Statistics for all: Estimating confidence
In the first post in this series, I promised to share a quick and dirty trick for determining how much confidence you can have in a test score. I will. But first, I want to show you a bit more about what confidence means when it comes to educational and psychological tests.
Let’s start with a look at how test scores are usually reported. The figure on the left shows three scores, one at level 8, one at level 6, and one at level 4. Looking at this figure, most of us would be inclined to assume that these scores are what they seem to be — precise indicators of the level of a trait or skill.
But this is not the case. Test scores are fuzzy. They’re best understood as ranges rather than as points on a ruler. In other words, test scores are always surrounded by confidence intervals. A person’s true score is likely to fall somewhere in the range described by the confidence interval around a test score.
In order to figure out how fuzzy a test score actually is, you need one thing — an indicator of statistical reliability. Most of the time, this is something called Cronbach's Alpha. All good test developers publish information about the statistical reliability of their measures—ideally in refereed academic journals with easy to find links on their web sites! If a test developer won’t provide you with information about Alpha (or its equivalent) for each score reported on a test, it’s best to move on.
The higher the reliability (usually Alpha) the smaller the confidence interval. And the smaller the confidence interval, the more confidence you can have in a test score.
The table below will help to clarify why it is important to know Alpha (or its equivalent). It shows the relationship between Alpha (which can range from 0 to 1.0) and the number of distinct levels (strata) a test can be said to have. For example, an assessment with a reliability of .80, has 3 strata, whereas an assessment with a reliability of .94 has 5.
Strata have direct implications for the confidence we can have in a person’s score on a given assessment, because they tell us about the range within which a person’s true score would fall — its confidence interval — given the score awarded.
Imagine that you have just taken a test of emotional intelligence with a score range of 1 to 10 and a reliability of .95. The number of strata into which an assessment with a reliability of .95 can be divided is about 6, which means that each stratum equals about 1.75 points on the 10 point scale (10 divided by 6). If your score on this test was 8, your true score would likely be somewhere between 7.13 and 8.88 — your score’s confidence interval.
The figure below shows the true score ranges for three test takers, CB, RM, and PR. The fact that these ranges don’t overlap gives us confidence that the emotional intelligence of these test-takers is actually different**.
If these scores were closer together, their confidence intervals would overlap. And if that was the case — for example if you were comparing two individuals with scores of 8 and 8.5 — it would not be correct to say the scores were different from one another. For example, it would be incorrect for a hiring manager to consider the difference between a score of 8 and a score of 8.5 in making a choice between two job candidates.
By the way, tests with Alphas in the range of .94 or higher are considered suitable for high-stakes use (assuming that they meet other essential validity requirements). What you see in the figure above is about as good as it gets in educational and psychological assessment.
Most assessments used in organizations do not have Alphas that are anywhere near .95. Some of the better assessments have Alphas as high as .85. Let’s take a look at what an Alpha at this level does to confidence intervals.
If the test you have taken has a score range of 1–10 and an Alpha (reliability) of .85, the number of strata into which this assessment can be divided is about 3.4, which means that each stratum equals about 2.9 (10 divided by 3.4) points on the 10 point scale. In this case, if you receive a score of 8, your true score is likely to fall within the range of 6.6 to 9.5*.
In the figure below, note that CB’s true score range now overlaps RM’s true score range and RM’s true score range overlaps PR’s true score range. This means we cannot say — with confidence — that CB’s score is different from RM’s score, or that RM’s score is different from PR’s score.
Assessments with Alphas in the .85 range are suitable for classroom use or low-stakes contexts. Yet, every day, schools and businesses use tests with reliabilities in the .85 range to make high stakes decisions — such as who will be selected for advancement or promotion. And this is often done in a way that would, for example, exclude RM (yellow circle) even though his confidence interval overlaps CB’s (teal circle) confidence interval.
Many tests used in organizations have Alphas in the .75 range. If the test you have taken has a score range of 1–10 and an Alpha of .75, the number of strata into which this assessment can be divided is about 2.2, which means that each stratum equals about 4.5 points on the 10 point scale. In this case, if you receive a score of 8, your true score is likely to fall within the range of 6–10*.
As shown in the figure below, scores would now have to differ by at least 4.5 points in order for us to distinguish between two people. CB’s and PR’s scores are different, but RM’s score is uninterpretable.
Tests or sub-scales with alphas in the .75 range are considered suitable for research purposes. Yet, sad to say, schools and businesses now use tests with scales or sub-scales that have Alphas in or below the .75 range, treating these scores as if they provide useful information, when in most cases the scores — like RM’s — are uninterpretable.
If your current test providers are not reporting true score ranges (confidence intervals), ask for them. If they only provide Alphas (reliability statistics) you can use the table and figures in this article to calculate true score ranges for yourself. If you don’t want to do the math, no problem. You can use the figures above to get a quick sense of how precise a score is.
Statistical reliability is only one of the ways in which assessments should be evaluated. Test developers should also ask how well an assessment measures what it is intended to measure. And those who use an assessment should ask whether or not what it measures is relevant or important. I’ll be sharing some tricks for looking at these forms of validity in future articles.
*This range will be wider at the top and bottom of the scoring range and a bit narrower in the middle of the range.
**It doesn’t tell us if emotional intelligence is important. That is determined in other ways.
My organization, Lectica, Inc., is a 501(c)3 nonprofit corporation. Part of our mission is to share what we learn with the world. One of the things we’ve learned is that many assessment buyers don’t seem to know enough about statistics to make the best choices. The Statistics for all series is designed to provide assessment buyers with the knowledge they need most to become better assessment shoppers.
Guilford J. P. (1965). Fundamental statistics in psychology and education. 4th Edn. New York: McGraw-Hill.
Kubiszyn T., Borich G. (1993). Educational testing and measurement. New York: Harper Collins.
Wright B. D. (1996). Reliability and separation. Rasch Measurement Transactions, 9, 472.