At the heart of CTT is the idea of the true score for the construct we are measuring (e.g., math achievement) and its relationship with the observed score on a test.

- the observed test score
- true score for an individual
- random variability in the observed score caused by factors other than true ability (fatigue, stress)

The framework can't be falsified b/c the relationship can be understood conceptually but can't be formally tested using observed data.

`Error`

must be understood as it is central to understanding the framework.

- random error - specific to a time, place, examinee, or assessment, and balanced over the 4 factors
- systematic error - consistent across one or more of time, place, examinee, and assessment - leads to biased (upward or downward) observed scores

Core of CTT is the equation

X = T + E

Where

`X`

= the observed score on the scale

`T`

= the true score on the scale

`E`

= error

(Equation 3.1, p. 30)

- When we obtain a score on a math test (X
_{i}), we are really interested in the`true score`

(T_{i}). - T
_{i}represents the mean of a theoretical distribution of observed scores after repeated and independent assessments on the same test an infinite number of times. - we
*infer*T_{i}using X_{i}because we cannot test over and over again. - T is the expected value (population mean) of X, where the population is the scores from the student.
- if a student scores 91 on a math test, the score (X
_{i}) shows how much the students knows, measured on an imperfect test.- the score will be used to infer the student's understanding of math, T
_{i}, based on a single sample from the population of theoretically possible scores from the population. - ... the teacher needs to rely on the next best thing, the observed score

- the score will be used to infer the student's understanding of math, T

- an assessment can be anything from a full scale to a single item b/c all the models derived from Eq3.1 will allow us to gain a deeper understanding of the nature of the trait and the people.

- multiple assessments of a construct on the same individuals.
- If we have
*J*different assessments, eq3.1 would be

X_{ij} = T_{ij} + E_{ij}

where

- X
_{ij}= observed score for individual i on assessment j - T
_{ij}= true score for individual i on assessment j - E
_{ij}= error for individual i on assessment j

(Eq 3.2, p. 35)