Confused about mean and standard deviation for samples

[theBEAST]

Homework Statement

Here is a slide in my notes:

I am kind of confused about mean and standard deviation. So in my notes it says X1 to Xn are independent measurements. Then it says each independent measurement has a mean μ. But how is this possible, if they are independent measurements (in order words 1 measurement) how can you take the average of one measurement. Wouldn't it just mean the mean of one measurement is equal to the same value of the measurement? And is μ_{X_bar} is the average of all the samples X1 to Xn? Is what I have said so far correct?

Moving on, it says there is some uncertainty for each independent measurement, this kind of makes sense to me. How I see it is if we make a length measurement with a ruler, the uncertainty on it may be +/- 0.5 mm assuming the smallest increments on the ruler is 1 mm. Thus, each independent measurement will have the same uncertainty. Is my interpretation correct? I feel kind of lost right now.

 

[Ray Vickson]

Homework Statement

Here is a slide in my notes:

I am kind of confused about mean and standard deviation. So in my notes it says X1 to Xn are independent measurements. Then it says each independent measurement has a mean μ. But how is this possible, if they are independent measurements (in order words 1 measurement) how can you take the average of one measurement. Wouldn't it just mean the mean of one measurement is equal to the same value of the measurement? And is μ_{X_bar} is the average of all the samples X1 to Xn? Is what I have said so far correct?T

Moving on, it says there is some uncertainty for each independent measurement, this kind of makes sense to me. How I see it is if we make a length measurement with a ruler, the uncertainty on it may be +/- 0.5 mm assuming the smallest increments on the ruler is 1 mm. Thus, each independent measurement will have the same uncertainty. Is my interpretation correct? I feel kind of lost right now.

Independence just means that the measurements do not affect one another. If X_1 happens to be above the mean μ this will not in any way affect whether X_2 lies above the mean or below the mean, etc. Here, "mean" is a numerical value of a parameter that specifies the functional form of the probability distribution that generates the observations; it is the so-called "expected value" of any single measurement. For example, if I toss a fair coin and X = number of heads, then in a single toss I observe either X = 1 or X = 0, each with probability 1/2. The expected value is μ = 1/2. This does not imply that I can expect to get 1/2 a head on each toss; The word "expected" here has technical significance and does not need to match ordinary conversational meanings of the word.

 

[theBEAST]

Independence just means that the measurements do not affect one another. If X_1 happens to be above the mean μ this will not in any way affect whether X_2 lies above the mean or below the mean, etc. Here, "mean" is a numerical value of a parameter that specifies the functional form of the probability distribution that generates the observations; it is the so-called "expected value" of any single measurement. For example, if I toss a fair coin and X = number of heads, then in a single toss I observe either X = 1 or X = 0, each with probability 1/2. The expected value is μ = 1/2. This does not imply that I can expect to get 1/2 a head on each toss; The word "expected" here has technical significance and does not need to match ordinary conversational meanings of the word.

Oh okay, so does X mean sample? In other words X1 could be a sample of 4 measurements, X2 would be another sample of 4 measurements, etc?

Because for the central limit theorem, the sigma is the population standard deviation. So I am assuming that the value in the example (0.0564) is the uncertainty in the population of length measurements...?

 

[Ray Vickson]

Oh okay, so does X mean sample? In other words X1 could be a sample of 4 measurements, X2 would be another sample of 4 measurements, etc?

It depends on the context. In one group of measurements we might have X_i = number of heads on the ith toss or a coin (so X_1 = 0 or 1, X_2 = 0 or 1, etc.) In another experiment we might have X_i = total number of heads in the ith trial of tossing a fair coin 10 times, so X_1 = 0 or 1 or 2 or 3 or ... or 10, X_2 = 0 or 1 or 2 or 3 or ... or 10, etc. In another type of situation, X_i might be the diameter of the ith piece coming off a production line. In still another type, X_i might be the average diameter of the ith group of 5 objects coming off the production line, etc. What X_i is depends on how the experimenter or measurer wants to structure the situation.

 

[theBEAST]

It depends on the context. In one group of measurements we might have X_i = number of heads on the ith toss or a coin (so X_1 = 0 or 1, X_2 = 0 or 1, etc.) In another experiment we might have X_i = total number of heads in the ith trial of tossing a fair coin 10 times, so X_1 = 0 or 1 or 2 or 3 or ... or 10, X_2 = 0 or 1 or 2 or 3 or ... or 10, etc. In another type of situation, X_i might be the diameter of the ith piece coming off a production line. In still another type, X_i might be the average diameter of the ith group of 5 objects coming off the production line, etc. What X_i is depends on how the experimenter or measurer wants to structure the situation.

In this example, does X represent a single measurement and the standard deviation related to one measurement is 0.0564?

 

[haruspex]

In this example, does X represent a single measurement and the standard deviation related to one measurement is 0.0564?

The example given versus the general statement it exemplifies is a bit confusing.
Yes, in the example each Xi is a single measurement. It is not explained how it is known that the mean and uncertainty have the given values, you just have to take it on trust. And they are to be taken as precise and inherent to the entity being measured, not deduced from sample imprecise measurements.

There are several features not to like in the example.
First, what is meant by uncertainty here? It seems they mean one standard deviation, but I'm not aware of any such defined usage. To an engineer, it means the absolute limit on the error, but in that case having multiple samples would not lead to a revised estimate of uncertainty with the formula given.
Further, there could be a systematic error. If I measure a length using a gauge which is only calibrated to millimetres then I can only measure it to the nearest mm, giving a systematic error of +/- 0.5mm. I might get the same reading a hundred times, but the estimate of error does not diminish.