2020). Applied statistics: Concepts for counselors, second edition. Springfield, MO: Sunflower. AMAZON  Paperback ISBN-10: 168821772X, ISBN-13: 978-168217720    website: counselorstatistics


Please check out my website   www.suttong.com

   and see my books on   AMAZON       or  GOOGLE STORE

Also, consider connecting with me on    FACEBOOK   Geoff W. Sutton    

   TWITTER  @Geoff.W.Sutton    

You can read many published articles at no charge:

  Academia   Geoff W Sutton     ResearchGate   Geoffrey W Sutton 








Sunday, March 28, 2021

Skewed Distributions

 


Skewed distributions have one tail that is longer than the other tail compared to the "normal" distribution, which is perfectly symmetrical. 

Positive Skew

Below is an image of positive skew, which is also called right skew. Skew is named for the "tail." If you had statistics, you may have heard a professor say, "the tail tells the tale." The tail is the extended part of the distribution close to the horizontal axis.

The large "hump" area to the left represents the location of most data. In behavioural science, the high part often refers to the location of most of the scores. Thus, in positively skewed distributions, most of the participants earned low scores and few obtained high scores as you can see by the low level of the curve, or the tail, to the right.



Negative Skew

As you might expect, negatively skewed distributions have the long tail on the left thus, they are also called left-skewed distributions. A negatively skewed distribution of test scores illustrates an easy test--just what students want. Teachers used to talk about grading on a curve. You can see that such grading could be good or bad for students depending on what curve the teacher uses.



Skewed distributions are nonnormal by definition. 

Recall that in the normal curve, the mean, median, and mode are all at the same point in the middle of the distribution. The value of skew in a normal distribution is zero. 

In skewed distributions, the mode is at the high point and it represents the most frequent value or test score. The mean is pulled in the direction of the long tail and the median falls between the mode and the mean.

Common test questions ask what happens to the mean in skewed distributions. Keep in mind that the mean is "pulled" toward the tail. The mean is an average and, as such, it is most susceptible to extreme scores.

Skew and Data Analysis

Most statisticians accept small deviations from normality when analysing data using procedures designed for a normal distribution like the Pearson r, t tests, and the parametric ANOVAs

The question of acceptable ranges of skew will yield different answers from different sources. A range of +1.5 to -1.5 is a common rule of thumb. An important consideration is the "true" nature of the measured variable. Scientists may argue for flexibility in analysing data from a sample if the variable is known to be normally distributed in the population.

Skewed data can be adjusted and should be adjusted before using parametric tests. One method of adjustment is to convert all scores to logarithms and perform the data analysis on these transformed values.

If the data are too skewed and it is inappropriate to transform the data, then analysts should use nonparametric statistical methods.

Moments

In statistics, the concept of moments is taken from physics. Moments refer to central values. The first moment is found by calculating the value of the mean. The first moment is zero.

The second moment is seen in the calculation of variance, which uses squared values.

The third moment is found by calculating skew and the fourth moment results in the calculation for kurtosis.


Learn more about behavioural statistics in Applied Statistics Concepts for Counselors on AMAZON   or   GOOGLE








Learn More about analyzing data  in Creating Surveys on AMAZON or GOOGLE








Please check out my website   www.suttong.com

   and see my books on   AMAZON       or  GOOGLE STORE

Also, consider connecting with me on    FACEBOOK   Geoff W. Sutton    

   TWITTER  @Geoff.W.Sutton    

You can read many published articles at no charge:

  Academia   Geoff W Sutton     ResearchGate   Geoffrey W Sutton 


Friday, March 12, 2021

Dependent Samples Matched Pairs t test

 The Dependent Samples t test is used to test for significant differences between two sets of numerical data produced by the same organisms or organisms that are matched on all relevant variables.

In one example, a group of people who attend a workshop may complete a pretest and a posttest. A Dependent Samples t test can be used to compare the mean differences between the pretest and the posttest.

An example of a Matched Pairs t test can be used to compare two groups of people in a reading method experiment. A relevant variable would be reading ability. A reading test could be used to identify people with similar scores. One member of the pair is then randomly assigned to a new reading method group and the matching person is then assigned to the traditional reading group. At the end of the study, a Matched Pairs t test can be used to compare mean scores for the groups.

When the same person produces two sets of scores, each person is their own control. Because of the level of control, there is less variation than would be the case when using different people in each group.

When the groups contain different unmatched people or subjects, the groups are considered independent samples and an Independent Samples t test is used.

Applied Statistics Concepts for Counselors on AMAZON or GOOGLE

 






Learn More in Creating Surveys on AMAZON or GOOGLE







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Thursday, March 4, 2021

Foster Parent Experiences Measure

 

Scale name: Foster Parent Experiences Measure

 Scale overview

This set of questions uses different ratings for different items.

 Author(s) Denby, Ramona, Rindfleisch, Nolan, & Bean, Gerald. (1999).

 Items

The items assess different experiences of foster parents.

 

Response Type Likert-type scales with different ranges.

 Sample item

From the Training Effect variable: “ I felt competent to handle the types of children placed in my home.” See the PsycTESTS references for more sample items.

 

Reliability & Validity

Not included in the PsycTESTS entry.

 Availability

According to Science Direct, “Reprint requests should be addressed to Dr. Ramona Denby, University of Nevada Las Vegas, School of Social Work, 4505 Maryland Pkwy., Las Vegas, NV 89154”

 Permissions -- if identified

Test content may be reproduced and used for non-commercial research and educational purposes without seeking written permission. Distribution must be controlled, meaning only to the participants engaged in the research or enrolled in the educational activity. Any other type of reproduction or distribution of test content is not authorized without written permission from the author and publisher. Always include a credit line that contains the source citation and copyright owner when writing about or using any test.

