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4.05 Functional form of the normal distribution

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Basic Statistics

Université d'Amsterdam

4.7 (2,166 notes) | 150K étudiants inscrits

Cours 3 de 5 dans Méthodes et statistiques en sciences sociales Spécialisation

Visualiser le programme de cours

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

Compétences que vous apprendrez

Statistics, Confidence Interval, Statistical Hypothesis Testing, R Programming

Avis

4.7 (2,166 notes)

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DA

Aug 06, 2016

One of the best courses of statistics for the beginners. The concepts are well explained, the learning path well researched and above everything the R labs were ideal for the beginners.

AF

Jan 26, 2017

great introduction to statistics with no prior knowledge required. Although in parts has been challenging, for me is the right degree of difficulty to push an individual to learning.

À partir de la leçon

Probability Distributions

Probability distributions form the core of many statistical calculations. They are used as mathematical models to represent some random phenomenon and subsequently answer statistical questions about that phenomenon. This module starts by explaining the basic properties of a probability distribution, highlighting how it quantifies a random variable and also pointing out how it differs between discrete and continuous random variables. Subsequently the cumulative probability distribution is introduced and its properties and usage are explained as well. In a next lecture it is shown how a random variable with its associated probability distribution can be characterized by statistics like a mean and variance, just like observational data. The effects of changing random variables by multiplication or addition on these statistics are explained as well.The lecture thereafter introduces the normal distribution, starting by explaining its functional form and some general properties. Next, the basic usage of the normal distribution to calculate probabilities is explained. And in a final lecture the binomial distribution, an important probability distribution for discrete data, is introduced and further explained. By the end of this module you have covered quite some ground and have a solid basis to answer the most frequently encountered statistical questions. Importantly, the fundamental knowledge about probability distributions that is presented here will also provide a solid basis to learn about inferential statistics in the next modules.

4.05 Functional form of the normal distribution6:02

4.06 The normal distribution: probability calculations5:20

4.07 The standard normal distribution8:45

Enseigné par

Matthijs Rooduijn
Dr.
Emiel van Loon
Assistant Professor

From all the possible probability distributions, once stands out

because its the distribution that is encountered very frequently.

It's called, very appropriately, the normal distribution function.

In this video, I'll explain its most important properties.

The normal probability distribution is also called the Gaussian distribution.

It is symmetric, bell-shaped and characterized by it's mean mu and

standard deviation sigma.

The highest point of the distribution is located at the mean.

And its width is specified by the standard deviation.

Both mu and sigma are called parameters of the distribution.

The cumulative normal probability distribution has a sigmoidal shape,

where the mean is given at the probability value of 0.5 and

the sigma determines the steepness of the curve.

The shorthand for stating that the random variable X

has a normal distribution with parameters mu and sigma is this.

And this is the full equation describing the probability density of such

a variable.

It is a magnificent equation not because it may seem rather complex at first sight.

And, as far as I am concerned, also not because it contains three important

mathematical constants, pi, e and the square root of 2.

But it is special because the equation connects the statistical realm

to the material world.

The equation describes how particles distribute themselves by a process called

diffusion.

If you release a diffusion compound, for instance sugar, in a cup of tea,

then the concentration of the sugar will be distributed according to this equation.

And this applies not just to fluids, but also for instance,

to particles in the atmosphere, traffic in the street, and information in society.

At the same time, the Gaussian distribution is encountered frequently

because it is the distribution that you get if the effects or outcomes of

independent random processes are combined according to the central limit theorem.

However, let's not get carried away.

I'll try to explain the equation by taking it apart.

So this equation gives the probability density of a random variable X.

The function is a kind of exponential function with a constant in front and

a part in the exponent which contains small x,

the value that the random variable may take.

As you see in this part of the equation, the mean is subtracted from x and

is divided by sigma.

This is in fact the calculation of the z score.

So the values of the random variable are standardized before they enter the rest of

the equation.

Now let's focus on the constant in front of e.

The exponential function without a constant has a surface under the curve that is changing with the value of sigma.

But when multiplied with a constant it has a value of exactly one.

The value of the constant is in fact the height at the top of the curve.

This is the x equal to mu.

A somewhat counterintuitive property of the normal probability density is that it approaches zero

for very large positive or negative values

of x, but will never actually be zero.

This leads to the fact that the values a random variable can take will stretch from minus infinity

to plus infinity. All these values are possible outcomes albeit very small probability,

but still the sum of all probabilities will be one.

To finalize,

let's go back to the two parameters mu an sigma which determine the location and shape

of the normal curve completely.

Here you see the probability distribution of time spent traveling

from home to work on a week day

for men in Western Europe.

On average the travel time is thirty minutes with a standard deviation of six minutes.

And this is the curve for women in the same countries

with a smaller mean but a larger standard deviation.

What you see is that the peak gets lower if the curve gets wider.

Another property of the curve is

that the values and the units at the y-axis change

if you change the unit along the x-axis.

For example, if you'd expressed the time in hours instead of minutes

the probability density values change from a probability per minute

to a probability per hour, hence these increase 60 fault.

Let me summarize what I've explained in this video.

The normal or Gaussian probability density function is a symmetric, bell-shaped curve

and its corresponding cumulative function has a sigmoidal shape.

The location and shape are fully described by two parameters,

the mean and standard deviation. The mean determines the center of the curve,

the standard deviation determines its width.

The wider the curve, the lower its peak by necessity,

because the surface under the curve always equals 1.

This is the short annotation to state that a variable x is normally distributed

with a mean of 63 and a standard deviation of 12.

And this is the equation of the normal distribution in which you can identify the value of the random variable x,

and the two parameters, mean and standard deviation.

The equation is not only describing a probability distribution,

it's also describing the outcome of many processes in the material world,

where some form

of diffusion is important.

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