Kafli 3

Differences, Consistency, and the Meaning of Test Scores

• Measurement relies on understanding differences (variability), consistency (covariability), and how to interpret test scores.
• These concepts are essential for measurement theory, test evaluation, and application.
• Understanding builds a strong base for psychometrics and understanding psychological test scores.

Three Key Building Blocks

• Variability, the degree of differences within a set of test scores or among values of a psychological attribute, must be understood.
• The importance of this concept and how to quantify it is described
• Consistency or covariability is the degree to which variability in one set of scores is consistent with variability in another set.
• The importance of covariability, and explaining ways to quantify it are both described

The Nature of Variability

• Psychological measurement is based on the idea that people may differ in behavior/psychological traits.
• Differences can be explored to understand the source and meaning of psychological differences.
• Measurement rests on detecting differences even when studying a single individual.

Types of Differences to Measure

• Interindividual variability refers to differences between people in how they react
• Example is the difference between students SAT scores
• Intraindividual variability refers to differences within a single person that are consistant
• Example is a psychiatric patient's symptom changes in response to treatment
• Psychological differences must exist and be quantifiable for psychological measurement to occur

Importance of Individual Differences

• Individual differences are critical in psychology, particularly in understanding behavioral variability.
• Measuring these differences is a fundamental aspect of behavioral sciences in research.
• Understanding the causes of individual differences and understanding if they can be measured
• Individual difference assumption is made to assist in employer hiring efficacy
• Variability detection attempts are made by college admissions and clinicians in therapy

Individual Differences and Psychological Measurement

• Measurement is based on the fact that psychological differences can be observed in the measurement process.
• Psychometric concepts, such as reliability and validity, rely greatly in the ability to quantify the differences in populations.
• Experimental psychology depends on measuring individual differences to see effects of experiences on behavior
• Clinical settings require differences be measured to diagnose psychological pathology and observe any affect of therapy

Variability and Distributions of Scores

• Measuring psychological differences relies on the fact that test scores can vary.
• When test scores are obtained from a group of people or at different points in time it becomes a distribution of scores.
• Variability refers to the differences within a distribution of different scores from the population.
• To quantify these differences becomes a key element in behavioral research

Variability and Distributions of Scores

• Psychometrics involves understanding variance and covariance and also their relationship.
• Variance quantifies variability or individual differences in a distribution of scores.
• Covariance quantifies the connection between variance in one set of scores and another.
• Detection of Distribution of test impacts measurement concepts in psychology
• When tested, some will score low, some high, some in between.

Distribution Description

• Statistics describes a distribution in a meaningful way.
• Central tendency, variability, and shape are concepts that stage a discussion of psychometrics.
• Central tendency looks at which score best represents the set of scores and identifies commonalities
• Variability describes the differences among peoples test scores.
• Shape can be applied by judging if scores are evenly distributed around the mean

Central Tendency

• Central tendency is perhaps the most basic aspect of test scores.
• It describes what the "typical" score is and what describes the entire population best
• Mean, median, and mode are statistical values can be used to reflect central the population
• The arithmetic mean (X) represents this typical population score.
• The equation for the mean is: Mean = X = ΣX/N

Variability - Quantifying Difference

• Quantifying the degree to which people in a group differ from each other can be achieved by measuring which each person deviated.
• Thus needing to quantify the amount of variability within a distribution of test scores.
• Several statistical values can be used to quantify variability within a distribution of scores.
• Variance and standard deviation are common indexes of variability in behavioral research especially as it pertains to psychometric theory
• Variance measures the average degree to which scores deviate from the mean

Variance Computation

• Variance is a core concept in psychometrics and it comes from summarizing in three easy steps
• First, the deviation of the group mean can be discovered by subtracting the mean of the sample from the score, with the equation: (X – X)
• Table 3.1 of the provided document provides a strong breakdown of this step.
• Then, with each result values above and below the mean can be noted.
• Second, the squared deviation can be computed squaring each result from the first step, with the equation (X – X)².
• Finally, the variance (s2) of the population can be calculated with the mean of the squared deviations with the equation: Variance = s² = Σ(X – X)²/ N

Standard Deviation

• The numerator of the variance means that the variance is itself a type of mean
• It is a measure of variability.
• Standard deviation (s), is simply the square root of the variance.
• As an index of variability, the standard deviation reflects variability in terms of the size of raw deviation scores unlike variance which reflects squared deviation scores, making SD more intuitive

Interpreting the Concepts of Standard Deviation and Vaariance

• The size of the variance is determined by the degree to which scores in a distribution differ from each other
• Example variance can be low, 1.67, which means scores are very tightly clusted around the mean rather than spread out over a wide area

Factors to Consider for Standard Deviation and Variance

• Neither can be less than 0, which indicates the scores in the distribution do not vary at all
• Positive value means there is a substantial about of variability.
• Theres no way to describe it as largely small but needs a form of measurement like SAT score
• A variance is most interpretable when put into some kind of context.
• Is only truly meaningful when the 2 distributions are based on the same type of measure

