There are two ways to make an estimate of a person’s abilities. A point estimate (a single number) is precise but usually wrong, whereas an interval estimate (a range of numbers) is usually right but can be so wide that it is nearly useless. Confidence intervals combine both types of estimates in order to balance the weaknesses of one type of estimate with the strengths of the other. If I say that Suzie’s expected reading comprehension is 85 ± 11, the 85 is the point estimate (also known as the expected score or the predicted score or just Ŷ). The ± 11 is called the margin of error. If the confidence level is left unspecified, by convention we mean the 95% margin of error. If I add 11 and subtract 11 to get a range from 74 to 96, I have the respective lower and upper bounds of the 95% confidence interval.
Calculating the Predicted Achievement Score
I will assume that both the IQ and achievement scores are index scores (μ = 100, σ = 15) to make things simple. The predicted achievement score is a point estimate. It represents the best guess we can make in the absence of other information. The equation below is called a regression equation.
If X is IQ, Y is Achievement, and both scores are index scores (μ = 100, σ = 15), the regression equation simplifies to:
Predicted achievement = (Correlation between IQ and Achievement) (IQ – 100) + 100
Calculating the Confidence Interval for the Predicted Achievement Score
Whenever you make a prediction using regression, your estimate is not exactly right very often. It is expected to differ from the actual achievement score by a certain amount (on average). This amount is called the standard error of the estimate. It is the standard deviation of all the prediction errors. Thus, it is the standard to which all the errors in your estimates are compared. When both scores are index scores, the formula is
To calculate the margin of error, multiply the standard error of the estimate by the z-score that corresponds to the degree of confidence desired. In Microsoft Excel the formula for the z-score corresponding to the 95% confidence interval is
For the 95% confidence interval, multiply the standard error of the estimate by 1.96. The 95% confidence interval’s formula is
95% Confidence Interval = Predicted achievement ± 1.96 * Standard error of the estimate
This interval estimates the achievement score for 95% of people with the same IQ as the child. About 2.5% will score lower than this estimate and 2.5% will score higher.
You can use Excel to estimate how unusual it is for an observed achievement score to differ from a predicted achievement score in a particular direction by using this formula,
=NORMSDIST(-1*ABS(Observed-Predicted)/(Standard error of the estimate))
If a child’s observed achievement score is unusually low, it does not automatically mean that the child has a learning disorder. Many other things need to be checked before that diagnosis can be considered valid. However, it does mean that an explanation for the unusually low achievement score should be sought.
This post is an excerpt from:
Schneider, W. J. (2013). Principles of assessment of aptitude and achievement. In D. Saklofske, C. Reynolds, & V. Schwean (Eds.), Oxford handbook of psychological assessment of children and adolescents (pp. 286–330). New York: Oxford