SCOPES domain = Social/relationships

 Reference(s)

Denby, Ramona, Rindfleisch, Nolan, & Bean, Gerald. (1999). Predictors of foster parents' satisfaction and intent to continue to foster. Child Abuse & Neglect, 23(3), 287-303. doi: 10.1016/S0145-2134(98)00126-4

Denby, R., Rindfleisch, N., & Bean, G. (1999). Foster Parent Experiences Measure [Database record]. Retrieved from PsycTESTS. doi: http://dx.doi.org/10.1037/t20685-000

Reference for using scales in research:

Creating Surveys on AMAZON or GOOGLE

 


 

 

 


Reference for clinicians on understanding assessment

Applied Statistics Concepts for Counselors on AMAZON or GOOGLE

  


 






Resource Link:  A – Z Test Index

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Checkout My Website   www.suttong.com

  

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Positive Home Integration Scale (PHIS) (Foster Homes)

Scale name: Positive Home Integration Scale (PHIS)

Scale overview

The PHIS is a 9-item Likert-type rating scale.

“This article introduces a youth-reported measure (Essential Youth Experiences [EYE]) developed to assess the experiences of foster youth in their home environment and their critical relationships across a number of service systems.” (From the Abstract)

Author(s) Kothari, Brianne H., McBeath, Bowen, Bank, Lew, Sorenson, Paul, Waid, Jeff, & Webb, Sara Jade. (2018

Items = 9

Response Type

Ratings from 1 to 10 with two anchors (they vary with the item) and a midrange label (somewhat).

Subscales

None identified in the PsycTESTS source.

Sample item

To what extent do you feel that you are treated with kindness in your (foster) home?

 Reliability & Validity

See the article reference for psychometric properties.

 Availability

The full set of 9 items with the 10-point descriptive labels can be found in the PsycTESTS reference below.

 Permissions -- if identified

Test content may be reproduced and used for non-commercial research and educational purposes without seeking written permission. Distribution must be controlled, meaning only to the participants engaged in the research or enrolled in the educational activity. Any other type of reproduction or distribution of test content is not authorized without written permission from the author and publisher. Always include a credit line that contains the source citation and copyright owner when writing about or using any test.”

 SCOPES domain = Social/relationships

Reference(s)

 Kothari, B. H., McBeath, B., Bank, L., Sorenson, P., Waid, J., & Webb, S. J. (2018). Positive Home Integration Scale [Database record]. Retrieved from PsycTESTS. doi: https://dx.doi.org/10.1037/t76186-000

 Kothari, Brianne H., McBeath, Bowen, Bank, Lew, Sorenson, Paul, Waid, Jeff, & Webb, Sara Jade. (2018). Validation of a measure of foster home integration for foster youth. Research on Social Work Practice, 28(6), 751-761. doi: https://dx.doi.org/10.1177/1049731516675033

Reference for using scales in research:

Creating Surveys on AMAZON or GOOGLE

 


 




 

 

Reference for clinicians on understanding assessment

Applied Statistics Concepts for Counselors on AMAZON or GOOGLE

 


 

 




Resource Link:  A – Z Test Index

  

Links to Connections

Checkout My Website   www.suttong.com

  

See my Books

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FOLLOW me on

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Wednesday, March 3, 2021

Family Stress Measure (Foster Parents)

 


Scale name: Family Stress Measure

 

Scale overview

The five questions on the Family Stress Scale were rated on a 5-point Likert scale ranging from never to very often.

 

Author(s) Geiger, J. M., Hayes, M. J., & Lietz, C. A. (2013)

Items = 5

 Response Type: Likert-type, 5-point

 Sample item

How often have you experienced any of the following events:

1. Severe difficulties with your foster child's biological family/parents?

 Reliability/ Validity

Not reported in PsycTESTS entry.

 Availability

The 5-items are listed in the PsycTESTS entry.

Geiger, J. M., Hayes, M. J., & Lietz, C. A. (2013). Family Stress Measure [Database record]. Retrieved from PsycTESTS. doi: https://dx.doi.org/10.1037/t25361-000

 Permissions -- if identified

“Test content may be reproduced and used for non-commercial research and educational purposes without seeking written permission. Distribution must be controlled, meaning only to the participants engaged in the research or enrolled in the educational activity. Any other type of reproduction or distribution of test content is not authorized without written permission from the author and publisher. Always include a credit line that contains the source citation and copyright owner when writing about or using any test.”

SCOPES domain = Social/relationships

Reference(s)

Geiger, Jennifer Mullins, Hayes, Megan J., & Lietz, Cynthia A. (2013). Should I stay or should I go? A mixed methods study examining the factors influencing foster parents' decisions to continue or discontinue providing foster care. Children and Youth Services Review, 35(9), 1356-1365. doi: https://dx.doi.org/10.1016/j.childyouth.2013.05.00

 Reference for using scales in research:

Creating Surveys on AMAZON    or   GOOGLE  Worldwide

 


 

 

 

Reference for clinicians on understanding assessment

Applied Statistics Concepts for Counselors on AMAZON or GOOGLE

 


 

 




Resource Link:  A – Z Test Index

 

Key Words: Parenting, Foster Parenting, Foster Parenting Stress, Foster Family Stress

  

Links to Connections

Checkout My Website   www.suttong.com

  

See my Books

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Read published articles:

 

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  ResearchGate   Geoffrey W Sutton 

 

 

 

 

variance and standard deviation

Variance is a measure of the dispersion of values in a distribution of values.  In psychology and behavioral science statistics, the varianc...