Importance in Effects

• The importance of variance and standard deviation lies mainly in their effects on other values that are more directly interpretable
• Its a foundamental concept in things like the corelation coefficient, relability coefficient, etc
• Previously taken stats courses may be odd about The "N - 1" versions of equations to help find inference of statistics since the equation doesnt change all too much

Distribution Shapes

• A distribution can be represented graphically by a bell curve, with the horizontal x-axis representing test scores and the vertical y-axis representing proportions.
• The curve represents the amount of the population with people in certain values
• Curves can be symmetric or mirrored

Importance in Curves

• A normal distribution is when a cureve has equal sides and it is very important in stat procedures and concepts
• Many procedures are vaild only if scores are distributed normally in samples or are drawn from them by population
• It is impossible to have scores in true normality, only approximated

Curves in Action

• Shape can be assessed by quantifying with the amount of skew.
• Like that of asymmetrical form of the distribution and the measure of value to show
• A perfectly symmetric distribution has a skew value of 0
• Asymmetrical skews with values below/above can also exist

Quantifying the Association or Consistency Between Distributions

• An equally important concept is the degree to which two distributions of score vary in a corresponding manner
• We might find that people vary in IQ scores and also in GPA
• Two variables should be considered for questions of covariability.

Interpreting the Association Between Two Variables

• To know the direction of the relationship, determine if people who obtain high scores on one typically do/do not on the other
• Positive or direct association means people match scores
• Negative or inverse association means one variable increases while the other falls
• Many find consistency a useful form of measurement: A strong association can show how that people stay the same over different variables

Scatterplots

• Useful and visually show correlations between two variables with an upward/downward treand
• Indicates if each one goes in line with the variables in full
• A general way to asses the way different associations work in full

Covariance Beginnings

• An index to create a interpretable measure relating to variance to a quantitative one-
• Contrast: a measure showing relation in score association, this begins the path to determining a score base better

Covariance In Action

• Earlier indications of relations will require at least 2 people per score
• Each can also be pared again with others that show how different test show results for others.
• Computing the covariance between two distributions shows the same relation in a reverse view of the initial

Three Step Computing

• Compute devation as such, take the values with their labels to form something more complete
• The second we compoute the amount of values or products of said measure in the new deviation, then
• Cross these scores by taking the amount set in the two new deviation in relation to one another
• Third and final step, compute the mean of the new cross-product

Direction of Association

• For the final discusion, two important issues are up for considiration: direction and magnitude of the covaliance-
• It gives direct intel of what direction the product is
• if the relation is postitive then there is said to be "the relation between the two points remains"
• And by the number, or negative, it can mean how the points shift in position or overall

Magnitude and Scale

• Unfortunatly, covariance alone can can not find intel about magnitude and instead two factors show this effect-
• strength of the association has impact on the actual way the covariance shows itself
• or in the effect that both large values of the covariance is stronger rather than weak
• or just large relation means they don't correlate
• This is to say covariance is an important part of seeing relation but limited alone to test

Varience and Covariance Matrixes

• Before heading into the relations, its important to talk about using matrixes as statistical value in analysis
• By some metric, we gain an understadning of patterns which are then put into a square amount of numbers
• By using this type of variance we can obtain an upper leff corner for better matrix intel

Core Properties of Those Matrixes

• Are organized one way by knowing the four of a vriable in tow
• Second shows the vaiance to the top left corner for total relations
• Third shows how cells show colavarance between pairs set in relation to the others-
• Fourth shows how cell diaganols run indentical to others around the cell.

Core Variance

• We see though then they have an intrest when we take in relation the points.
• Then can see how we relate one point of info, then see relation
• But it can be tricky if not done right.

Correlation Introduction

• The correlation has been said and is intended to be set to create a index-
• Can be understood as a linear association for such-
• Are all but given here between points
• Each has to take into points in a equal

Coefficient of Correlation

• Can show the reflection or direction is all but clear
• Will have the value set to some thing more set in value to begin
• is what great benefit comes to those who know it

Correlation Factors

• Will be set to what is strong and set to what is weak
• As such it remains intrest

Binary Coding

• If coded to just this type will provide a more well seen data pull if we use this 0/1
• If we were to use binary data well, it might require some special test too-
• The fact is binary is a lot like others depending on the response of the item.

Basic Tests

• A set if equal and how so.
• So take this all into new formating in the end and be sure to not lose it to time fully over
• Keep the same tests to look great and the variance

Interperting a Score

• As taken earlier, we want to create a test to test how and how can we test more accurately then now?
• The biggest test is that tests show results then the results need interpertation or else why test?

Key Facts of Scoring

• Two facets can decide or mean scoring values over time for each level-
• The first quantative being the low to high score relations
• And The second: to better build and rebuild in relations

Needed - Referencing

• For that said test scores in the same form as that of others take the same test after all-
• This builds a great understanding of how each score might look as taken later in a more full form.