Interesting thread. Probability and Statistics is not a particular branch of mathematics that I understand very well. I suspect there are some folks who claim to understand it and yet will be unable to explain in plain language what they think they understand about it. A super basic example of probability and statistics that is used as an introduction is a dice. When you roll a dice there are 6 possible results. The probability of getting anyone side of the dice when you roll it once is one out of six possible outcomes. Rolling a 1 has a 1/6 probability. If you roll a dice a thousand times the resulting distribution is that your roll will even out and accumulate about 166 or 167 of each side of the dice. There is no bell curve to this result, yet. There is simply a histogram and the expectation is that for the six sides of the dice the expectation is that all of the outcomes will occur equally distributed between 6 separate outcomes. So far, about 10 minutes into my first lecture, I was keeping up in my one college class on P&S. Then I learned that the "expected value" of rolling a dice was 3.5. I was never able to catch-up after that. In plain English, the expected value of rolling a dice does not exist. You might roll any result from 1 through 6 with equal probability. So the first thing you have to do is rephrase the definition of what 3.5 really means. What it really means is that if you add up the result of each roll and divide by the number of rolls then the result of this math will be 3.5 provided you roll that dice an infinite number of times. However, P&S has another handy rule of thumb that claims that you really only need to roll it 30 times, and you'll end up about as close as you need to get to 3.5. This is a bogus rule of thumb by the way. 30 samples are sometimes enough and sometimes they aren't. Now suppose you roll two dice and add the result. It can be anywhere from 2 to 12, and now, the mathematics of an expected value begins to work in sync with plain English, because the expected value is now 7, 3.5 on average for each dice added together equals 7. Now there are a total of 36 possible combinations, outcomes or actual values for the sum of rolling these two dice. If you now a square box and fill it full of these probabilities you'll have 36 sections each with a probability of 1/36 occurring on any given single roll of the pair of dice. However, the probability of the sum of the roll will be the sum of the diagonals within the square. There is only one of 36 ways to roll a two - snake eyes. And the only way to roll a 12 is similar, double sixes. There are two ways to roll a 3 or an 11, 3 ways to roll a 4 or a 10, 4 ways to roll a 5 or a 9, 5 ways to roll a 6 or an 8, and 6 ways to roll a 7. If you plot the number of times that any given outcome occurs in a given number of rolls then you will start to approximate a "continuous probability distribution of a real valued random variable." Here is a graph that I put together using LibreOffice Calc with 1000 iterations and it still doesn't quite smooth out to a normal distribution. The peak at 7 is way too high. And this is another interesting feature of the mathematics of random variables. Besides the fact that just one dice alone won't be enough to generate anything approximating a bell curve, it turns out that a truly random number generating algorithm is also near impossible to construct. Restraining orders are managed here in Harris County by the family division of the district court. There are about a dozen courts that handle these cases and who knows how many attorneys may participate in the process as the requestor for a protective order. I suppose there may be a database available to collate the data for statistical analysis and if so then someone could very well crunch the numbers to assign how many cases were granted an order versus the ones that were dismissed by the judge. The problem with using a concept such as a standard deviation and a normal probability distribution is that there are only two outcomes, the order is granted or dismissed. And the probability of the outcome seemingly ought not to be 1/2 & 1/2. It seems to me that most folks wouldn't hassle themselves with going to court over nothing, so I tend to think there should be more outcomes that grant protective orders than dismiss them. Supposing that some sort of analysis was sufficiently formulated to establish that the probability of granting a protective order should be x and then the probability of dismissing it would be 1-x. 60/40 or 80/20 or whatever. Now there is enough to have built up a coin that can land heads or tails, maybe weighted to land heads - it's still not enough to get to the minimum amount of stuff needed to approach the dataset needed for a bell curve. All you can do is count up cases that fill one bin or another. The application of this type of math really needs to have some range of more or less continuous applicability along the x axis and frequency is what is plotted on the y axis. In these examples, https://www.varsitytutors.com/hotmath/hotmath_help/topics/normal-distribution-of-data the curves would be the number of batteries, raspberries and people on the y axis and time, weight and height on the x axis. You could probably plot some type of court cases using this method. Civil cases that involve financial awards might be some for example. Something like neighbors that sue other neighbors for property damage maybe could be plotted in dollars along the x axis and the number of awards would be the y axis. This may or may not then follow a "normal" distribution and only if it did would the curve then have the proper shape for the variance / standard deviation to apply. Some distributions have two or more humps and the deviation and mean have to be accounted for separately.
There isn't anything strange about his at all. Gamblers face this all the time. The answer to the likely value of a hand, that is, what one might expect from betting a dollar on that hand in that situation is always going to be fractional. It's clearly a standard part of polling data, and many other real world situations, too. This just shows that one needs to take great care in matching the methodology to the question at hand. There are all sorts of questions to ask about the outcomes of court order requests. I don't see you identifying on. And, guessing what percentage should be denied seems particularly arbitrary. NO real world data are as nicely shaped as what is in your first course level of statistics. So, one needs to go on to learn more about statistical analysis.
