UAH v6.1 Global Temperature Update for September, 2025: +0.53 deg. C

They don’t even understand what sample size is! Single measurements of different things don’t create a large sample size. It creates a large number of samples of size 1. Thus when you try to calculate the standard deviation of the sample means it becomes the standard deviation of the concatenated samples.

SEM = SD/sqrt[n] –> SEM = SD

since sqrt[1] = 1

“Sample size means the size of the sample – it’s as simple as that.”

It means a SAMPLE of the same thing. As per the start of this sub-thread you can’t take a measurement of cholesterol from one person and a measurement of blood pressure from a different person, concatenate them into a single data set and say you have created a sample of size 2.

Similarly you can’t take a measurement of temperature from Las Vegas and a measurement of temperature from Miami, concatenate them into a single data set, and then say you have one temperature sample with a size of two. What you actually have is two temperature samples of size one. Therefore the SEM becomes the SD of the two data points since the sqrt[1] = 1.

It’s like measuring the bore size of a 350cu Chevy, a 426 Chrysler hemi, and a 40hp/4cylinder Farmall tractor engine and saying you have a sample size of 3. And then saying the accuracy of the average bore size is the SD of the three measurements divided by sqrt[3]. SD/sqrt[3] is *NOT* the accuracy of the average, it is how precisely you have located the average value – two entirely different things. You actually have three samples of size one – so again the SEM is the SD of the values.

You are, once again, exhibiting the meme of a blackboard statistician that thinks “numbers is just numbers”.

“ A random independent sample can provide accurate information about the population”

No, it can’t, not if the sample is of reasonable size. The clue is that you have to ASSUME that the SD of the single sample is the same as the SD of the population in order to calculate an SEM. You have no good way to tell if the sample SD is the same as the population SD, you just have to ASSUME that it is. The kicker is that if the SD of the sample *IS* the same as the SD of the population then the SEM is probably useless since it’s highly unlikely that a sample that has a different mean than the population will have the exact same SD as the population.

The clue here is that the SEM is defined as the standard deviation of a set of sample MEANS, “means” as in plural. If you only have one sample then you are only ESTIMATING what the actual SEM is. Estimating a value actually ADDS to the measurement uncertainty, i.e. increases the SEM. How much it increases it is the big question.

None of this actually changes what the SEM is – a metric for how precisely you have located the population mean. It is *NOT* the accuracy of the population mean. Even with VERY LARGE systematic measurement uncertainty the SEM can be made very small by using large sample sizes.

This is the big problem with climate science. Climate science assumes that a small SEM implies an accurate mean value. It doesn’t imply that at all. It requires the application of your typical meme of “all measurement uncertainty is random, Gaussian, and cancels”.

“t represents how many units from a population”

““Single measurements of different things don’t create a large sample size.””

bellman: “Wrong.”

Nope. Not wrong. Single measurements of different things are *NOT* units from the same population. Single measurements of different things are single samples of many different populations.

You have *still* not figured out that you can’t take a corral full of a mix of Shetland ponies and quarter-horses and say that single measurements of each form a sample of a “population”. The values at best will form a multi-modal distribution in which standard deviation is useless as a statistical descriptor. It’s no different than mixing single measurements of temperature in the NH and SH together – you wind up with a multi-modal distribution in which the SD is useless. If the SD is useless then the SEM is useless as well since it is derived from the SD.

Single measurements of different things under different environmental conditions will *NOT* produce a Gaussian distribution of random values that can be assumed to cancel. If they don’t cancel then the SEM can’t be used as the measurement uncertainty either. And it is the SEM that gets divided by sample size, not the measurement uncertainty.

Single values of different things represent a sample size of 1. That single measurement is the total population for that “different thing”.

“It describes how close any one sample of a specified size is likely be to the population mean. It is not describing the mean of multiple samples.”

The concept of the SEM is built upon the central limit theorem. The CLT theorem holds that the means of multiple samples, when combined into a data set, will tend to a Gaussian distribution. The mean of that Gaussian distribution will thus be the best estimate of population mean and how precise that estimate will be is determined by the size of the samples.

A single mean from a single sample can *NOT* generate a Gaussian distribution of sample means. Therefore the CLT doesn’t apply. That means that an SEM GUESSED at by using the SD of the single sample as the population SD has a huge uncertainty of its own built in because you simply can not know how closely the SD of the single sample represents the population SD. If you *do* know the difference between the actual population SD and the sample SD then the SEM becomes useless because knowing the SD of the population means you also know the average of the population.

If the SEM obtained from a single sample has a in-built uncertainty then how can it accurately represent the uncertainty of the average? What factor do you use to add to the SEM to represent its own in-built uncertainty?

“It very much does. It allows you to assume that given a large enough sample the sampling distribution will be close to Gaussian.”

Huh? The CLT says you can get a Gaussian distribution from a non-Gaussian distribution if you just use enough data points from the non-Gaussian distribution in your sample? That means that if you have the entire population and your “sample” is the entire population that the CLT will make it a Gaussian distribution somehow?

What in Pete’s name kind of magic does the CLT do in order to change a non-Gaussian distribution into a Gaussian distribution?

“Sampling distribution.

You keep demonstrating that you do not understand what you are talking about.”

You can’t have a sampling distribution made up of means from multiple samples without having multiple samples.

One sample, made up of elements extracted from a population, REMAINS ONE SAMPLE WITH ONE SAMPLE MEAN. No sample distribution!

The CLT simply doesn’t apply when you have one sample no matter how many elements you have in that sample. The CLT does *NOT* say that any single sample, regardless of size, will tend to a Gaussian distribution. That would imply that a single sample of large size extracted from a very skewed parent distribution would tend to Gaussian instead of the skewed distribution of the parent distribution. If that were the case then sampling would simply be unusable.

Daily summer temps in Kansas average about 27C with a σ=6 and SEM of 4. Daily winter temps average about 3C with a σ=6 and an SEM of 4.

If this doesn’t represent a bi-modal distribution then I don’t know what would. Carry these into monthly averages and you will *still* have a bi-modal distribution of temperatures. . And the σ and SEM will carry forward as well for each mode. You won’t actually know the monthly average to within 4C for either mode or for the combined distribution. In fact the σ for the combined distribution will go to 13C and the SEM goes to 7C for the bi-modal distribution. The SEM does *NOT* go to 13/sqrt[12] < 4.

Conclusion? Thinking that two daily observations can adequately represent the daily temperature distribution is ignorant at best and fraudulent at worst. Thinking that averaging averages can reduce the SEM of the underlying data is ignorant at best and fraudulent at worst.

Anomalies don’t help. Anomalies don’t change the σ and SEM of the data, they merely shift the distribution along an axis. Smaller numbers “sound” like they are more accurate but they really aren’t. It’s because the measurement uncertainty doesn’t get shifted down, it actually goes UP. Changing 10C +/- 0.5C to 5C by subtracting 5C doesn’t change the measurement uncertainty to less than 0.5C. You just wind up with 5C +/- 0.5C, the accuracy actually gets *worse* since the relative uncertainty goes from .5/10 = 5% to 0.5/5 = 10%!

Climate science gets around this by just assuming all measurement uncertainty is random, Gaussian, and cancels. So do you. You just won’t admit it.

“And there, once again, is your problem. A sampling distribution is a probability distribution. The probability distribution tells you the probs ility of getting a single sample mean, and tells you what you would expect to happen if you took an infinite number of samples. But you do not need to estimate by literally taking a large number of samples.”

