Choose the correct answer from the options given below:

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: According to Classical Test Theory (CTT), total variance is equal to the sum of true variance and error variance.
Reason R: Error variance and true variance are independent of each other as per CTT.
In light of the above statements, choose the most appropriate answer from the options given below:

Correct option is A

The correct answer is Both A and R are correct and R is the correct explanation of A.
1. Assertion A is correct: According to Classical Test Theory (CTT), the total variance observed in a measurement is composed of two components—true variance (the variance attributed to the actual trait or characteristic being measured) and error variance (the variance caused by factors that are not related to the trait being measured, such as measurement errors).
2. Reason R is correct: In CTT, true variance and error variance are considered independent of each other. The true variance reflects the actual variation in the trait being measured, while error variance reflects random fluctuations or inaccuracies that are not related to the trait itself.

Information Booster

Classical Test Theory (CTT)

1. Overview of CTT:
Classical Test Theory (CTT) is a framework used to understand the relationship between a true score (the actual measurement of a trait or ability), the observed score (what is actually measured), and the error in measurement. CTT is one of the earliest theories of measurement and has been foundational in the field of psychometrics.
CTT posits that the observed score XX is the sum of the true score TT and the error score EE, expressed as:
X=T+E
where:
XX = observed score
TT = true score
EE = error score

2. Total Variance in CTT:
In CTT, the total variance observed in test scores is the sum of two components:
True Variance: Variance that is attributable to the actual differences in the characteristic being measured (e.g., intelligence, skill level, etc.).
Error Variance: Variance that is due to errors in measurement (e.g., random fluctuations, biases, or inconsistencies in the testing process).

The equation for total variance is:
Total Variance=True Variance+Error Variance\text{Total Variance} = \text{True Variance} + \text{Error Variance}

3. Independence of True Variance and Error Variance:
According to CTT, true variance and error variance are independent of each other. This means that the variability in the true score does not affect the variability in the error score, and vice versa. This assumption is important for the reliability of measurement because it assumes that random errors do not systematically affect the true variance of the measure.

4. Reliability in CTT:
Reliability in CTT refers to the proportion of the total variance in observed scores that is due to true variance, which is a measure of consistency and accuracy of a test.

Formula for reliability:
Reliability=$$True VarianceTotal Variance=1−Error VarianceTotal Variance\text{Reliability} = \frac{\text{True Variance}}{\text{Total Variance}} = 1 - \frac{\text{Error Variance}}{\text{Total Variance}}$$​​
A high reliability means that the test is consistent and that the observed scores reflect the true scores more than the error.