I don't know if this will be helpful, because I'm not a math enthusiast, either. But I recently was presented with this term, "standard of deviation," which I did not understand, initially, either. It was, however, in regards to intelligence, more specifically, to I.Q. scores. Apparently 15 pts. is the S.O.D., in I.Q. And the way I heard one person express it (who has an extremely high I.Q., at any rate) was that, if your I.Q. is a full S.O.D., 15 pts., higher than someone else's, it is like you are at a whole different level of intelligence, on the floor above that other person, as it were.
Good example. SD describes how tightly results of a specific test are centered around the average. It is a measure of what the curve of test results looks like. Different IQ tests could show different SDs. So, those studying IQ can use SD as one tool for testing IQ tests! One could work to determine if a specific IQ test is biased by various population characteristics, for example. In science in general, if the SD is large it may indicate that the hypothesis being tested is weak or a fail.
You can use standard deviation and the cental limit theorem’s 68, 95, 97.7 rule to determine the probability of where people will score on IQ tests. For example, 68 % of people will score within one standard deviation which corresponds to scoring between 85 and 115 and 95% of people will score between 70 and 130 and 97.7% of people will score between 55 and 145. You can also use a Z table to find the probability of scoring less than any particular score, more than any particular score, and between any two particular scores.
I don't see this as a good example, as it ignores aspects of the test itself, the method of testing and the subjects (animate or inanimate) being tested. For IQ tests, many argue that they are always wrong, as they are estimates based on information taken by some testing procedure. At the very best, they may be designed for a specific culture and go through serious testing of the test to try to detect whether it is returning meaningful results. One does not know ahead of time what the shape of the curve is going to be.
I don't see the utility of standard deviation in most settings. A given applicant to join an organization has to have an IQ of, say 125 to get hired. Until he takes the test (even assuming they are unbiased), you don't know what his score will be, regardless of the probabilities. I think maybe the organization could say, "We need 500 people randomly chosen from the community with an IQ of 125. Therefore, we know we need to test X number of applicants." Similarly, Raytheon could manufacture a battery and maybe need to know what percentage of them will be bad when determining the cost of production. But in that case, couldn't they use a simple percentage?
These look pretty good, I guess. But in every case, you're still just guessing as to what will actually happen. Especially on the weather. The insurance example seems most useful https://www.statology.org/standard-...ard Deviation in,house prices they can expect.
That is interesting and helps with getting a mental image of relative scoring, but you aren't saying that this rule means that the numbers of people below the statistical mean are going to reflect the number above it, necessarily, for every standard of deviation, are you? That seems like an idealized example of symmetry. That is, 100 could still be the statistical mean with, hypothetically, more of a the 68% falling below the mean by 1 deviation or less, if say, pointwise, you saw an excess of the next two S.O.D.s, above the mean, right? I hope I expressed that clearly-- do you know what I mean (that is, what I'm saying)?
Are there companies that actually do IQ testing on applicants? Yikes! As for the battery example, it does make sense in manufacturing to simply have a cut off point. The battery has to meet some test or head for recycling (I hope!) But, I'd bet stats are used in determining what those levels actually are and in identifying what it is that causes the manufacturing process to meet or fail to meet the objectives.
I don't know about companies, but it's understandable. I don't want people with average or below average IQ's working around hazardous materials. Someone else mentioned IQ above. But somebody is checking distribution of IQ's across the population, otherwise we wouldn't know that 100 is the average. Jordan Peterson has videos talking about the military's use (at least in the past) because they figured out, according to Peterson, that people below an IQ of 83. He says this goes back to WWI. They now use the ASVB test, but it's pretty much the same thing, I think. https://work.chron.com/military-intelligence-testing-23740.html https://www.todaysmilitary.com/joining-eligibility/asvab-test/asvab-sample-questions I for one have no objection to the military requiring that inductees answer most or all of those questions correctly.
100 being the average is basically somebody's idea of what the average should be called. It's just not the case that all the various IQ tests would result in this 100 number. I'm fine with our military having criteria for applicants. I've hired a LOT of people in my career in engineering. For that, "IQ" was never a question, but believe me applicants did get grilled. BUT, that's not the issue here.
What is "the issue"? I don't care about IQ snd I didn't bring it up. I just want to know what standard deviations are (answered already) and their practical application. I imagine that no one with an engineering degree from an accredited uni has an intelligence quotient low enough to cause any employer (who is an engineer himself) any concern.
Yes, this thread has way too much IQ stuff. But, IQ was proposed as an example - and that example needed some help.
When a new IQ test comes out they are standardized using a sample of test-takers and the the average of the test results is set to 100 and their standard deviation is set to 15.