A single sample does not make up a “sampling distribution” or a “probability distribution”. It makes a single sample of multiple elements from the parent distribution. It is *still* a single sample!

“It’s really strang that you’ve become an expert in statistics yet have never heard of the concept of estimating a confidence interval from a single sample.”

Once again – a single sample does *NOT* create a sample distribution. It creates a single sample made up of chosen elements from the parent distribution.

It was *you* that provided the link about “guessing” vs “estimating”. It says: “To guess is to believe or suppose, to form an opinion based on little or no evidence, or to be correct by chance or conjecture.”

One sample of a parent distribution is LITTLE TO NO EVIDENCE. The CLT cannot be used in such a case. All you have to offer is that you believe or suppose that a single sample always adequately represents the parent distribution standard deviation which also implies that you already know the parent distribution average. That’s a GUESS!

It’s just part and parcel with your ingrained meme that all measurement uncertainty is random, Gaussian, and cancels.

You forgot to add in the introduction to the wikipedia link:

“In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. For an arbitrarily large number of samples where each sample, involving multiple observations (data points), is separately used to compute one value of a statistic (for example, the sample mean or sample variance) per sample, the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample (i.e., a set of observations) is observed, but the sampling distribution can be found theoretically.” (bolding mine, tpg)

“In many contexts” simply doesn’t include measurements. It’s an outgrowth of the statisticians worldview that “numbers is just numbers” and that “multiple observations” have no measurement uncertainty.

It goes on to say: “Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean x¯ for each sample – this statistic is called the sample mean. “

Again, multiple samples.

From your second link:

“The sampling distribution of a statistic is the distribution of all possible values taken by the statistic when all possible samples of a fixed size n are taken from the population. It is a theoretical idea—we do not actually build it.” (bolding mine, tpg)

Once again you need multiple samples to form a sampling mean.

It goes on to say:

“We take many random samples of a given size n from a population with mean µ and standard deviation σ. Some sample means will be above the population mean µ and some will be below, making up the sampling distribution. ”

Again, “many random SAMPLES”.

A single sample does *NOT* give you a sampling distribution. Not even based on your own links.

Are you now going to try and claim that you were misunderstood? That you didn’t mean a single sample but the mean of multiple sample means?

“The population standard deviation is usually estimated from the sample SD, but that hardly amounts to “guessing”.”

Of course estimating is guessing! If it wasn’t then your estimate would always be the true value which means it wouldn’t be an estimate!

It’s why measurements are given as “best estimate +/- measurement uncertainty”. The “best estimate” is a guess at the true value. It’s why any subsequent measurement that falls into the measurement uncertainty interval can be considered to be a best estimate as well!

You *still* have the blackboard statistician’s bias that the average is always the “true value”. It isn’t. Never has been and never will be. That belief is a result of statistical analysis teaching methods that never address measurement uncertainty. It’s a fundamental lack of understanding as to what a distribution is and what the standard deviation represents. It’s where the meme of “all measurement uncertainty is random, Gaussian, and cancels” originates.

“Then why not call it an estimate. You know full why why you try to suggest it is just a guess.”

Because guessing ADDS uncertainty, it doesn’t lessen it! Calling it “estimating” doesn’t change that simple fact.

Your link is just garbage. Picking the winner of a horse race based on past performance of the horses is STILL guessing. The problem is that you can *never* have perfect knowledge of all of the confounding factors. That’s where your link goes wrong.

From the link: “The distinction between the two words is one of the degree of care taken in arriving at a conclusion.”

Pure BS. You can take all the care you want, it doesn’t matter. If you don’t *know* the true value then you are guessing at what it is. An informed guess is *still* a guess. And guessing adds uncertainty, it doesn’t lessen it.

From the link: “To guess is to believe or suppose, to form an opinion based on little or no evidence,”

That’s like saying you *could* guess that a horse not in the race might win the race! It simply doesn’t apply to measurement and metrology. The measurement uncertainty interval limits the choices one can make, just like knowing what horses are actually in the race. Estimating the true value within that measurement uncertainty interval *is* just a guess.

“The same pathetic lies”:

No, it isn’t. You simply don’t understand the concept of probability. You assume the average of a Gaussian distribution is the value that will *always* happen, that it has a probability of 100%. The truth is that using the average *is* a BEST ESTIMATE. It is *not* the only possible estimate nor is it the true value. It is *still* just a GUESS as to what the true value is.

“You are making the same contradiction as Pat Frank. You are talking about a 95% interval but also claiming there is no corresponding distribution.”

Do *YOU* know what the distribution IS?

We are back to where you started with this garbage. The *uncertainty* interval is part of the Great Unknown! Until you can accept that very basic fact you’ll never understand metrology. Your meme of “all measurement uncertainty is random, Gaussian, and cancels” is so ingrained in your brain that you can’t even conceive of the reality of uncertainty. You simply don’t know that the true value is the mid-point of the interval, i.e. the “average” value, which would be the case if the distribution is random and Gaussian. You can’t even conceive of the truth that the distribution might be asymmetric and skewed where the true value might be clear at one end of the interval.

Saying that the 95% interval is where the true value probably lies is *NOT* telling you anything about where in that interval the true value is because you simply don’t know what the distribution inside the interval is.

There is a reason that the experts say that any subsequent measurement that falls within the interval can be considered as acceptable and reasonable. If you *know* the distribution then you would have to say “that result is less reasonable than this one” instead of just saying “that result is reasonable”.

“That only makes any sense if you think the standard deviation is describing a probability distribution, and a Gaussian one at that.”

Knowing where 95% of the measured values exist is *NOT* assuming anything about the probability distribution of the measurements. Again, that distribution is simply part of the Great Unknown!

The standard deviation only tells you the average distance of the values from the mean. It really tells you nothing about the distribution itself. Even a highly skewed distribution will have a mean and a standard deviation.

If yo u*really* know the distribution inside the interval then you should be able to provide the 5-number statistical description for the distribution. Can *you* do that?

“An individual measurement is not a value attributed to the measurand. The values that it’s reasonable to attribute to the measurand are defined by the “experimental standard deviation of the mean”. Not the range of measurement values.”

Your lack of reading comprehension skills is showing again. If a measurement, *any* individual measurement, lies within the uncertainty interval then it IS a value that is reasonable to attribute to the measurand. If an individual measurement lies outside the uncertainty interval then it is an indication that the measurement process needs to be examined to determine why. Is the environment different? Is there an unidentified systematic uncertainty? Have all contributing factors been included in the uncertainty budget? Does the uncertainty interval need to be revised?

There *is* a reason why experimental measurement results require that the same measurand be measured multiple times using the same instrument under the same environmental conditions. You simply cannot determine a usable uncertainty interval by measuring different things a single time under different environmental conditions. It’s a fundamental reason why measurement uncertainty gets larger when considering single measurements of different things under different environmental conditions instead of reducing based on the SEM. Averaging single measurements of different things using different instruments under different environmental conditions does *NOT* reduce measurement uncertainty as you, bdgwx, and climate science believe.



“which only makes sense if you are assuming a Gaussian distribution.”

Nope. The standard deviation tells you the spread of the data around the average, regardless of the shape of the distribution. If you are using the average as the best estimate then the SD will tell you the spread of the data around that value.

“Could you please for once explain exactly what you mean by that?”

Your crystal ball is extremely cloudy. It’s useless for accurate predictions. It represents the Great Unknown. All you know is that your future exists somewhere in the cloudy ball.I suppose you frequent carnival fortune tellers who claim they can penetrate the Great Unknown and tell you what your future holds?

