A clothes line isn't big enough, we need a Ferris wheel.
[ENTJ] Let's talk statistics.
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A clothes line isn't big enough, we need a Ferris wheel.
I think the mean (100) is already implied in your IQ score, Tony. So I’m not sure the z-score adds anything.
@Pin did say in the OP that the z-score measures how far “one deviates from the norm”, which seems like a much broader notion that just intelligence. If that’s the case, I would be curious to hear a more precise description of it and what (apart from IQ) it measures.
o2b
Permanent Fixture
Aren't z-scores mainly used for significance testing. I think it's just a reason for some journal to justify their results eg. a z-score of 1.32 which is significant (quote stats journal, 2011) etc. So as someone said previously, to state that something has deviated from the norm and it was significant (according to chosen journal article). It quantifies the difference but whether it means anything is how that is interpreted.
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Statistical significance depends on the standard we use and why. A z-score of 1.32 might not be statistically significant. Statistical significance depends on the alpha level used in our field of study.
True, the 1.32 was just a random number. I believe it's also used to assess normality with skew and kurtosis of the sample.
"A z-test is applied for normality test using skewness and kurtosis. A z-score could be obtained by dividing the skew values or excess kurtosis by their standard errors."
As the standard errors get smaller when the sample size increases, z-tests under null hypothesis of normal distribution tend to be easily rejected in large samples with distribution which may not substantially differ from normality, while in small samples null hypothesis of normality tends to be more easily accepted than necessary. Therefore, critical values for rejecting the null hypothesis need to be different according to the sample size as follows:
Kim, 2013
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3591587/
Also, when I said it's typically used in significance testing, I guess it's highly dependent on the sample.
For example, the weight difference between dogs of 8 years and 12 years was not significant p = 0.77, however the z-score for 12 year old dogs in the sample was 7.65 which according to (Kim,2011) is significant. Perhaps the 12 year old dogs in this sample were not measured correctly by the examiner causing a significantly skewed distribution, further investigation of 12 year old dogs is required.
If you relate this back to IQ, it's going to tell you how you fair in a certain sample eg. Your z-score in a pack of 30 y/o males might be different than 68 y/o retirees. But in this circumstance it says less about you and more about 30 y/o males unless your IQ is exceptionally different from your cohort.
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A Z score is an alternative way of expressing SD's difference from the mean. The trouble is that the philosophy of statistics can easily get lost in playing with the concepts. We normally take 3*SD as significant, by convention. That's because each statistical analysis is an experiment to determine if something is more likely to be different or the same as something else. So saying that, on specific measurement, @Ren is 2 * SDs from the mean is insufficient evidence to say he's actually different from the mean. On the other hand if you yourself are measured as 3 * SD from the mean, then this would be taken as significant and evidence that you are different. The break point is conventional though and varies according to circumstances. So if an experiment were conducted to prove Einstein was wrong about gravity, and it's a noisy experiment with lots of scatter and contoversy in the results, we might only accept it as a clear indication at 5 *SD because Einstein is so right about so many things. I don't think in these situations the Z value is adding anything much in the way of succinctness in the terminology.
But the creator of the Z measure came up with a formula to judge the financial viability of a company - I must say it looks to me as about as valid as the Drake equation and looks quite different to the basic concept related to SD.
o2b
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Hey @John K
Well, there is a way of determining the extent to which a population actually is normally distributed, realizing there are other distributions that reflect populations, such as Poisson. Once it is established that a population is normally distributed, the percentage of the population that is at or beyond any deviation from the mean is known with an extremely high level of certainty. (Realizing that only a population size of infinity can provide 100% certainty.)
Well, there is a way of determining the extent to which a population actually is normally distributed, realizing there are other distributions that reflect populations, such as Poisson. Once it is established that a population is normally distributed, the percentage of the population that is at or beyond any deviation from the mean is known with an extremely high level of certainty. (Realizing that only a population size of infinity can provide 100% certainty.)
Certainly, but that's only true if you have details for the whole population. In practice this is very very rarely the case - what you are doing is comparing one sample with another.
But of course there is also the theorum that says that the sample mean from any distribution is approximately normally distributed - that's a most unexpected and extremely useful fact that makes the normal distribution far more than just one of many.
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