Mathman said:Here is a "population" of five numbers: 9, 10, 10 10, 11. The average (arithmetic mean) is 10. So far so good.
Here is another population: 0, 5, 10, 15, 20. The average of this population is 10, too.
So how are the two populations different, if they both have the same average? Well, in the first population, all the numbers are close together. They are all very close to the mean. In the second population they are spread apart. Some of them are a long way from the average. The variance measures, "how far from the mean are these numbers?"
So for each number, we look and see how far from average it is. The numbers 9, 10, 10, 10, 11 are, respectively, -1 unit, 0 units, 0 units, 0 units, and 1 unit away from the average of 10. Because we don't really care about the + or - sign, we square these differences, then take the average. That's the variance:
Application If the data are normally distributed then we expect about 68% of the data to be within plus or minus one standard deviation of the mean, and we expect about 95% of the data to be within +/- 2 standard deviations of the mean. So for our first population we expect about 68% of the data to be in the range 10 +/- 0.63 -- that is, between 9.37 and 10.63 -- and we expect 95% of the data to be in the range 10 +/- (2*0.63), or within the interval 8.74 and 11.26.
degrees of freedom
the concept of bias
The standard deviation has many uses besides just as a descriptive measure of variation. For instance, if you want to estimate the mean of a population by knowing the mean of a sample, a 95% confidence interval for the true mean is given by the formula,
true mean = sample mean +/- 1.96 sigma/(square root of n),
where sigma is the standard deviation.
Mathman said:Michelle skates her SP at three different events. I time the performances with my Citizens' watch. The times are
2:38, 2:40, and 2:42.
So the average time is 2:40, but once she went under by 2 seconds and once she went over by 2 seconds.
Gezando, actually there is a reason (in fact, two) why I don't like to use that convention about nested parentheses. The first is that I want to reserve braces { } to enclose the elements of a set, and I want to reserve brackets [ ] for html commands, like bold, color, etc. I don't want to confuse the GS vBoard software, LOL.
The second reason is that not all programming languages read these symbols as parentheses in mathematical or logical statements. But they do all handle nested parentheses ((((((((...)))))))) very easily. It's just we poor humans that need help in matching up which pairs of parentheses go together.
Exercise. In seven performances , Michelle's SP went
2:45, 2:47, 2:47, 2:48, 2:46, 2:47, 2.47
for a mean of 2:47 and a standard deviation of 1.3 seconds. What is the probability that she will go over the new time limit of 2:50 in Moscow?
Answer: About 1 per cent.
bronxgirl said:OK, we could discuss latest theories in antibiotic resistance patterns, and maybe I'd stand a chance here
That's the very point I tried to make with the judges at Dortmund, but they didn't listen to me!gezando said:The 2 seconds + or - observed by you is a function of the Citizens (watch) not MK