” I’ve said that this can be an accepted way of handling epistemical uncertainty, but it is not the way any of the texts you use treats measurement uncertainty.”

More malarky! It’s *exactly how all the texts treat measurement uncertainty. It’s why you can’t extend measurement resolution beyond what your instrument and its measurements provides – anything past that limit is part of the Great Unknown! It’s why the stated value is an ESTIMATE, and not the actual true value!

It’s why the GUM says: “Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of
the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively” (bolding mine, tpg)

“It tells you the average (biased) spread. It does not tell you the percentage of the distribution within a given interval.”

Exactly what do you think we’ve been telling you? You are now reduced to just repeating what you’ve been told already! It’s why, in a skewed distribution the mode can bias a single sample’s determination of the average! It’s why the SD of a single sample cannot be assumed to be the same as the SD of the parent distribution. The CLT and the LLN tells us that *multiple* samples, even if they are smaller than the single sample, will have a sample distribution that tends to Gaussian and give a better approximation of the mean of the parent distribution.

There are two main issues here.

1. Even if multiple samples give a better indication of the parent average, that average may not mean anything if the parent distribution is skewed.
2. You can’t just assume the parent distribution of a set of measurements is Gaussian. The actual distribution is part of the Great Unknown.

It’s why your memes of “all measurement uncertainty is random, Gaussian, and cancels” coupled with “a single sample is always iid with the parent distribution” are so wrong-headed.

“So to be clear, when you say the great unknown you just mean a probability distribution”

NO, NO, No, …..

It means you simply do not *know* what the probability distribution *is*! Why is that so hard for you to get into your head? You are just stuck on assuming everything is Gaussian. Even if the distribution *is* Gaussian, that doesn’t imply that a measured value that is different from the mean is not a “true value”. *ANY* value in the distribution could be the “true value” just like a long shot horse *could* be the winner in a race!

All the measurement uncertainty interval is for is to tell you if a measured value is reasonable or not. It can *NOT* tell you which measured value is more of a “true value” than a different measured value.

That true value is just part of the Great Unknown. Always will be.

GUM: “Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively” (bolding mine, tpg)

“unknowable quantities” — part of the Great Unknown!

“Have you ever tried putting these words into a form that makes sense?”

I have. Twice. Here it is again. In a skewed distribution the mode is the most frequent value. As you say, the mean, the mode, and the median can all be different. When you are sampling guess what value has the highest probability of being chosen, even randomly? THE MODE.

That means the value of your mean of your single large sample will be biased away from the parent mean and toward the parent MODE!

In fact, this can happen even with multiple smaller samples of smaller size. The bias just won’t be as pronounced. The SEM of the multiple smaller samples will partially account for this. With a single sample you *have* no SEM from a sampling distribution.

“The SD of a sample cannot be assumed to the same as the population, regardless of the distribution.”

If the SD of a sample cannot be assumed to be the same as the population then how can it be used to calculate an SEM as you assert?

“Of course the average of multiple sample means of a given size will be more accurate than a single sample of the same size”

Oh, now you are changing your story, eh? So will one large sample of 400 will estimate the mean of a parent as well as 20 samples of size 20?

Now tell us how we’ve all misunderstood what you were saying!

“As I suspected you are just engaging in handwaving and intuition to arrive as a false conclusion.”

There is no handwaving or intuition involved at all – except from you so that you can dismiss it.

from copilot:

———————————————————
does a skewed parent distributon affect samples taken from the parent distribution

Yes, a skewed parent distribution does affect the samples taken from it — especially when the sample size is small. The shape of the population influences the variability, symmetry, and reliability of sample statistics like the mean and standard deviation.
📉 How a Skewed Distribution Affects Samples1. Sample Mean Variability Increases

• In a skewed population, sample means are more likely to vary widely from the population mean.
• For example, in a right-skewed distribution (like income), many samples may underestimate the mean unless they include extreme high values.

2. Sampling Distribution Reflects Skew for Small Samples

• When sample sizes are small, the sampling distribution of the sample mean tends to mirror the skewness of the population.
• This means the sample means themselves will be skewed, not symmetric.

3. Central Limit Theorem (CLT) Mitigates Skew with Larger Samples

• As sample size increases (typically n≥30n \geq 30), the sampling distribution of the mean becomes approximately normal, even if the population is skewed.
• This makes the sample mean more reliable and less sensitive to the skew of the parent distribution.

4. Bias in Small Samples

• While the sample mean remains unbiased in expectation, small samples from skewed populations can produce systematically misleading estimates in practice.
• This is especially problematic in fields like insurance or finance, where extreme values are common.

————————————————-

The most important part here is Item 1: in a skewed population sample means (implying multiple samples or you wouldn’t have “means”, you would only have one mean) the means of those samples will vary widely from the population mean.

This is where the CLT comes in. With *multiple* sample means of adequate size those sample means will form a Gaussian distribution even if the population is skewed.

But, and this is a *BIG* but, the standard deviation of those sample means will still indicate a wide interval of uncertainty for the actual population mean.

With just one sample you will have no way to estimate what the uncertainty interval associated with your estimate of the population mean actually is.

You can argue till you are blue in the face that all you need to estimate the statistical descriptors of a parent distribution is one single sample but it’s just not the case. And the size of that single sample doesn’t matter.

The CLT requires multiple samples of the parent distribution and that is all there is to it. Without the CLT at play all you can do is what you are doing, just ASSUME with no justification that the sample is iid with the parent. Just like you always assume that all measurement uncertainty is random, Gaussian, and cancels. That way you can ignore the uncertainty associated with your guess at the parent mean and standard deviation – your guess will always be 100% accurate.

“Yes a small sample size will have less certainty with a skewed distribution”

Your lack of reading comprehension skills are showing again!

copilot says:

• In a skewed population, sample means are more likely to vary widely from the population mean.

Do *YOU* see the word “small” in there anywhere? I sure don’t!

If all you have is ONE SAMPLE, then you are assured that the skewness will cause the mean of that sample to vary widely from the population mean.

“But you were claiming that with a large sample size the mean would be closer to the population mode.”

Nothing copilot said disagrees with my assertion. In fact, copilot’s statement

• “When sample sizes are small, the sampling distribution of the sample mean tends to mirror the skewness of the population.”

implies that the SAMPLING DISTRIBUTION, even with small sizes, will tend to mirror the skewness of the population! THAT IS EXACTLY WHAT YOU WANT!

You seem to be saying that it is *better* to have a sampling distribution that does *NOT* mIrror the skewness of the parent population!

AND, we are right back to you trying to assert that a single sample can create a sampling distribution. ONE VALUE DOES NOT MAKE A DISTRIBUTION!

“Yes, that’s what more uncertainty means.

How do you measure the uncertainty of a sample distribution when you only have one value and no sample distribution?

“It’s in the passage before it.”

And here you are, cherry picking AGAIN!

“ampling distribution”

For the umpteenth time – HOW DO YOU GET A SAMPLING DISTRIBUTION WHEN YOU ONLY HAVE ONE SAMPLE WITH ONE MEAN?

Are you *ever* going to actually answer that simple question?

“the sampling distribution of the sample mean tends to mirror the skewness of the population.”

Have you even got the faintest of clues as to what this sentence is actually saying?

“But it’s amusing how you keep dismissing statistics, yet then take some statistically generated text as an authority.”