Please cite something about that. I mean, standard deviation has a definition. It is a measure of the dispersal of the result of the test, calculated in a defined way. Standard deviation is the name of a formula. You can't just change that dispersal. It is what it is.
I'm new to this and in no position to argue, but I think that since: "The Empirical Rule states that 99.7% of data observed following a normal distribution lies within 3 standard deviations of the mean. Under this rule, 68% of the data falls within one standard deviation, 95% percent within two standard deviations, and 99.7% within three standard deviations from the mean", see https://www.davidsongifted.org/gifted-blog/iq-and-educational-needs/ .... it follows that the standard deviation is not "set to 15," but rather that the majority of the population (68%), falls within one standard deviation of the mean of 100 (IQ 85-115). It happens to be 15, but if the average IQ were 150, the standard deviation would be ... well, something other than 15 on either side of 150. If I understand correctly, and I doubt that I do, anytime you have a mean of 100 in a set of normally distributed data, IQ or other wise, the standard deviation is 15. It's the empirical rule producing the "15," as opposed to it being assigned. What I cannot see is the forest; way too many trees!
This sounds like a method of comparing two different charts. It's like overlaying the new test over some other test, centering the high points and then stretching the x axis so the SD points overlay each other. But, that that method doesn't account for the actual shape of the data in the new test. Besides that, the sample audience given the test may simply give different results due to how the sample was selected or to test question design. The new test may not even show a bell shaped curve. It could have more than one high point, for example. There is no law that says a test may have only one high point. Maybe the test audience includes a group that understood the questions on the test and another group that didn't. OK, but let's be careful not to be circular. >>The data is the boss. It is what it is, regardless of rules. A "normal distribution" is defined to be the case where the data happen to show specific characteristics as you pointed out. The "empirical rule" you quoted applies to a "normal distribution" - a distribution your test results may or may not demonstrate. For any test I run, I don't know ahead of time whether the results will be a "normal distribution". I would have to look at the data and see if it matches the criteria for being such a distribution. You won't know if that IQ test give to some number of people results in a normal distribution until you calculate the mean and the standard deviation, etc. and then determine whether it qualifies as a "normal distribution". An IQ test someone designed and then applied to some group of individuals COULD result in a distribution that shows more than one high point or has the left side stretching out farther than the right side does, or has some other characteristic that doesn't qualify as being a normal distribution. One has to live with the result. The data is the boss. One can study why it turned out that way, what it may mean about the test, what it may mean about the way the audience was selected, etc., but you don't get to change the way people answered the questions during the test in order for some "rule" to be fulfilled.
Okay, thanks. I also explored this very problem with a friend in the context of baseball. Suppose -- just work with me -- you have 500 pitchers in a league and every single one of them -- except one -- is discovered to have an e.r.a. of exactly 3.5, the one outlier having an e.r.a. of 8.0. That's an abnormal distribution, and if you plot it out, it's going to make for a funny looking curve. I don't say slider (rimshot). But I get lost when you say this: You won't know if that IQ test give to some number of people results in a normal distribution until you calculate the mean and the standard deviation, etc. and then determine whether it qualifies as a "normal distribution". I didn't know that you could even have a 'standard deviation" to calculate absent a normal distribution.
When intelligence quotient (IQ) tests are initially standardized using a sample of test-takers, by convention the average of the test results is set to 100 and their standard deviation is set to 15 or 16 IQ points. When IQ tests are revised, they are again standardized using a new sample of test-takers, usually born more recently than the first. Again, the average result is set to 100 "The cognitive impact of the education revolution: A possible cause of the Flynn Effect on population IQ". Intelligence.
Well, in reading the wiki on the Flynn Effect, it seems this is the study of claimed sustained increases in IQ of various populations over time. So, the method you mention does have to do with a method of comparing test outcomes rather than measuring IQ. That wiki doesn't really describe the method well, nor does it justify the normalization method. Besides that, the wiki seems to present a pretty lax attitude toward designing and testing new IQ tests. As I've understood it, that's a huge issue. I'd trust this published study of the Flynn Effect more than the wiki. It uses metanalysis of large numbers of existing IQ tests as well as other testing that has been used over time. https://www.pnas.org/content/115/26/6674#sec-3
That is an interesting conundrum. Can a binomial distribution be a normal distribution? You can have as many trials as you want with a binomial distribution but there are still only two variables. You can have a million trials but there are still only two variables. And something magical happens when you have a sample size over 30.
@ Le Chef: Here's the basic information you're seeking. For a characteristic which produces a graph which is roughly in the shape of a bell curve, 1 standard deviation includes the central 68% of the numbers. 2 sd's includes the central 95% and 3 sd's includes the central 99.5%. These numbers should be considered as approximations only. Be aware that not very many sets of numbers form a 'true' bell curve. Also, the smaller the group of numbers, the less certain we can be about them and their 'true' distribution. [And yes, statisticians have ways of dealing with sample size, too.] Hope this helps. Regards, stay safe 'n well.
