Approximating a bell-shaped curve: The central limit theorem

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From the course by University of Cape Town

Understanding Clinical Research: Behind the Statistics

715 оценки

University of Cape Town

Understanding Clinical Research: Behind the Statistics

715 оценки

If you’ve ever skipped over`the results section of a medical paper because terms like “confidence interval” or “p-value” go over your head, then you’re in the right place. You may be a clinical practitioner reading research articles to keep up-to-date with developments in your field or a medical student wondering how to approach your own research. Greater confidence in understanding statistical analysis and the results can benefit both working professionals and those undertaking research themselves. If you are simply interested in properly understanding the published literature or if you are embarking on conducting your own research, this course is your first step. It offers an easy entry into interpreting common statistical concepts without getting into nitty-gritty mathematical formulae. To be able to interpret and understand these concepts is the best way to start your journey into the world of clinical literature. That’s where this course comes in - so let’s get started! The course is free to enroll and take. You will be offered the option of purchasing a certificate of completion which you become eligible for, if you successfully complete the course requirements. This can be an excellent way of staying motivated! Financial Aid is also available.

From the lesson

Building an intuitive understanding of statistical analysis

There is hardly any healthcare professional who is unfamiliar with the p-value. It is usually understood to have a watershed value of 0.05. If a research question is evaluated through the collection of data points and statistical analysis reveals a value less that 0.05, we accept this a proof that some significant difference was found, at least statistically.In reality things are a bit more complicated than that. The literature is currently full of questions about the ubiquitous p-vale and why it is not the panacea many of us have used it as. During this week you will develop an intuitive understanding of concept of a p-value. From there, I'll move on to the heart of probability theory, the Central Limit Theorem and data distribution.

Introduction to the central limit theorem: The heart of probability theory1:52

Asymmetry and peakedness: Skewness and Kurtosis4:21

Learning from the lotto: Combinations4:24

Approximating a bell-shaped curve: The central limit theorem4:39

Meet the Instructors

Juan H Klopper
Dr
Department of Surgery

And so we get to the central limit theorem.

Now something magical and beautiful's going to happen right before your eyes.

I've mentioned that we've got a population of 10,000 individuals.

And for some variable, we know the absolute value of all of those 10,000.

I could take blood of all 10,000 certain human beings,

I could take blood for all 10,000 of them, and I know for

some blood result the exact value for all 10,000 of them.

Now imagine also that those values are completely random.

I don't follow some pattern.

Any person in that 10,000 can have for that blood result any value.

But let's just for arguments sake stick those values to be between 30 and

40 just for the graph that I'm gonna show you.

So all 10,000 of them are going to have different values between 30 and 40 for

that blood result.

Okay, now, let's say we take 30 random samples, 30 random people.

Total random, 10,000 standing in front of us, and we take 30 of them,

and we're going to look at the results.

We're going to calculate in the mean of those 30 results.

Now imagine we can do that over, and over, and over, and over again.

Every time I'm going to draw 30 different ones.

Now with some combinations, just how many you can come up with choosing 6 from 47.

Now imagine choosing 30 from 10,000.

That would be an enormous number.

Of different combinations of thirty people you could do.

Take 30 people, get their values, calculate a mean.

Now have a look at this.

There's all our 10,000.

Histogram there we see the values were between 30 and 40 and we see how many

individuals fell into the different little categories from 30 to 31, 30 to 32, etc.

So a completely random, different spread that could have any value.

But now, what is going to happen if we start plotting all those means?

So I take my 30 individuals, I calculate the mean, I've got a mean on the table.

Send them back, take another 30 at random.

Now you all know now, there's going to be billions, and billions,

and billions of combinations from those 10,000.

And I could have at random drawn any 50 of them and

that's what happens in our research.

We take 50 people at random.

And that was just a mean for them, or

the difference between the means of two groups or three groups or whatever.

We just could have done so many, 30 different, or

50 different or 100 different.

There would be so many different means we could work out.

But imagine we did that for those 10,000, drawing 30,

drawing 30 and doing the means.

And we start placing them on the table, we start stacking them.

Different means will occur more commonly than others.

And here's the beauty.

No matter how skewed that original data was,

no matter skewed an individual set of 30 was, if we could

do all those of means of all those countless,

countless, countless, countless samples of 30, this is what it's going to look like.

The distribution of those means is going to be

beautifully symmetrically bell-shaped.

And that's why we can use statistics.

And that's why we can use the p-value, the geometrical area under a curve.

Because if we stack up all the means of all the possible means,

it's going to look like this and that is the central limit theorem.

We can now say that the 30 we found from a study or if we read a study in the literature.

The 30 individuals in that study or the 60 individuals in

that study of a difference between two groups is just one of

the countless possible others and that's what the statistics does.

It takes those values, uses some of those values, and then draws the curve for you.

The mathematics would take some representation, draw that curve for you.

And your city, or the city that you're reading about for

some week on a much larger set of possible values.

And if the value that you found in your study

should not have occurred very commonly.

If in actual fact the probability of it was less than 0.05 we return

that significant because it would have been rare to find that specific result.

And there you have it, the central limit theorem.

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