I’ve given you MULTILPLE quotes that are *NOT* AI generated. They *all* say the same thing. In order to have a sampling distribution you need multiple values which form that distribution.

Yet here you are, still claiming that one sample with one mean forms a sampling DISTRIBUTION.

You are a troll, pure and plain.

from wikepedia:

“In other words, suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the probability distribution of these averages will closely approximate a normal distribution.” (bolding mine, tpg)

==> “performed many times” implies multiple sample means

————————————————————-

from researhdatapod.com

“In statistics, the CLT bridges individual data and population parameters. It allows us to make inferences about the population from a sample by demonstrating that, with a large enough sample size, the distribution of sample means becomes normally distributed. This principle supports many inferential statistical methods, making it one of the most widely used theorems in the field.” (bolding mine, tpg)

“distribution of sample means” – you need more than one for a distribution
———————————————————–

from Taylor, Appendix E

Table E1. Results of a large number of experiments, α = 1, 2, 3, …, each of which consists of N measurements of a quantity x. The ith measurement in the αth is denoted by x_αi and is shown in the ith column of the αth row.”

“For each of the infinitely many experiments, we can calculate the mean and variance. For example, for the αth experiment (or αth “sampling”) we find the sample mean xbar_α by adding all the numbers in the row of the αth experiment and dividing by N.”

–> to simplify, what Taylor is doing is getting N measurement samples, each of which consists of multiple measurements – i.e. N samples of the measurand. This is confirmed by the following:

“We want to show that, based on N measurements of x, the best estimate for the true width σ is the sample standard deviation of the N measurements. We do this by proving the following proposition: If we calculate the sum of squares SS_α for each sample α and then average the sums over all α = 1, 2, 3, …, the result is (N-1) times the true variance σ^2:”

When Taylor says “N measurements” he is talking about N sampleS of the measurand.

Your assertion that a single sample extracted from a parent distribution will be iid with the parent distribution, and will be Gaussian as well, just doesn’t pass the common sense test.

My guess is that you are playing your usual game of Equivocation. You continually use the term “uncertainty” for both multiple measurements of the same thing using the same device under the same conditions” and for single measurements of different things using different devices under different conditions. You bounce back and forth between the definitions of “uncertainty” as needed in the moment without ever stating that you are doing so.

That’s probably what is happening here. You use the term “sample” to mean one experiment with multiple measurements and to mean “a collection of experimental measurement means obtained from multiple experiments”.

You change the definition as needed in the moment = the classic Equivocation argumentative fallacy.

It’s either that or it’s your lack of reading comprehension skills leading to an inability to differentiate what is meant when you see the word “sample”.

“You are lost in the trees. All you are doing is repeating explanations about what a sampling distribution means, not how you can use it on practice. “

You just keep on believing that ONE SAMPLE data set extracted from a parent distribution will be iid with the parent distribution.

Just don’t ask me to use anything you design that might affect human safety.

““A sample” singular. That’s all you need to use the CLT to make an inference about the population.”

You are either using the Equivocation argumentative fallacy our you truly believe that one set of data extracted from a parent distribution will *always* be iid with the parent distribution.

Again, just don’t ask me to use anything you design that might affect human safety.

“Do you seriously think he’s saying you have to perform an infinite number of experiments in order to establish the uncertainty of the mean?”

I’ve never once used the term “infinite”. All I’ve ever asserted is that one data set extracted from a parent distribution can’t form a DISTRIBUTION. All you get from one extracted data set is ONE MEAN. One mean does not a sample distribution make. There *IS* a reason for performing multiple experiments thus obtaining multiple samples of the parent distribuiton.

“No he isn’t. He is very obviously talking about a sample of size N, that is N individual measurements of the thing being measured”

What he is saying is SPECIFICALLY laid out in the text. Your lack of reading comprehension skills is atrocious.

Taylor: “Results of a large number of experiments, α = 1, 2, 3, …, each of which consists of N measurements of a quantity x.”

Taylor: “For example, for the αth experiment (or αth “sampling”) we find the sample mean xbar_α by adding all the numbers in the row of the αth experiment and dividing by N.”” (bolding mine, tpg)

Each experiment IS A SAMPLE of the parent distribution of size N with an associated mean. “large number of experiments” *IS* multiple samples (experiments) providing multiple sample means. Their means bracket the mean of the parent distribution. Those multiple sample (experiment) means provide a distribution. The mean of ONE of those experiments will *NOT* form a distribution, it is just one value.

“αth experiment” – multiple samples

Learn to read. Your cherry picking simply leaves you with no actual knowledge of what is actually being said.

If your ONE SINGLE SAMPLE is not iid with the parent distribution then it can’t give you an accurate representation of either the mean or the standard deviation of the parent distribution.

It is truly just that simple.

You can argue till you are blue in the face. The CLT requires a distribution of sample means to work, commonly known as the sampling distribution. One sample can only give you one mean. One mean does not make a distribution.

wikipedia: “For example, consider a normal population with mean μ and variance σ2. Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean x¯ for each sample – this statistic is called the sample mean. The distribution of these means, or averages, is called the “sampling distribution of the sample mean”. This distribution is normal N(μ,σ2/n) (n is the sample size) since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not (see central limit theorem).” (bolding mine, tpg)

All you’ve done in this trolling sub-thread is basically say that all of the references I’ve given you are wrong. That all you need is ONE sample to find the mean and standard deviation of the population.

Like I’ve said several times, I hope I never have to use something you’ve designed that can affect human safety.

“I’ve explained to you many times why this is meaningless, explained that you don’t understand what iid means, given you the correct meaning, and explained why you would not what the sample to be independent of the population, and you just keep repeating the same nonsense.”

iid is INDEPENDENT, SAME MEAN, SAME STANDARD DEVIATION, AND SAME SHAPE.

You simply won’t admit this. All you ever focus on is the “independent” piece! And all that the term “independent” means is that each selected value from the parent for use in the sample does not depend on any previous choice of value from the parent. It has nothing to do with the sample being independent of the parent!

You have *NO* guarantee that your single sample has the same mean, same standard deviation, and same shape as the parent distribution. And the CLT won’t apply because you only have one value to use because you only have one sample.

With multiple samples the mean of the sample means converges on the mean of the parent distribution due to the CLT. And the interval of uncertainty around that estimate of the population mean is the standard deviation of the sample means.

With one single sample you have to ASSUME the standard deviation of the sample is the same as the population standard deviation in order to estimate how accurate your estimate of the population is. You get stuck in a circular loop – assume the sample SD is accurate so the mean is accurate because the assumed SD is accurate.

It’s painfully obvious what you are doing. You are calling a sampling distribution a “sample”. Your single “sample” is actually made up of multiple samples, each with a mean and standard deviation. Just as with “uncertainty” you can’t bring yourself to use unequivocal terms. It would tend to invalidate your ability to defend your crazy assertions. You can’t bring yourself to admit your assertion that a single SAMPLE accurately estimates a parent mean does 1. not meet the requirements for the CLT to apply and, 2. does not guarantee being iid with the parent.

Equivocation, i.e. changing the definition of a term without notification. Using the term “sample” to describe a single set of data extracted from a parent distribution AND as multiple sets of data extracted from a parent distribution *is* equivocating. And it is one of your favorite tactics.

“Saying the sample is iid with the population is meaningless, unless you are treating each as a random variable.”

If the single sample does *NOT* match the mean, standard deviation, and shape of the parent distribution then it *cannot* accurately estimate those values for the parent distribution.

Your assertion that it can is just garbage.

