Evaluating the Normal Distribution

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来自 Duke University 的课程

Introduction to Probability and Data

2474 个评分

Duke University

Introduction to Probability and Data

2474 个评分

课程 1（共 5 门，Specialization Statistics with R）

This course introduces you to sampling and exploring data, as well as basic probability theory and Bayes' rule. You will examine various types of sampling methods, and discuss how such methods can impact the scope of inference. A variety of exploratory data analysis techniques will be covered, including numeric summary statistics and basic data visualization. You will be guided through installing and using R and RStudio (free statistical software), and will use this software for lab exercises and a final project. The concepts and techniques in this course will serve as building blocks for the inference and modeling courses in the Specialization.

从本节课中

Probability Distributions

<p>Great work so far! Welcome to Week 4 -- the last content week of Introduction to Probability and Data! This week we will introduce two probability distributions: the normal and the binomial distributions in particular. As usual, you can evaluate your knowledge in this week's quiz. There will be <b>no labs</b> for this week. Please don't hesitate to post any questions, discussions and related topics on <a href="https://www.coursera.org/learn/probability-intro/module/VdVNg/discussions?sort=lastActivityAtDesc&page=1" target="_blank"><b>this week's forum</b></a>.</p>

Normal Distribution17:19

Evaluating the Normal Distribution2:57

Working with the Normal Distribution5:22

与讲师见面

Mine Çetinkaya-Rundel
Associate Professor of the Practice
Department of Statistical Science

Now, let's talk a little bit about how to

evaluate whether a distribution is nearly normal or not.

Here we have a histogram, and what we call a

normal probability plot of a sample of 100 male heights.

So let's focus on the normal probability plot

and talk about what are we really looking

for here, and how can it help us evaluate whether a distribution is normal or not.

On a normal probability plot, data are plotted on the y-axis.

And theoretical quantiles, that means quantiles that follow a normal

distribution, are what would be expected under

the normal distribution, are plotted on the x-axis.

If there is a one-to-one relationship between the data and

the theoretical quantiles, then the data follow a nearly normal distribution.

Since a one-to-one relationship would appear as a straight line on a scatter

plot, the closer the points are to a perfect straight line, the more confident

we can be that the data follow a normal model.

So when looking at a normal probability plot, we're looking for straight lines.

Constructing a normal probability plot requires calculating percentiles and

corresponding z-scores for each observation in your data set, which

can be quite tedious, especially if you have a

large sample, which is what we like to work with.

Therefore, we generally

rely on software when making these plots.

Here's an example of data that do not really follow a normal distribution.

How it appears on the histogram and how it appears on a normal probability plot.

These are height of NBA players from the 2008 and 2009 season.

Since NBA players tend to be disproportionately

taller compared to the general population, the distribution

of their heights is left-skewed.

On a normal probability plot, left skew appears as points

bending down and to the right of the normal line.

We can also see that these points have jumps,

and that's actually due to rounding when reporting heights.

Just like with histograms, normal probability

plots also reveal shapes of distributions.

In a right-skewed distribution, points bend up and to the left of the line.

If the distribution's left skewed, like the one we saw

earlier, points bend down and to the right of the line.

Distributions with short tails, this means a distribution

that's narrower or skinnier than the normal distribution,

follow an S shaped curve.

While those with long tails, so a distribution that's wider than the normal

distribution, starts below the line, bend to follow it, end above it.

We can also use the 68-95-99.7% rule to evaluate normality by

assessing whether the distribution follows what's required by this rule.

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