05 04 Variance data example

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來自 Johns Hopkins University 的課程

Statistical Inference

2647 個評分

Johns Hopkins University

Statistical Inference

2647 個評分

課程 6（共 10 門，Specialization Data Science）

Statistical inference is the process of drawing conclusions about populations or scientific truths from data. There are many modes of performing inference including statistical modeling, data oriented strategies and explicit use of designs and randomization in analyses. Furthermore, there are broad theories (frequentists, Bayesian, likelihood, design based, …) and numerous complexities (missing data, observed and unobserved confounding, biases) for performing inference. A practitioner can often be left in a debilitating maze of techniques, philosophies and nuance. This course presents the fundamentals of inference in a practical approach for getting things done. After taking this course, students will understand the broad directions of statistical inference and use this information for making informed choices in analyzing data.

從本節課中

Week 2: Variability, Distribution, & Asymptotics

We're going to tackle variability, distributions, limits, and confidence intervals.

05 01 Introduction to variability4:47

05 02 Variance simulation examples2:46

05 03 Standard error of the mean7:12

05 04 Variance data example3:33

與講師見面

Brian Caffo, PhD
Professor, Biostatistics
Bloomberg School of Public Health
Roger D. Peng, PhD
Associate Professor, Biostatistics
Bloomberg School of Public Health
Jeff Leek, PhD
Associate Professor, Biostatistics
Bloomberg School of Public Health

And so let's actually go through a data example.

Here I'm going to use the father son data from the using our data set, and

I'm going to grab the, the son's height, and

n is going to be the number of observations like always.

If I were to plot a histogram of the son's height, I get this distribution right

here and I overlayed the histogram with a continuous density estimate.

It's quite Gaussian looking.

Now this density estimate is an estimate of the population density.

We don't have the population density because we

didn't collect an infinite amount of data.

Variability of this histogram, which is what the sample variance is calculating.

It's estimating the variability of son's height

from whatever population this was drawn from.

Let's assume that it was a random sample.

Let's just go through a couple of numbers we can calculate here.

So here I took variance of x, variance of x divided by n,

standard deviation of x, standard deviation of x divided by square root n.

I rounded all the numbers to two decimal places.

So 7.92 and 2.81, the variance of x and the standard deviation of x,

are simply talking about the variability in son's heights from this data set,

which are estimates of the variability.

The population variability of sons heights if you're willing to

assume that these sons are a random sample from some meaningful population.

I like 2.81 in this case over 7.92 because

7.92 is expressed in inches squared and 2.81 is expressed in inches, so

I like to work in the units rather than the units squared.

0.01 and 0.09 are no longer talking about the variability in the children's heights.

It's talking about the variability in averages of ten children's heights.

So 0.09 is probably the most meaningful one, and it's the standard error.

Or in other words,

the standard deviation in the distribution of averages of n children's heights.

And a, again, in this case it's an estimate of that, but

it's the best estimate we have from the data that we have.

So let's summarize what we know because we covered a lot of

somewhat complicated topics in this lecture.

And I would say, fundamentally, what differentiates understanding

statistics from not understanding statistics is understanding variability.

So if I were to say, what's the most important lecture, it might be this one.

So let's summarize what we know.

The sample variance is an estimate of the population variance.

The distribution of the sample variance is centered at what it's estimating.

This is a good thing.

This means that it's unbiased.

And it gets more concentrated about what it's estimating as you collect more data.

Again, this is a good thing.

This means if we go to the trouble of collecting more data,

we get a better estimate, that the distribution of the sample variance is

more concentrated about what it's estimating.

We also know a lot about the distribution of sample means.

We know where it was centered at from the last lecture, but

we also know in this lecture that the variance of

the sample mean is the population variance divided by n and the square root of it,

sigma divided by square root n, is the so called standard error.

These quantities represent how variable averages are drawn from this population.

And it turns out that we can say a lot of about the distribution of averages

from random samples even though we only get to look at one on a given data set.

And this gives a lot of work, a lot to work with and

it forms a lot of the foundation of ways in which we can perform

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