“That’s what it means in the context of taking a random sample. But you are trying to apply it to the distribution of the sample in relation to the population.”

If each piece of the data in the parent distribution is not independent of all other pieces then the data selected for the single sample cannot obtain independent data no matter what selection process is used. Some of the data selected will not be independent from other pieces of data that is selected.

If the parent distribution are MEASUREMENTS then they should be random and independent of each other. If they are not such then a correlation term must be included in the propagation of measurement uncertainty. This makes the results for the mean from the single sample even more uncertain.

“you still don’t get that a mean of means will have the same value as the mean of all values.”

Your single sample will *not* be the mean of all values or it isn’t a sample at all but the parent distribution! Nor is the mean of the sample means actually the value of the mean of the parent, i.e. the same value as the mean of all values. The standard deviation of the sample means is the estimated interval in which the actual parent mean might lie. It is a metric for how precisely you have located the parent mean.

“If you think the standard deviation of a large sample is a poor estimate of the population standard deviation, it also follows that the standard deviation of a set of sample means will be a poor estimate of the standard deviation of the sampling distribution.”

The standard deviation of the sample means is *NOT* an estimate of the standard deviation of the parent distribution. It is a metric for the interval in which the mean might lie. It’s a metric for how precisely you have located the mean of the population.

You simply do not understand what the SEM *is*.

Your assertion is that the single sample will have no uncertainty associated with it regarding the statistical descriptors of the parent so no uncertainty interval surrounding the estimate of the mean. That’s based on your assumption that the standard deviation of the single sample is also the standard deviation of the parent.

If the standard deviation of the single sample is *not* the same as that of the parent you are left with no way to estimate the uncertainty interval surrounding your estimate of the parent mean because you have already assumed that you know the parent standard deviation which implies that you also know the parent mean. You *must* have a sampling distribution (i.e. multiple samples) in order to estimate that uncertainty interval – i.e. the true SEM.

“No. You assume it’s an estimate of the population standard deviation. How good an estimate depends on the nature of the population distribution and the sample size.”

If the single sample standard deviation is an estimate (i.e. a guess) then it will have an uncertainty interval. How do you find that uncertainty interval from an assumed accurate value for the population SD? You will wind up in the same circular logic if you assume it is the (sample SD)/N – “assume the sample SD is accurate so the mean is accurate because the assumed SD is accurate. “

As usual, you leave left out measurement uncertainty. With one single sample your equation for the SEM should be (sample SD)/N +/- SD_uncertainty. How do you come up with that SD_uncertainty factor?

“No, that’s just an estimate of the sampling distribution.”

It’s a metric for SAMPLING ERROR. One single sample can’t tell you what the value of the sampling error is!

“The only circle is in your brain. You don;t assume the mean is accurate – that’s why you divide by (N – 1).”

If the estimate of the SD is inaccurate then you can’t correct the inaccuracy by dividing by (N-1). You are *still* assuming that the SD of the single sample is the same as the SD of the population!

” If I’ve ever called a sampling distribution a sample, then point out where I said it”

I didn’t say that *you* actually said it. You are using the argumentative fallacy of Equivocation to avoid actually saying it. Just like with the term “uncertainty”. You never actually define what uncertainty you are using, uncertainty in the estimate of the mean or propagated measurement uncertainty – that way you can always say that the reader didn’t understand what you actually said.

“I’ll take that as an admiration you were wrong to use the term iid.”

Huh? Saying that a sample is *NOT* iid because it doesn’t have the same mean, standard deviation, and shape of the parent is wrong?

Your lack of reading comprehension skills is showing again.

“The point is that a sample will tend to match the population, tending to get closer to the parent as sample size increases.”

if taking multiple samples is too expensive then how is taking one sample which approaches the size of the parent population not too expensive as well?

“That’s why you want to know the SEM.”

You are caught in your circular logic again. You can’t find an SEM from one sample without ASSUMING the sample is iid with the parent. i.e. the sample SD and the Parent SD are the same. If the sample is iid with the parent then the SEM should be zero. You can’t get out of this circular logic without assuming the sample is *NOT* iid with the parent. But that means that the SEM calculated from the sample SD is not accurate either – and you have no way to estimate how inaccurate it is. The CLT doesn’t help. The CLT says the sampling distribution gets more accurate for the population mean as the sample size grows regardless of the shape of the parent distribution, the samples do not have to be iid with the parent distribution.

“Your willfully misunderstanding what I’m saying. The mean of means will have the same value as the mean of all values in the complete sample.”

If you have a complete sample then you have the parent distribution! The SEM will be zero. And how does that jive with your excuse that taking multiple samples is too expensive to do? You seem to be saying that measuring everything is less expensive than measuring just part.

“They are not, at least not in the way you mean it. The population is the set of all possible values you are sampling. “

Those values will be of the form: “estimated value +/- measurement uncertainty”. As usual you are ignoring the measurement uncertainty by using your meme of “all measurements are random, Gaussian, and cancel”.

Your sample will consist of values in this form. This is accounted for in multiple samples by the CLT approaching a Gaussian distribution where it is justified to assume that the standard deviation defines the interval in which the population mean will lie – just like measuring the same thing multiple times with the same instrument under the same conditions let’s you assume the standard deviation of the values is the measurement uncertainty of the mean.

“Yes. If there’s a dependency on your sampling that needs to be accounted for. In measurements that might be because of s systematic error. In other cases it might be a bias caused by the sampling process.”

With just one sample how to you measure this bias without assuming the sample is iid with the parent? Meaning no sampling error!

“Not this again. A standard deviation is not an interval. It can be used to establish an interval.”

The standard deviation describes the interval around the mean in which the possible values might lie. You’ve been given images showing this multiple times. The standard deviation is the square root of the variance. The solution to a square root *always* has a positive and negative solution. √4 = +2, -2 Thus a +SD and a -SD. You are stuck in the blackboard statistics world that the mean is the only possible value of a distribution. It isn’t. A probability distribution implies that *any* value in the distribution is possible. +SD is an interval. -SD is an interval. +SD to -SD is an interval.

GUM: “uncertainty (of measurement)
parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand

NOTE 1  The parameter may be, for example, a standard deviation (or a given multiple of it), or the half-width of an interval having a stated level of confidence.”

The dispersion of values includes those values below AND above the mean. Not just the mean. And not just the values on one side of the mean.

“Numbers is just numbers” is garbage from the blackboard world.

“This whole discussion is about how much uncertainty there is in a sample.”

And there is no way to ascertain that uncertainty from a single sample without getting into a circular loop. You certainly haven’t laid out any way to ascertain it without the circular loop of: Sample is iid with parent –> mean is iid with parent –> SEM is zero

“Speaking of Taylor, seeing as you did all his exercises, can you point me to any exercise or example where he illustrates the need to take multiple samples in order to determine the SDOM? All the ones I see only take a single sample. E.g. exercise 4.17.”

I keep telling you that you need to study the entirety of Taylor’s book and work through all the Examples. But you never do. You just continue to cherry pick things you think support your assertions but which, when taken in context never do.

See Examples 5.31 – 5.32.

In 5.31 you are given 40 measurements, four rows of 10 measurements each.
——————————————–
(a) Compute the standard deviation σ_t,for the 40 measurements. (b) Compute the means {,,…, ¢;) of the four measurements in each of the 10 columns. You can think of the data as resulting from 10 experiments, in each of which you found the mean of four timings. Given the result of part (a), what would you expect for the standard deviation of the 10 averages 1,,…, ty)? What is it?

