I need to be able to understand and use the term "standard deviation" in the following context: Assume there are 1,000 judges in a state who grant or deny requested relief, in a given area of law, at certain rates. For example, some judges, let's say just 10 of them, grant only 1% of the applications they receive for restraining orders. Others, let's say 10 again, grant 99% of the applications before them. That's 20 out of the 1,000 who are at the extremes. The approval rates for the remaining 980 judges are clustered around an average approval rate of, say, 60%. If you charted these stats on a bell curve, there would be ever decreasing numbers of approval rates as you moved toward either of the two extremes (1% and 99%) of the curve. So my question is, how do you express these differences in terms of standard deviations? Or do you? Be gentle: I did not study math beyond trig, which I passed but never really understood. Yes, I've read the wiki article on standard deviations. Did not understand. If you use words like "logarithm", "pi", "cosine", or "algorithm", I won't understand. Assume you are talking to a dummy. Because you are. Thanks!
A 5 second google will give you the answer https://www.mathsisfun.com/data/standard-deviation-formulas.html
Are you sure there aren't other factors involved in the process? Are all judges equally likely to get any of the applications? Are the applications all equivalent? Are all outcomes equivalent? After all, success rate is an important measure. Do applicants have any way of affecting which judge rules on an application? Do you think the audience to whom you will present your standard deviation argument will be swayed by a standard deviation argument? Math is not going to consider any of the factors other than the numbers fed in. If you want to fire a judge, there are probably better arguments to be made.
Standard Deviation is even simpler than that equation: Picture a number line in algebra where 0 is at the center of the number line and numbers to the right of zero are positive while numbers to the left of 0 are negative. Now picture a normal curve. A normal curve is just like a number line in algebra except instead of 0 at the center the mean is at the center with observations to the right of the mean being positive and observations to the left of the mean being negative. However, let's say you're doing a study on height. The mean height is 5' 9" therefore anyone shorter than 5'9" will be a negative number. However there is no such thing as negative height so what can you do??? You square all the observations in your study which makes all your observations positive. However, squaring the terms misrepresents their distance from the mean. So what can you do??? You take the square root which gives you the observations actual distance from the mean. Then you just divide the sum of all those observations by the number of observations in the study giving you the average distance all the observations in your study are from the mean which is the standard deviation.
I think you need the actual value of each observation to calculate the standard deviation however you can determine the percentage of how many observations are within one standard deviation, two standard deviations and three standard deviations of the mean using the central limit theorem and the 68% 95% and 97.7% rule. For example, 680 of your observations will be within one standard deviation. 950 of your observations will be within two standard deviations. and 977 of your observations will be within 3 standard deviations.
OK. But, I think it's important to indicate that the standard deviation is a measure of the dispersion in a dataset. So, if a plot of the data looks like a bell curve with a high center and short tails right and left, the standard deviation will be very different than if the bell curve has a low center and long tails to right and left. (Recognizing that the data may not be shaped as a bell curve, of course.)
My thoughts exactly I do not have a huge background in statistics but I remember my first lesson was “choose the right statistic”
You cats are all smarter than I. But it's not about firing a judge. It's about whether you can speak in terms of standard deviations when describing the wide range of approval rates. If 20 of the judges have a higher than average approval rate on the applications submitted and 20 have a lower rate, cantt that be plotted on a bell curve?
Ha, no. I respect your question, but I'm 69 years old, not in school, and not going back. I just hear and read "standard deviations" come up in discussions and in articles and I never know exactly what they mean. Or even inexactly. Also I was going to an interviewast week (now over) with a panel of judges and I suspected a statistical question might come up that could be answered in terms of bell curves and deviations. Even after all the above answers, I still don't know the purpose of couching an explanation of these disbursements of data in terms of standard deviations.
