Calculating confidence intervals and percentiles

This post was inspired by a question posed on the Provisional Psychologists Forum Australia Facebook group.  Some cognitive ability test batteries provide interpretive information at the subtest level (eg Woodcock Johnson Tests of Cognitive Abilities – Fouth Edition; WJ-IV) while others only provide it at the higher index level (eg Wechsler Intelligence Scale for Children – Fifth Edition; WISC-5).  So while you will still get a score, such as 9 on Block Design, you won’t get the associated confidence interval and percentile rank.  The rationale for this is usually that the interpretation is more reliable at the index level and the test publisher may prefer you to only interpret at that higher level.  But the absence of such information does not mean it is not able to be calculated quite easily, providing you have a little extra information. For the first time you can use some of the stuff you learned in undergrad stats!

Here is the first equation, lets assume your Wechsler subtest score is 7:

z_score = (score – mean) / standard deviation

With Wechsler scales, the maths is pretty easy, knowing that the mean is 10 and standard deviation is 3.

z_score = (7 – 10) / 3
z_score = -1

Converting a z_score to a percentile rank can be done mathematically but there is a much easier way. Open Excel, put your new z_score in the top left cell, and in the cell to the right type in “=NORMSDIST(A1)” without the “”. This will return the percentile rank of 0.1586, or ~16th percentile.

The confidence interval is a little bit more complex but still really simple. What we are trying to account for in the confidence interval is the measurement error (or standard error of measurement, SEM), which changes depending on the subtest. Let’s assume that the reliability of our subtest was .85. Here is the next equation you need:

SEM = standard deviation x square root(1 – reliability)

SEM = 3 x square root(1 – .85)

SEM = 1.16

So now we can construct a confidence interval (CI). A 68% CI is the score +/- 1 SEM, which in this example is 5.84-8.16. A 95% CI is the score +/- 2 SEM, or 4.68-9.32. This assumes that reliability and measurement error is uniform across the range of the construct, and that the construct is normally distributed. This is often not quite the case, but it is close enough for our purposes. So in the end you can conclude that your subtest score of 7 had a 95% confidence interval ranging from 4.68 to 9.32, which is higher than 16% of the population. You can do this for any scale, as long as you have the mean, standard deviation, and reliability coefficient.