———————————————-

Why do you think Taylor wants you to find the average of 10 experiments with four measurements each if not to get a distribution of sample averages?

If you read Section 5.7 he says:

“Because the calculated quantity (x_bar) is a simple function of the measured quantities x_1, …, x_n, we can now find the distribution of our answers for x_bar by using the error-propagation formula. The only unusual feature of the function (5.64) is that all the measurements x_1, …, x_n happen to be measurements of the same quantity, with the same true value X and the same σ_x.”

Thus this is the restrictive, limiting situation where only random measurement uncertainty exists for multiple measurements of the same thing using the same instrument under repeatable conditions.

He goes on to say: “Thus, after making many determinations of the average x_bar of N measurements, our many results for x_bar will be normally distributed about the true value X.”

Remember, this whole section is about proving that in the case of solely random measurement uncertainty for a single measurand, x_best is the average value of multiple measurements of the same thing by the same instrument under repeatable conditions.

The method of obtaining multiple samples, finding the mean of each, and then determining the mean of those sample means applies to *any* case, not the limited, restrictive case.

You are *still* stuck in the blackboard statistical world where everything is random, Gaussian, and cancels. And a single set of measurements is always iid with the parent.

Someday you need to learn to list out *ALL* of the assumptions and requirements needed for your assertions to apply. A “measurement uncertainty budget” if you will. But I’m not going to hold my breath waiting for it to happen.

“Demonstrating that you can read every word and still not understand the point.”

You simply cannot read, can you? The ENTIRE purpose of the example is to compare and contrast the population statistical descriptors with the statistical descriptors determined from four samples of size 10!

“There’s no suggestion that this is how you would take the uncertainty of the mean, when you already have a sample of 40, with a SEM of it’s sd / √40.”

The 40 pieces of data is the POPULATION. You have four samples of size 10 from the population!

“And you keep ignoring all the examples where the uncertainty of a single sample is taken, something you claim is impossible. Look at exercise 5.33. Eight measurements as the sample”

You *still* can’t read!

From 5.25:

“(a) Assuming these measurements are normally distributed, what should be your best estimates for the true value and the standard deviation? ”

From Section 5.5:

————————————–
Now is a good time to review the rather complicated story that has unfolded so far. First, if the measurements of x are subject only to random errors, their limiting distribution is the Gauss function G_x,σ(x) centered on the true value X and with width σ. The width σ is the 68% confidence limit, in that there is a 68% probability that any measurement will fall within a distance o of the true value X. In practice,
neither X nor σ is known. Instead, we know our N measured values x_1, …, x_N where N is as large as our time and patience allowed us to make it. Based on these N measured values, our best estimate of the true value X has been shown to be the mean x_bar = Σx_i/N , and our best estimate of the width σ is the standard deviation σ_x of x_1, …, x_N
as defined in (5.45).
Two further questions now arise. First, what is the uncertainty in x_bar as an estimate of the true value of X? This question is discussed in Section 5.7, where the uncertainty in x_bar is shown to be the standard deviation of the mean, or SDOM, as defined in Chapter 4. Second, what is the uncertainty in σ_x, as an estimate of the true width σ? The formula for this “uncertainty in the uncertainty” or “standard deviation of the standard deviation” is derived in Appendix E; the result proved
there is that the fractional uncertainty in σ, is

(fractional uncertainty in σ_x = 1/ √2(N-1)

————————————————-

I gave you an excerpt from Appendix E to study. Apparently you didn’t bother. I didn’t figure you would.

Here it is again:

“We want to show that, based on N measurements of x, the best estimate for the true width σ is the sample standard deviation of the N measurements. We do this by proving the following proposition: If we calculate the sum of squares SS_α for each sample α and then average the sums over all α = 1, 2, 3, …, the result is (N-1) times the true variance σ^2:”

The issue here is that, yes, you *can* estimate the population statistical descriptors from a single sample BUT the uncertainty of the result is significantly greater than using multiple samples, even if the samples are smaller in size!

You try to use examples based on using a single set of measurements to find a best estimate of the true value of a measurand used in a SINGLE experiment where the measurement uncertainty is assumed to be random, Gaussian, and cancel to justify using ONE single sample for everything, whether the parent distribution is Gaussian, skewed, or anything else.

You are *still* cherry picking instead of actually studying the subject. And you *always* wind up making assertions that have no actual relationship with reality.

“Typical Tim. He finally accepts he was wrong, but buries it under so much waffle, and then implies it’s what he was trying to tell me all along.”

Typical bellman. Can’t read for context, just a master cherry picker! You left out my statement after the “BUT”.

“The issue here is that, yes, you *can* estimate the population statistical descriptors from a single sample BUT the uncertainty of the result is significantly greater than using multiple samples, even if the samples are smaller in size!” (bolding mine, tpg)

If you don’t mind highly INACURRATE estimates then, yes, you *can* GUESS at what the mean of the parent is from a single sample. BUT you leave yourself no way to calculate just how inaccurate your GUESS is.

*YOU* asserted that a single sample allows ACCURATE estimates of the parent mean. That assertion is just plain garbage.

“But he still doesn’t get that the only way multiple samples area better estimate is becasue you are taking a much larger sample.”

Which also applies to your SINGLE sample! Again, you can’t read!

Multiple samples of SMALLER size gives a more accurate estimate than a SINGLE larger size!

Even a large single sample mean can be modulated by the mode in a skewed distribution. Multiple smaller samples minimize the modulation and more accurately determines the mean of the parent.

You are *still* trying to argue that a single sample is better than multiple samples. It’s a garbage assumption only useful in blackboard statistical world and in climate science.

“you will probably get a better estimate by taking 30 such samples, but it will only tell you how uncertain an individual sample of size 30, not the uncertainty of the sample of size 900 you’ve actually taken.”

The SEM is a metric for SAMPLING UNCERTAINTY. Multiple samples allow you to accurately estimate this. A single sample does not. A single sample requires assuming the sample SD is equal to the parent SD – which then leads to also assuming that the mean from the single sample is accurate since it’s derived from an assumed accurate parent SD.

Taking 30 samples of 30 costs no more time than one sample of 900.

Multiple samples allow a metric to be developed from the standard deviation of the sample means.

Your single sample of 900 gives you ONE VALUE for the mean. You have no way to accurately estimate how close that one mean is to the population mean.

You didn’t bother to work out Taylor’s example did you?

Why do you think everyone defines a sampling distribution

from illinois.edu: “We just defined a few statistics that we could use to summarize a sample. What would happen if we took a second sample from the same population? Realistically, we would get a different set of observations in our second sample. From this different set of observations, we could recalculate the sample statistic. Would we get the same statistic? We probably would not calculate the exact same value for the statistic. Instead, we would likely get something similar to our first calculation. What do we actually know or expect for different values that our sample statistic can take from sample to sample? We can answer this question by studying sampling distributions.” (bolding mine, tpg)

With just one value from one sample for the mean you *must* assume it is accurate. You have no other choice since you have no method for estimating how uncertain it might be. Assuming the sample SD is the same as the parent SD just *adds* uncertainty which, again, you have no way to estimate.

It isn’t a matter of it not being a probability distribution. It’s a matter of NOT KNOWING what the probability distribution *is*.

Beyond that, even if you could somehow *know* the probability distribution you *still* won’t know the actual true value. This goes back to the issue that you claim you don’t think the average is the true value – BUT THE FACT IS THAT IS *EXACTLY* what you think. It’s obvious in everything you assert!

it goes to the fact that a specific horse may have a low probability of winning a race – BUT IT STILL MIGHT WIN! That horse might just be the true value for that particular measurement of which horse is the fastest!

