A conceptual introduction to psychometrics, development, analysis, and application of psychological and educational tests, by G. J. Mellenberg (first edition) – Summary chapter 4

Giving scores to different responses (e.g. agree = 3) is called scoring by fiat and has no theoretical justification. The observed test score is derived from the item scores by taking the unweighted or the weighted sum of the item scores. The construct score is derived from the item responses under the assumption of a latent variable response model.

The unweighted sum of item scores is the sum of all the item scores of a person j. It uses the following formula:

The weighted sum of item scores can be used because some items need to weigh heavier than others. This uses the following formula:

‘w’ denotes the weight of an item. The formula for the population mean of observed scores is the following:

It is the sum of the (un)weighted score divided by the number of test takers. ‘ε’ denotes expectation and ‘p’ denotes that the expectation is taken with respect to the population of test takers. The formula for the test variance is the following:’

It means the sum of (the test score of person j – the expected score) squared divided by the number of test takers. The expected score is the population mean of observed scores. The formula for the population standard deviation is the following:

It is the square root of the test variance. The population mean test score is estimated by using the sample mean and uses the following formula:

It is the sum of the observed test scores divided by the number of test takers. The item mean of item k uses the following formula:

It is the sum of item score on item k divided by the number of test takers. For dichotomous items, the item mean is equal to the proportion of that item. The formula for item variance of item k is the following formula:

It is the sum of the item score k minus the mean of item k squared. The item test variance uses the same formula, except it uses test scores of test taker j, instead of item scores. The item standard deviation is the square root of the item variance. Item variance for item k for dichotomous items can be calculated using the following formula:

The correlation between item k and item l can be calculated in the following way:

It is the sum of item score k minus the mean of item k times item score l minus the mean of item l divided by the square root of the sum of item score k minus the mean of item k squared times the item score l minus the mean of item score l squared.

The correlation between item k and item l for dichotomous items can be calculated in the following way:

It is the proportion of item k and item l minus the proportion of item k times the proportion of item l divided by the square root of proportion of item k times (one minus the proportion of item k) times the proportion of item l times (one minus the proportion of item l).

Covariance is the measure of how much two variables vary together. The covariance between item k and item l can be calculated using the following formula:

It is the sum of item score k for test taker j minus the mean of item k times the item score l for test taker j minus the mean of item l divided by number of test takers minus one. The covariance between item k and item l for dichotomous items can be calculated using the following formula:

It is the number of test takers divided by the number of test takers minus one times the proportion of item kl minus the proportion of item k times the proportion of item l. The test variance can also be calculated by summing up everything in the variance-covariance matrix. The variance-covariance matrix is a matrix where the covariance between all items are calculated.