Normal Distributions

00:00 Okay. Here we go.

00:03 So far, we learn that a distribution of a data set shows us the frequency at which possible values occur within an interval.

00:10 We also said that there are dozens of distributions.

00:14 Experienced statisticians can immediately distinguish a binomial from a Poisson distribution, as well as a uniform from an exponential distribution in a quick glimpse at a plot. In this course, though, we will rather focus on the normal and students T distributions due to the following reasons.

00:33 They approximate a wide variety of random variables.

00:37 Distributions of sample means with large enough sample sizes could be approximated to normal. All computable statistics are elegant.

00:47 Decisions based on normal distribution insights.

00:49 Have a good track record.

00:51 If it sounds like gibberish now.

00:53 I promise that things will be much easier once we get started.

00:58 Here is a visual representation of a normal distribution.

01:02 You have surely seen a normal distribution before, as it is the most common one.

01:07 The statistical term for it is Gaussian distribution, but many people call it the bell curve, as it is shaped like a bell.

01:15 It is symmetrical and its mean median and mode are equal.

01:20 If you remember the lesson about skewness, you would recognize it has no skew.

01:25 It is perfectly centered around its mean.

01:29 All right. It is denoted in this way and stands for normal.

01:34 The tilt sign shows it is a distribution, and in brackets we have the mean and the variance of the distribution.

01:41 On the plane, you can notice that the highest point is located at the mean because it coincides with the mode.

01:48 The spread of the graph is determined by the standard deviation.

01:53 Now let's try to understand the normal distribution a little bit better.

01:58 Let's look at this approximately normally distributed histogram.

02:02 There is a concentration of the observations around the mean, which makes sense as it is equal to the mode.

02:08 Moreover, it is symmetrical on both sides of the mean.

02:13 We used 80 observations to create this histogram.

02:16 It's mean is 743 and its standard deviation is 140. Okay, great.

02:25 But what if the mean is smaller or bigger? Let's first zoom out a bit by adding the origin of the graph.

02:32 The origin is the zero point.

02:34 Adding it to any graph gives perspective.

02:38 Keeping the standard deviation fixed or in statistical jargon, controlling for the standard deviation.

02:44 A lower mean would result in the same shape of the distribution.

02:47 But on the left side of the plane.

02:51 In the same way, a bigger mean would move the graph to the right.

02:55 In our example, this resulted in two new distributions, one with a mean of 470 and a standard deviation of 140 and one with a mean of 960 and a standard deviation of 140.

03:11 All right, let's do the opposite.

03:14 Controlling for the mean.

03:15 We can change the standard deviation and see what happens.

03:20 This time the graph is not moving but is rather reshaping.

03:25 A lower standard deviation results in a lower dispersion.

03:28 So more data in the middle and thinner tails.

03:32 On the other hand, a higher standard deviation will cause the graph to flatten out with less points in the middle and more to the end or, in statistics, jargon.

03:41 Fatter tails.

03:44 Great. These are the basics of a normal distribution.

03:48 In our next lesson, we will use this knowledge to talk about standardization.

03:53 Stay tuned.