It’s just like the meme of “all measurement uncertainty is random, Gaussian, and cancels” is so ingrained in your brain that you can’t even discern how it determines every assertion you make. It’s the same for the average being the true value – it’s so ingrained in your brain that you don’t even realize how it flavors your assertions!

Just like the horses in a horse race determines the possible winner of a race, the measurement uncertainty interval determines the possible values of the measurand. You just don’t know which value is the “true value”. It’s part of the great unknown. You’ll never understand why some people bet on a long shot horse or on a pair of deuces in a poker game.

“You could have saved a lot of effort if you had just stated that, rather than using weird phrases like “the Great Unknown”. The point still stands. If you say:”

I have stated it *MANY* times. What do you think the words “you don’t know” mean?

“You’re still struggling with the concept that if something is uncertain, it has no certainty”

No, this is EXACTLY what we’ve been trying to pound into your brain for two years!

The uncertainty interval only shows that you do not KNOW the true value. It doesn’t mean the true value doesn’t exist! The true value of a measurand CERTAINLY exists, that doesn’t mean that *you* know what it is!

“That’s your lie. I do not think that. I’ve never thought it. It’s an insane thing to think. It exists only in what’s left of your mind.”

You can deny it all you want. It just shines through in everything you post.

“You do understand the irony of you making repeating that inane sentence in all your posts, and then claiming I’m the one who has it stuck in their brain – don’t you?”

I am not the one asserting that a single sample of a parent distribution is always Gaussian!

” Why do you think I would describe an uncertainty interval, or confidence interval or whatever, if I think that I knew the true value.”

Because you assert that an ESTIMATE is not a guess! That the average is not a GUESS but is the actual value!

“Because they’re hopeful idiots.”

See what I mean? You don’t believe that anything besides the average will ever happen! So why bet on something that isn’t the average!

“ If one assumes”

Your lack of reading comprehension skill is showing again.

The standard deviation tells you the spread of the data around the mean regardless of the shape of the distribution. In a skewed or multimodal distribution, however, the mean is of limited value. Therefore the SD is of limited value as well.

You *do* realize that you just undercut your assertion that a single sample will always be iid with the parent, right?

“If each input’s are estimated from the mean of several measurements, then their associated estimated standard deviations, are those of the mean. See example H2.”

Example H2 is based on repeatable measurements, i.e. multiple measurements of the same thing under the same conditions.

Perhaps you should read *all* of H2 and stop cherry picking.

“If f is not a linear function, then the results of approach 1 will differ from those of approach 2 depending on the degree of nonlinearity and the estimated variances and covariances of the Xi.”

You are *still* stuck in thinking that all measurements are random, Gaussian, and cancel leaving the SEM as the measurement uncertainty. That isn’t even true if the observations are used for determining the measurement uncertainty of a non-linear function!

This whole forum is based on learning about CLIMATE SCIENCE where the measurements used are *NOT* experimental observations, i.e. multiple measurements of the same thing using the same instrument under repeatable conditions. Yet you continue to try and force the discussion into analyzing experimental observations that are considered to be random, Gaussian, and whose individual measurement uncertainties cancel.

You are the prototypical internet troll – wasting everyone’s bandwidth on garbage which has no actual application to the subject at hand.

“ Or you can use methods that directly estimate the uncertainty of the standard deviation.”

What method might that be? Could it be the propagation of meaureement uncertainty as we’ve been trying to get you to learn?

“That factor you never take into account is that multiple samples just mean more sampling. If you take 20 samples of size 20, you have to make 400 measurements, and if you can afford to do that you can just as well treat it as a single sample of size 400.”

A single sample of 400 is *NOT* equivalent to 20 samples of size 20. The CLT addresses this and you don’t seem to be able to get that into your head!

A single sample of 400 is *not* guaranteed by any math to be Gaussian if the parent distribution is Gaussian. This means that you simply don’t know if the average of that single sample adequately represents the parent average. Nor do you know if the sample SD is the parent SD. You just have to *assume* that it is.

If you take 20 samples of size 20 the CLT says that you *will* generate a Gaussian distribution from the means of those 20 samples. And the standard deviation of those sample means tells you how closely you have located the parent average.

“Not what I had in mind, but if you can explain how to use measurement uncertainty propagation, to estimate the uncertainty of the standard deviation a sample, please elaborate.”

Even after literally years of having this explained to you, you continue to fail to grasp even the most basic concepts of metrology.

“a sample” is *not* the same as “multiple samples”. The measurement uncertainty of the total length of two 2″x4″x8′ boards, each measured one time using different devices, ADDS to a total measurement uncertainty. The total length is *not* used to determine the experimental standard deviation of the measurements of “a sample”.

Technically there should be a measurement uncertainty budget prepared for each board including things like manufacturing tolerance, instrument measurement uncertainty, moisture content, temperature, etc. Those then contribute to the measurement uncertainty of the total length. The measurement uncertainties ADD.

“More words of wisdom from someone who admits he knows nothing about statistics.”

In other words you have no refutation for the assertion.

“Yes, in the sense that a sample of size 400 has a SEM equal to SD / √400. Whereas each sample of size 20 has a SEM of size SD / √20.”

You *STILL* don’t get it! SD / √400 requires using the assumption that the sample SD is the same as the population SD. If that assumption holds then you *also* know the population mean since it is used to determine the population SD. In that case the SEM is useless. It implies that you already know the population SD and mean.

The CLT can be applied when you have 20 samples. I.e. the means of those 20 samples will approach a Gaussian distribution, the mean of which is a better estimate than the mean of a single sample where you have to ASSUME what the SD is. The mean of the sample means gives a smaller sampling error than the mean of one sample where you have to ASSUME its SD.

It’s why I asked you what factor you have to apply to SD / √400 to account for the fact that it is based on a GUESS – which you just blew off and never answered.

“It’s not guaranteed, but it’s almost certain to be as close as makes no difference. Much more likely than your sample of 20.”

Malarky! You are literally suggesting that ALL single samples will tend to a Gaussian distribution. Even single samples from a highly skewed parent distribution.

And it’s not just a sample size of 20. It’s 20 SAMPLES of size 20. The math, i.e. the CLT, applies. The CLT does *NOT* apply to a single sample!

“The distribution of the population and the sample is irrelevant. You should have figured that out by now. The adequacy of that sample mean is determined by σ / √400, i.e. σ / 20.”

The distribution of the population and of the sample CERTAINLY applies! You have *NO* idea of what the sampling error actually is from one single sample. The CLT simply doesn’t apply!

Your assertion that all samples are Gaussian is just part and parcel of your meme that all measurement uncertainty is random, Gaussian, and cancels. You just can’t get that out of your head!

“It says the sampling distribution will tend to a Gaussian distribution as sample size increases.”

Bullshite! The CLT says that the mean of the sample means will tend to be Gaussian EVEN FOR SMALL SAMPLE SIZES. The SD associated with those sample means will be wider with small sample sizes than with larger sizes. Meaning the sampling error will be worse for small sample sizes than for large sample sizes.

But the CLT does *NOT* say that single samples will tend to be Gaussian as you assert!

“And then, as you only have 20 samples, there’s no “guarantee” that those 20 samples will accurately represent the sampling distribution.”

The SD of the multiple sample means *IS* the sampling distribution! It *is* the SEM!

