Showing posts with label var. Show all posts
Showing posts with label var. Show all posts

## Thursday, January 20, 2022

### 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 variance is typically a reference to the extent to which numerical values vary around the arithmetic mean of a data set. Theoretically, the values vary around a population mean but in most cases, researchers work with samples.

In statistics, write sigma squared for the population variance σ2

Write final form sigma squared for the sample variance ς2

In reports, write VAR for variance.

How it works

If we have a set of different numerical values such as scores on a test we can calculate a mean, which is the average of all the scores divided by the number of scores.

The difference of one score from the mean is a deviation score. X is a score and the Greek letter mu μ is the symbol for the population mean.

In a sample, which is what we normally have in psychology, we subtract a score X from the sample mean M. Thus, X - M = the deviation score.

If a person earns a 7 on a test where the mean is 10 then their score is 3 points below the mean. The deviation score is -3 (minus three).

To find an average deviation for all the scores on the test, we must subtract each score from the mean. We end up with a set of positive and negative values.

We want to find an average but if we add the positive and negative values we will end up with zero. To solve the problem,

square each deviation score

add them together to get the sum of the squares (SS)

then divide by the number of scores (n) to get the variance.

The variance = the SS divided by n

Standard Deviation (SD)

The standard deviation is the average unit by which scores are distant from the mean.

Find the standard deviation by taking the square route of the variance.

In reports, write SD for standard deviation.

Examples in tests

A common mean for IQ tests is 100 and a common standard deviation is 15 points.

In personality tests, a common standard score is a T-score. The mean is T = 50 and SD = 10.

Key concepts in this post

deviation score

mean

sum of squares

standard deviation

variance

Applied Statistics Concepts for Counselors on   AMAZON or   GOOGLE

Related post

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

You can read many published articles at no charge:

Academia   Geoff W Sutton     ResearchGate   Geoffrey W Sutton

### Factor Analysis Principal Components Analysis

Factor analysis (FA) is a statistical method of reducing a large set of data to a smaller set by identifying patterns in the data that have common characteristics. Factor analysis is sometimes called data reduction or dimension reduction.

The original numerical values in the data set are observed variables (also called manifest variables) such as the items in a large survey or test. Factor analysis may find patterns characterized by a shared statistical relationship representing a factor, which is also called a dimension. A researcher examines the content of the items linked to this factor and chooses a factor label such as verbal skills for related items on an intelligence test.

The factors may be treated as variables in additional research. These are secondary variables. Because they are created from the observed variables, they are considered latent variables. For example, if 5 items on a personality test are associated with one factor labeled "agreeableness" then agreeableness is a latent variable.

The set of identified factors is referred to as the structure of the data set. If the data are from a test then researchers refer to the structure of the test.

Factors are identified based on the variance they account for in the data. The amount of variance explained by a factor is represented by an eigenvalue. Researchers look for eigenvalues of 1.0 or more to consider a factor to be a valuable contribution to explaining the underlying structure of a data set.

Not all factors are equal. That is, when more than one factor have been identified, they will contribute differently to explaining the variance in the data set.

Different kinds of Factor Analysis

Exploratory Factor Analysis (EFA). When researchers do not know the structure of a data set, they use EFA to discover the set of factors.

Confirmatory Factor Analysis (CFA).  When researchers wish to test a hypothesis about a data set, they perform CFA. For example, if they believe their forgiveness questionnaire contains one factor called forgiveness, they can examine the structure to see if one factor best accounts for the data set. If one factor is the best solution then they have found support for their hypothesis.

Principal Components Analysis (PCA) is a common form of confirmatory factor analysis.

Factor Analysis is important to understanding tests in Counseling and Psychotherapy. See

Applied Statistics Concepts for Counselors on   AMAZON or   GOOGLE

Factor Analysis is often used to reduce the data collected from survey research.

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

You can read many published articles at no charge:

Academia   Geoff W Sutton     ResearchGate   Geoffrey W Sutton

### Post Hoc Tests and Data Analyses

A post hoc test is a statistical test used to determine if a pair of values are significantly different from each other after the primary ...