Hello Everyone,
I wanted to ask if IQ Scores are always normed with the z-score formula ([x-M]/SD) or are they sometimes normed with percentiles (Rank/[n+1])?
@Lion They use z-scores, not percentile ranking. Your raw score gets compared to the normed sample using (score - mean)/SD, then converted to the IQ scale (M=100, SD=15). Percentiles are calculated from the IQ score afterward, not used for the actual norming. Using rank/(n+1) would make scores depend on sample size which defeats the purpose of standardization. The z-score method keeps everything consistent - a 115 always means +1 SD above average regardless of which normative sample or age group.
Okay so if it shouödnt depend on sample size couldnt be the age norms in the higher range wrongly represented?
I think when it comes to the extremes, smaller sample sizes mean fewer people in the tails of the distribution, making those scores less precise and reliable. This is why test publishers aim for large norming samples, but even then, very high or low scores should be interpreted more cautiously than scores near the average. Some tests even extrapolate beyond their actual sample data at the extremes, which introduces additional uncertainty.
You can always derive a percentile from a Deviation IQ score, but you can’t accurately derive the precise, standardized Deviation IQ score from a raw rank/percentile alone without assuming a perfectly normal distribution.
It’s always z-scores because percentile rankings do not always have the same interval between them. If you look at a normal distribution (like the one pictured below), you can see that the space between percentiles is not equal across the scale. They’re bunched up towards the middle and spread out at the extremes. This means that using them as the foundation to calculate IQs would distort the results: instead of being distributed normally (which is how the raw scores on most subtests are), percentiles would force the scores to have a rectangular distribution. It would compress the scores at the extremes and spread out the scores in the middle.
z-scores (labeled here as “Standard Deviation”) don’t have this problem. Because they always have the same interval, they don’t force the distribution to any particular shape. That means that the normal distribution of the original raw data is maintained when those scores are converted to IQs.
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