There are several specific formulas to determine such. And you are incorrect. Even a small population can fit a normal curve, if that is the population used for norming. What you are talking about it generalizability, and that is a different concept altogether. I have already explained the percentages of populations that are represented in the various SDs. That gives you the information you are requesting.
A small population can fit a normal curve, but the likelihood of doing so perfectly is pretty low. I'll demonstrate that if you want me to.
So what you're saying is that in a given test distribution,
exactly 68.2689% of scores must fall within 1 SD, 95.4500% must fall within 2 SDs, and 99.7300% must fall within 3 SDs? What if it's off by a thousandth of a percent? Is it suddenly invalid? See how unlikely that is? That's why I'm asking for a range.
The graph of a Gaussian is a characteristic symmetric "bell curve" shape that quickly falls off towards plus/minus infinity. A normal distribution of test scores does not fall off quickly toward zero. I have already explained this to you. Therefore, it approximates a normal distribution.
Gaussian and normal are interchangeable terms in statistics. To quote one of my books, "Probability and Stochastic Processes" (Yates, Goodman) on p. 137, "Because they occur so frequently in practice, Gaussian random variables are sometimes referred to as
normal random variables." The probability density functions and everything else about them are exactly the same. More information here:
Normal Distribution -- from Wolfram MathWorld
Sweetie, I have my terms straight. And the skew is not a parameter. Statistical parameters are values, usually unknown, and therefore has to be estimated, used to represent a certain population characteristic.
Skewness is indeed a value that can be calculated from a distribution. Here's a list showing how to calculate them for various distributions.
Z table - Normal Distribution
Note that for normal, Laplace, uniform, and student-t distributions, the skewness is 0. Speaking of student-t distributions...
Student-t is not a distribution. It is a converted raw score. Chi-Square is a statistical hypothesis test and can only be used when the null pyhpothesis have found to have been true in previous statistical analysis. We are not even concerned with hypothesis testing in this situation. There is no hypothesis to be tested. We are determining the validity of an assessment.
I opened up my book as you suggest. Right now I am staring at a picture of a student-t distribution (also known as Student's t-distribution) on p. 232 of "Probability and Random Processes with Applications to Signal Processing" (Stark, Woods). Here's a picture of one that looks exactly like what's in my book:
Student's t-Distribution -- from Wolfram MathWorld. And right now, I'm staring at a picture of a chi-square distribution on p. 236. Here's what it looks like:
Chi-Squared Distribution -- from Wolfram MathWorld
Evidently, your understanding of the rest of the terms you have thrown out is fairly superficial as well. A skewed distribution is not a normal distribution. You need to return to your statistics text book. If a distribution is skewed, it is not normal. That is a very basic concept.
Then it's a good thing I said this: "It is not an actual normal distribution, but it's similar".
Now, if you're going to talk down to me and to the others here while touting your own expertise, you should be the one who knows what you're talking about. Instead, you're claiming that normal distributions can have two modes, Gaussian distributions are just approximations of normal distributions, and there's no such thing as a student-t or chi-square distribution. Furthermore, I didn't intend to get in a big debate about the field of statistics. I just wanted to find out from you, the forum expert in industrial and organizational psychology, the specific criteria used to deem a test invalid based on the distribution. What you've given me so far is unclear, but if I understand it right, it means the vast majority of tests will be invalidated based merely on chance. If so, then what good are your methods? If you're the expert around here and you're this far off the mark, are you beginning to see why more people don't trust your Board of Industrial and Organizational Psychology?
And please don't call me "sweetie". Only my mom and my wife may do that.