Ohhh gotcha. I'm sorry I have had lots of students try to get help online from strangers before so I had to ask. The way I always understood standard deviations is that they are a way of measuring how far an answer is away from the average. It's important for regressions. But I think you know that. Honestly I'm in school now and I never understood what a standard deviation is or why it's needed. Go with Ryobi's answer. They make the most sense.
Okay, hold it ... do you use "the 68% 95% and 97.7% rule" in all calculations of the standard deviation? I thought you plotted out the saya and tha amount if disbursement determined the deviation. I think I am maybe confusing the forest for the trees, or vice versa.
Well, you plot data on a chart. The result MIGHT look like a bell curve to some greater or lesser extent. But your example points to a larger problem related to how sampling is done. As we've seen with various efforts to pretend that ivermectin works, that vaccines don't work, etc., one can find ways of collecting data that simply don't allow for finding a truthful answer to certain questions. It's long been said that one can lie with statistics. In fact, one can make huge mistakes by accident, simply by not taking care that sampling techniques result in data that can be legitimately used to address a question you want answered.
This gets into how SD can be used. One can decide what confidence intervals are appropriate for the decision being made.
The 68% 95% 97.7% (empiricle rule) is part of the Central Limit Theorem. The CLT is what the Normal Distribution is based upon. You do take your data and put it into the equation and solve for the standard deviation. But you can tell a lot of things about your data with the empiricle rule and the standard deviation. For example, we know the mean and standard deviation for IQ tests. 100 is the mean score on IQ tests and 15 is the standard deviation. Therefore, using the empiricle rule we know 68% of the people who take the IQ test will score between 85 and 115, and we know 95% of the people will score between 70 and 130 et cetera.
Since your data points are at the extremes, you wouldn't be able to calculate the standard deviation of that sample. Imagine if you did calculate the standard deviation from those data points, your standard deviation would be huge.
The value of the normal distribution is practically zero when the value lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied.
Thanks. I regret to report that I don't even understand some of the terms you are using, and I really am trying. For example, what does "highly unreliable" mean? Can you give me an example? And do you mean "unreliable" as in "it doesn't tell us anything useful"? I feel like I am trying to play Mozart without having learned Three Bind Mice first.
That is beautiful and I don't even know why. Seriously. I appreciate you guys trying to help me, but I just can't see the big picture here. Nor do I understand the terminology, like even the word "standard". I sense it doesn't have the same meaning of "standard" in the sense of "basic" or "typical." I think I am understanding that standard deviations help to make predictions, but some things are not susceptible to statistical analysis because you have to make sure that all the samples are from the same bin. You can't be throwing apples into a crop of oranges when establishing the mean weight or diameter of oranges from a given crop. Is that the idea?
Interesting. Standard deviation is a measurement technique in statistics. I suspect someone gave that technique a name. It is what it is, not somehow enhanced by the name it was given. As for your comment on data, I TOTALLY agree. For example, some of the most serious forms of blunder on issues surrounding COVID come from using data that isn't well enough selected to answer the question at hand, no matter what math is applied.
Highly unreliable would be your sample standard deviation (The standard deviation from the sample you actually took) does not reliably tell you what the population standard deviation may be. In Inferential statistics (inferential statistics is taking a sample from the population and using the sample to approximate the values in your sample to the population you are studying.) For example, your population may be the weight and standard deviation of all the apples in an orchard. In that case you could take a sample of say 30 apples from the orchard and calculate the weight and sample standard deviation of those apples and use that to make an inference about the weight and standard deviation of all the apples in that orchard.
I think you hit the nail on the head. I'm not sure, but I think in the OP,s original post with the either or scenario of a judge either granting or not granting relief the proper statistic would be a Binomial distribution.
Yes, using statistics to answer questions requires more than the application of math. Reasonable analysis starts with a valid approach to sampling. So many times on this board we see cases where the sample is not appropriate for answering the question at hand.
Yes, I'm afraid I opened it with an inappropriate hypothetical sampling. Thanks for the great answers, some of which I almost halfway understood.