“It’s likely to be very close. Much close than your 20 samples of size 20.”

Nope. Even moderate sized samples (30 to several hundred) can miss the parent distribution statistical descriptors if the parent is skewed. The mode, the most frequently encountered value, can pull the average indicated by a single sample away from the true average because it is more likely to show up in a single sample than the actual average. That also means the SD obtained from the single sample will not be accurate.

All you are doing here is trying to defend the illegitimate assumptions made by climate science to justify ignoring measurement uncertainty and its propagation.



“One sample is a value taken from the sampling distribution. You can estimate what that sampling distribution is from your one sample. This is explained in any elementary text book.”

No, you can’t. You admitted that when you posted that the SD of a skewed distribution may not have the 68% of values!

You are still stuck in your catch-22. If ONE SAMPLE IS A VALUE taken from the sampling distribution then your sample size is 1 (ONE)! That means that n = 1. If n = 1 then there is *NO* SD to use in the calculation of the SEM!

“You can demonstrate that the CLT works by using a computer to generate multiple samples. But that doesn’t mean you would do that in the real world, becasue in the real world sampling costs money and can have ethical implications.”

Total and utter bullshite! Why do you think scientists perform MULTIPLE experiments with MULTIPLE observations trying to confirm theoretical hypotheses?

Why do you think MULTIPLE items are pulled off of production lines and tested to use in quality control processes?

You *truly* have no idea of how the real world works do you?

“The only important assumptions are that the sample is iid”

How do you *KNOW* a sample is iid with the parent distribution? This is an assumption that you simply cannot make in the real world of measurements! It’s the typical fantasy world of the blackboard statistician where this assumption works!

“That the population has a finite standard deviation, and for the student-t distribution that the population is normal.”

And here we are, once again, with you assuming everything is Gaussian!

“It does not mean that all samples, or for that matter measurement uncertainties are random, Gaussian.”

You just said above that the population has to be normal, i.e. Gaussian, for your assumption to work. So how do you *know* that a distribution of single measurements of different things taken by different instruments under different conditions is *normal*?

“I keep forgetting you are psychic. You obviously know my mind better than I do, as I’m sure I’m always considering non-Gaussian distributions, including for measurement uncertainties.”

You’ve already admitted that the SD of a skewed distribution doesn’t give 68% of the possible values. You’ve admitted elsewhere that the mode of a skewed distribution can bias the mean of a sample. How then can the sample be iid with the parent distribution?

It’s obvious that you do *NOT* consider non-Gaussian distributions in *any* of your assertions.

“They will be the same in both cases. If you have 400 values and take their average you will get exactly the same mean as if you split the 400 values into 20 samples,”

This is *NOT* what anyone is saying. The 20 samples have to be taken from the parent distribution! Not 20 samples taken from a different sample!

If you make your sample of 400 values into the parent distribution then of course 20 samples of *that* parent distribution should mimic the mean of the 400 values. That’s what the CLT says!

But that does *NOT* mean that the 20 samples will mimic the parent distribution the 400 values are drawn from!

Did you *really* think you were going to fool anyone with this trick?

“This is just getting desperate. Firstly, I was talking about non-Gaussian distributions, not specifically skewed ones. You know, those types of distributions you claim I don’t believe exist.”

Now you are just waffling! Do you really think that measurement uncertainty is never skewed? You’ve been given enough examples of where uncertainty *is* skewed true that it should have sunk in.

Basically you are trying to say that *all* measurements fit your blackboard, limited examples. Hogwash of the worst kind!

“And more importantly, that’s the whole point of the CLT. It says that with a sufficiently large sample size, the sampling distribution will be close to a Gaussian distribution.”

Now you have changed your tune. Did you really think this was going to fool anyone?

Once sample does *NOT* a sampling distribution make. One sample of 400 values produces ONE value, not a sampling distribution.

Did you finally figure out that you were wrong?

“Maybe you should have read the introduction”

Your lack of reading comprehension skills is showing again.

“In statistics, the CLT bridges individual data and population parameters. It allows us to make inferences about the population from a sample by demonstrating that, with a large enough sample size, the distribution of sample means becomes normally distributed.”

I have bolded the part that applies here. A *single sample* does not give a “distribution of sample means” – PERIOD, EXCLAMATION POINT.

You are confusing the rule that multiple IID random variables used as observations converge to the actual parent distribution. This simply doesn’t apply to single measurements of different things using different devices under different conditions. It may not even apply to multiple measurements of different things using the same device under the same conditions if the different things produce non-iid observations.

In fact, it may not even apply to multiple measurements of the same thing using the same device under the same conditions unless the (usually unstated requirement) measurements are made over a short period of time. If this is not adhered to then there is no guarantee that unidentified random fluctuations will not ruin the IID assumption. See Bevington for why sample sizes cannot be unlimited, random fluctuations will produce outliers that bias the statistical descriptors.

You are fighting a losing battle here. You keep trying to use blackboard examples with highly restrictive requirements (which you never bother to list) for describing how the real world of metrology works.

There *are* situations where non-iid samples can converge to the population mean but certain conditions have to be met. You’ve not done anything to show how these conditions are met or even list out what the conditions *are*! And, in fact, they *all* only apply where you have multiple samples!

Your reading comprehension skills are atrocious.

Why do you keep ignoring “ the distribution of sample means”?

The sample size has to be large enough to adequately fit the requirements for the CLT to hold. That is *all* that is being said about “a sample”. But you need *multiple* samples of sufficient size to create a distribution of sample means for the CLT to apply.

You are still caught up in a catch-22. It seems you want to claim that your single sample is actually a set of sample means. If that is true then your sample size becomes 1 (one) and the SEM becomes the SD of the data set!

So, do you have a data set of sample means or a data set of observations drawn from a parent distribution?

“t’s just another way of saying the sampling distribution of the mean.”

what does the word “distribution imply?

“It’s a probability distribution giving you the probability of any specific mean”

What does the word “distribution imply?

“You can use a large number of samples to estimate it if you can afford it, but it isn’t necessary as any text book will explain how you can estimate it from a single sample.”

The issue isn’t being able to estimate it, it’s how ACCURATE that estimate is. If that single sample isn’t iid with the parent then your use of the single sample will produce an INACCURATE estimate.

The CLT says that if you use multiple samples you will get an accurate estimate of the population mean and the accuracy will be given by the standard deviation of the means of the multiple samples.

With a single sample you don’t get a standard deviation of sample means, you have only ONE value with no way to estimate just how accurate that one value is.

“Not so. The CLT is a description of how the sampling distribution tends to normal as sample size increases. There is no margic number where it suddenly becomes true.”

Your lack of reading comprehension skills is showing again. I did *NOT* say there is a breakpoint, I *said* there is an adequate size that is needed.

And you need MULTIPLE means in order to even have something that forms a distribution, let alone a Gaussian one. A single sample, regardless of how large it is, does not create a distribution.

“Stop making stuff up. That is not what I’ve claimed.”

Then where does your sampling distribution come from? Do you think a single value, all by itself, forms a distribution?

“events “

i.e. plural

One event is not “events”.

One sample doesn’t create a sample distribuiton.

*I* didn’t miss it. What do you think the long shot horse is? You have a problem relating with the real world!

Nope. You keep saying that ONE EVENT is all you need. One sample, one event. One measurement of a measurand.

One event does not a distribution make. EVENTS make a distribution.

Are you now admitting that ONE single sample of a parent distribuiton will *not* be either iid with the parent distribution or Gaussian?