topic Re: How do I interpret Central Limit Theorem? in Discussions

How do I interpret Central Limit Theorem?

AAzad — Thu, 04 Jan 2024 15:11:12 GMT

The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.

For continued process verification (CPV) or other statistical analysis, we assume that n=30 or more will ensure normal distribution and this "n" (sample number) is generally represented by the number of batches. For example, assay of an API (active pharmaceutical ingredient) in a drug product from 30 different batches is considered normally distributed. According to above definition, the assay of each batch has to be calculated mean of adequate number of samples (sample size) for each and every batch and not the composite sample assay or assay of one tablet or capsule. Hence the assumption of normal distribution on the basis mean coming from one individual assay is not right. In this regard, dissolution can be considered better candidate for the application of central limit theorem. For each mean value of dissolution time point, at least we use 6 units or more (L2, L3). Any comment will be appreciated.

Re: How do I interpret Central Limit Theorem?

Mark_Bailey — Thu, 04 Jan 2024 16:16:32 GMT

The CLT specifically says that the sum of random variables asymptotically approaches a normal distribution. The arithmetical mean happens to be the sum of the variables divided by their frequency, so the CLT applies to this mean as well.

The assumption you cite is a common 'rule of thumb' about the minimum sample size for asymptotic behavior. The sample size actually depends on the third moment, skewness. The literature contains articles that state a formula to calculate n, given the skewness. You could also use JMP to simulate the sampling distribution for different n.

Re: How do I interpret Central Limit Theorem?

dlehman1 — Fri, 05 Jan 2024 12:04:42 GMT

One addition to Mark_Bailey's comment. The rule of thumb for sample size and the CLT also depends on kurtosis. When kurtosis is high, much larger sample sizes are required before the sampling distribution of the mean approaches a normal distribution. As with most rules of thumb in statistics, they are reliable until they aren't. Simulating data to explore this is often a good way to see how the sample size is related to the sampling distribution.

I want to reiterate a complaint I have about the redesign of the Community (since it doesn't seem to belong anywhere else). To comment now requires that I get a code first. I used to save my login information and could quickly comment. Now, I had to create a new login because my university email quarantines the code for a day, and then go to my other email to get the code in order to comment. For me, at least, this is a more cumbersome process and makes me less likely to engage in the discussion. Also, I really don't see the importance of the multi-factor authorization here - it seems like an unnecessary concern. That is part of another ongoing complaint I have about IT departments in general - MFA has become the mantra of IT departments everywhere, regardless of how users feel about it. Frankly there are many websites that I don't care about my identity being exposed.

Re: How do I interpret Central Limit Theorem?

AAzad — Fri, 05 Jan 2024 14:28:47 GMT

Thanks a lot Mark. Could you please add the literature reference that you mentioned?

Re: How do I interpret Central Limit Theorem?

Mark_Bailey — Fri, 05 Jan 2024 15:45:59 GMT

One specific reference is Sugden, Smith, and Jones (2000) “Cochran’s rule for simple random sampling.” Journal of the Royal Statistical Society: Series B (Statistical Methodology). 62: 787–793.

Re: How do I interpret Central Limit Theorem?

Mark_Bailey — Fri, 05 Jan 2024 15:48:17 GMT

I do not mean to argue with your point about kurtosis. My understanding is that the issue arises due to asymmetry in the distribution. Kurtosis alone does not affect symmetry. Thanks for bringing up this point.

Re: How do I interpret Central Limit Theorem?

dlehman1 — Fri, 05 Jan 2024 16:05:55 GMT

Here is one paper I found on kurtosis and the CLT: https://www.umass.edu/remp/Papers/Smith&Wells_NERA06.pdf. The most usable reference I know of is in a textbook (Statistics for Business, Robert Stine and Dean Foster, Addison Wesley). In the 2011 edition of their book, they provide sample size conditions referring to skewness and kurtosis which I have found useful:

n> 10*skewness squared and n>10*absolute value of kurtosis, where the skewness and kurtosis are measured from z scores of the data. I once queried them about the origins of these conditions and the best they could recollect was that they based these on running many simulations. If anybody has a more precise reference, I'd be interested to see it.

Re: How do I interpret Central Limit Theorem?

hogi — Fri, 05 Jan 2024 16:06:29 GMT

it's still single factor: the password is replace by a verification code (sent via email)
Nevertheless - compare to a password that is save in the browser settings - it feels like MFA due to the multiple steps: open email app, copy code, paste code

@Ryan_Gilmore , any chance to specify a secondary (private) email address to get the verification code 2x in parallel?

I hope a Cookie will save the credentials for a loooong long time (> 1 year?)

same for Jmp activation via myJMP verification code ...

Re: How do I interpret Central Limit Theorem?

statman — Fri, 05 Jan 2024 17:55:50 GMT

Pardon my comments and please ignore if you prefer. Perhaps I don't understand your situation, but doesn't your situation (CPV) call for analytical statistical thinking, not enumerative (see Deming, W. Edwards (1975), On Probability As a Basis For Action. The American Statistician, 29(4), 1975, p. 146-152)? I suppose your concern for normality is because you want to use the appropriate estimates for central tendency and variation? I would think what you want is a sample that appropriately represents the central tendency and variation of the API of the process for the period of time you are assessing (over the variables in the process that change during that time period). How you sample the process can have a huge effect on conclusions you draw about the process central tendency and variation. I realize I may be in the minority here, but a sample size of 30 seems quite arbitrary. Consider these possible hypothetical situations:

30 samples randomly taken from the process making batches over multiple shifts and lots of raw material
30 samples from 1 batch, 1 shift and 1 lot of raw material
30 samples, 3 samples from 10 batches and 1 shift
30 samples, 1 sample from 30 consecutive batches over 3 shifts
30 samples, 1 from 30 batches, 10 batches from each of 3 lots of raw materials
30 samples, each sample measured 3 times for 10 batches

Each may give completely different estimates of mean and variation. Each will confound or separate different components of variation.

