True Score

Definition

True score can also be defined with respect to the idea of an observed score. In short, the observed score is the true score plus some error. Symbolically, the equation is

where Xo is the observed score or the participant's score on a measure (e.g., the participant's score on Prochaska's smoking decisional balance inventory) and Xt is the participants true score or what is really of interest. In addition, E represents the error term that can be comprised of measurement error (i.e., systematic error) and some random error (i.e., unsystematic error). For example, real body weight (i.e., the true score of our weight) is never really known as it fluctuates with time of day, gravitational pull, retention of water, clothing choice, and many other variables. The number viewed on a scale would be considered an observed score (the true weight plus some error). The observed weight on the scale is based on the real weight with the addition of the calibration of the instrument used to measure weight plus the other variables that might interfere with the scales ability to properly measure weight. Conceptually, the formula would be

or

Although weight can be measured on a scale, it is important to remember that it is estimated. Without knowing the amount of error or the true score, the observed weight is actually meaningless. The obtained weight (or measurement on a myriad of other psychological measures or tests) could be greatly flawed (i.e., have a large amount of error) or very precise (i.e., have a little amount of error). Without multiple measurements on the individual or test subject or replication studies, it is difficult to know how much error is in the measurement.

Assumptions of True Score Theory

Because of the theoretical nature of the true score, Xt, it is not possible to know its value. Although it cannot be directly measured, there are some basic assumptions (according to true score theory and classic measurement theory) concerning the true score. As mentioned earlier, the first assumption is that all observed variables contain some error. If there was no error, then the true score would equal the observed score Xo. Another assumption is that the true score and the error score are independent. In other words, the correlation between the true score and the error score is zero (signifying that the movement in the error score has no impact on the value of the true score). The error score E is considered completely random. For example, if a baby weighs 12 pounds (her true score) and is weighed multiple times, then the amount the baby's weight fluctuates (thereby showing the differential error values) does not impact the baby's real weight.

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