Standard Normal Distributions

00:00 Hey, let's talk about standardization.

00:04 Every distribution can be standardized.

00:07 Say the mean in the variance of a variable are mu and sigma squared, respectively. Standardization is the process of transforming this variable to one with a mean of zero and a standard deviation of one.

00:21 This simple formula allows us to do that.

00:25 Oc. Logically, a normal distribution can also be standardized.

00:30 The result is called a standard normal distribution.

00:34 In the last section, we explored shifts in the mean and the standard deviation.

00:38 So if we shift the mean by MU and the standard deviation by sigma for any normal distribution, we will arrive at the standard normal distribution.

00:49 Great. We use the letter Z to denote it.

00:53 As said previously, it is mean is zero, and it's standard deviation one. The standardized variable is called a z score and is equal to the original variable minus its mean divided by its standard deviation. Let's see an example that will help us get a better grasp of the concept. We'll take an approximately normally distributed set of numbers one, 2 to 3, three, three, four, four and five.

01:24 It's mean it's three, and it's standard deviation 1.22.

01:30 Now let's subtract the mean from all data points.

01:34 We get a new data set of minus two, minus one, minus one, 000, one, one and two.

01:45 Let's calculate the new mean.

01:47 It is zero.

01:48 Exactly as we anticipated.

01:51 Showing that on a graph, we have shifted the curve to the left while preserving its shape.

01:57 Clear. Ok, great.

02:02 So far we have a new distribution which is still normal, but with a mean of zero and a standard deviation of 1.2 to.

02:11 The next step of the standardization is to divide all data points by the standard deviation. This will drive the standard deviation of the new data set to one. Let's go back to our example.

02:25 Both the original data set and the one we obtained after subtracting the mean from each data point have a standard deviation of 1.2 too.

02:34 Remember, adding and subtracting values to all data points does not change the standard deviation.

02:42 Now let's divide each data point by 1.2 too.

02:47 We get -1.6, -0.8 to -0.8 to 000.

02:56 0.82. 0.82 and 1.63.

03:01 If we calculate the standard deviation of this new data set, we will get one.

03:07 And the mean is still zero.

03:10 In terms of the curve.

03:12 We kept it at the same position, but reshaped it a bit.

03:16 Great. This is how we can obtain a standard normal distribution from any normally distributed data set.

03:22 Using it makes predictions and inference much easier, and this will help us a great deal and what we will see next.

03:29 Thanks for watching.