From the above referenced paper:

“Analysis of variance, t-test, confidence intervals, and other statistical techniques taught in the books, however interesting, are inappropriate because they provide no basis for prediction and because they bury the information contained in the order of production. Most if not all computer packages for analysis of data, as they are called, provide flagrant examples of inefficiency.”

Re: How do I interpret Central Limit Theorem?

Ryan_Gilmore — Fri, 05 Jan 2024 18:27:21 GMT

For Community related issues, you can start a discussion in the Community Discussions board.

My JMP ID is the method being used throughout the JMP ecosystem, of which the Community is one part. This transition is the first step. Based on feedback such as this, we will investigate methods for improving the experience.

Re: How do I interpret Central Limit Theorem?

AAzad — Mon, 08 Jan 2024 20:29:50 GMT

Thanks a lot. Your points are valid, but I was looking for a simpler answer. I am an applied user of statistics without statistical background. It is basically statistical process controls (SPC) of pharma products in commercial manufacturing. In the process control charts, we get lot of warnings, specifically for Nelson Rules 1, 2 and 3, which are related to Special Cause of Variation. In the report, we need to explain all these warning signals and decide if we need to redesign the process to improve on those or those are due to other factors like inadequate numbers of sample size, or lack of compliance to independently and identically distributed randomly (IID) to ensure that we are not overreacting. Various articles in pharma on SPC and FDA guidance on SPC points to a sample size of 30 (mean from 30 different batches of a drug product) in order to make general assumption that the data are normally distributed. My question was: what should be the sample size of EACH of these means, say 30 means? Or, means for N1, N2, N3....N30 should be based on a sample size of what, like 5, 10, 20, 30, etc. or does not matter as long as you are using a mean sample size of N=30 (i.e. representing 30 different lots or batches) and above?

Re: How do I interpret Central Limit Theorem?

statman — Mon, 08 Jan 2024 22:20:31 GMT

Understood. Here are my further thoughts:

It is possible you are not correctly interpreting the control limits on your control charts. Let me make sure you understand the charts and their use. I will start with the Shewhart chart (aka X-bar, R charts). These charts require a rational and logical subgrouping and sampling strategy:

“The engineer who is successful in dividing his data initially into rational subgroups based on rational theories is therefore inherently better off in the long run. . .”

Shewhart

The purposes of the charts are to:

Understand whether the variation is special (assignable) or common (unassignable or random)
Determine which source of variation has greater influence on the metric being charted

The range chart is used to assess consistency/stability and hence predictability. Points beyond the control limits are evidence of "special cause" variation (Deming's term) or assignable (Shewhart) due the x's that vary within subgroup. These suggest the within subgroup sources may be acting unusually and un-predictably. It is worthwhile (economically) to study these "events". It also may not be wise to calculate control limits for the X-bar chart as the limits are un-stable or inconsistent.

The X-bar chart is a comparison chart. It compares the variation due to the x's changing between subgroup (the plotted averages) to the x's changing within subgroup (the control limits). This chart answers the question which component of variation is greater, the within (points are inside the control limits) or the between (points are outside the control limits). This is not evidence of inconsistency or instability as in the range charts (so not special per Deming).

I strongly suggest you read Shewhart and Wheeler's book on SPC and some of his published articles on Rational Subgrouping and Rational sampling (https://www.qualitydigest.com/inside/standards-column/rational-subgrouping-060115.html

https://www.qualitydigest.com/inside/statistics-column/rational-sampling-070115.html

Reference:

Shewhart, Walter A. (1931) “Economic Control of Quality of Manufactured Product”, D. Van Nostrand Co., NY

Wheeler, Donald, and Chambers, David (1992) “Understanding Statistical Process Control” SPC Press (ISBN 0-945320-13-2)

Woodall, William H. (2000), "Controversies and Contradictions in Statistical Process Control", Journal of Quality Technology, Vol. 32, No.4 October 2000

The last reference has a discussion associated with it.

Re: How do I interpret Central Limit Theorem?

MRB3855 — Tue, 09 Jan 2024 09:24:30 GMT

Hi @AAzad : WRT to the CLT...I'll restrict my comment to a short answer to your last question.

Q: My question was: what should be the sample size of EACH of these means, say 30 means? Or, means for N1, N2, N3....N30 should be based on a sample size of what, like 5, 10, 20, 30, etc. or does not matter as long as you are using a mean sample size of N=30 (i.e. representing 30 different lots or batches) and above?

A: It doesn't matter what the sample size is of each of the means. I can expand on this if you like.

And, as you can see from others' learned comments, there is a lot of nuance around your particular application. What I've answered here is the narrow question about the CLT.

Re: How do I interpret Central Limit Theorem?

Mark_Bailey — Tue, 09 Jan 2024 18:30:32 GMT

You don't mention how the run ruiles were selected for each chart. There are many rules to choose from. Like outlier tests, each is designed with the generating process (for the special cause) in mind. They should not be applied automatically to every chart. They will increase the rate of false alarms (decrease the average run length when in control).