# How to convert IRT theta score to a percentage score

I am trying to implement an adaptive test using 3PL IRT model. We need to screen the candidates and label their expertise as (beginner, intermediate, or expert).

We also need a percentage score for each examinee.

When the examinee is given a sufficient number of items, the initial estimate of ability should not have a major effect on the final estimate of ability.

The final estimate of ability is the last estimated theta value before the stopping rule. It can have values in the range -3 to +3.

Q: How do I convert this to a percentage score so that the end user can know their performance?

Q: How to determine the cut-score for each level of expertise? What are the things that I need to take into consideration while calculating this?

Theoretically in IRT, $$\theta$$ and $$b$$ (which are measured in the same scale) do not have lower or upper bounds. They can go from $$-\infty$$ to $$\infty$$. Sometimes a test is unable to capture the full extent of an examinee's knowledge because it doesn't have enough items in the lower or higher ends of the difficulty scale, for example. Anyway, it is not a straightforward task to convert from $$\theta$$ values to percentages.
One thing you can do (and I know of a national test in my country that does something similar) is to transform your data using a desired mean and standard deviation. For example, you can displace your mean from 0 to 500 by adding 500 to all examinees $$\theta$$ values and changing your scale accordingly. You can then present their scores as if they went from 0 to 1000 (which you know that theoretically is not the truth, it just means your scale is centered at 500).
• You can subtract $\mu_{\theta}$ and divide by $\sigma_{\theta}$ from all yout $\theta$ values, then add a new $\mu_1$ and multiply by a new $\sigma_1$ of your choosing to get a new scale for all $\theta$. In my example, $\mu_1=500$ and $\sigma_1$ is a value you could experiment with to get your max. and min. $\theta$ values within the desired bounds. – Douglas De Rizzo Meneghetti Jan 26 '19 at 4:13