Standard Normal Distribution

Now there is one very important distribution which is derived from the normal distribution, which we call as standard normal distribution but before understanding that, let's understand a basic concept which is of Z score.

Till now, whichever cases we have seen, we had to find out if my value first standard deviation away or second standard deviation away then what will be its probability? But, if I want such a situation where I don't know is it 1^st

standard deviation away or 2^nd standard deviation away.

Now, if I want to find out from my mean how much standard deviation away any of my values lies.

So, that is a special value or a special score which we call as Z score, we find with it.

Which means that Z score is such a value, which tells me how many standard deviations away from my mean, does my value lie.

So, let's see its formula.

Our Z score is calculated as x minus mew upon sigma.

In this we know all the terms, what is x? the value that we wish to find out how far is it from mean, mew will be my mean, sigma is my standard deviation.

To calculate Z score what we will do is first step we will only subtract mean from our value, in the second step we will divide that value by standard deviation.

Now, this Z score can be positive, it can also be negative and it can also be 0.

What is its significance? Let's see it once.

Suppose, my Z score is positive then this means that my x value lies to the right and it is more than mean.

If my x’s value is less than mean, then my Z score will be negative.

And if my x value is equal to the mean then my Z scores value would be zero.

We have seen this scenario and we have seen this formula.

Now, let's use this formula and see through one example.

Suppose in an academic exam which means whatever exam you have given, in that we have to use the Z score so that we can tell that the score that you have given in the class and the mean score of the class.

What is the relationship between the two or how much standard deviation away is it? In this kind of situation, we use a Z score.

Suppose we have given one entrance exam of the college and in there whatever is my data that is normally distributed, its mean is 82 and standard deviation is 5.

This means that the mean score of that entrance exam is 82.

Now, we have to know if any kid gets 90 marks in that exam, then what would be his Z score? Now to find what all things we will require, we will require the mean and standard deviation.

These two we have got from the question.

90 is my x’s value, which we have to find how far it lies.

If I put it into a formula I will get z’s value as 1.6.

So, this means that whatever are the marks of the kid, they are 1.6 standard deviations away from its means.

So, in this way, we use the Z score and we find out different values and how far away from the mean any value lies.

Now, we have seen how to calculate that score? By calculating every Z score normally, we also get one probability which is basically below or above the Z score.

 Every Z score is associated with the P value, which means the Z score that we have calculated, we can create its more significance if we check the area under the curve in that particular curve.

If I see the area under the curve, and if I see the value below my Z value even then I will get the probability and if I see the value on its right, even then I will get the probability.

But what would be that probability? It will be a cumulative probability, which means the total probability of what is the Z’s value, it tells me the Z’s value.

To find this Z table already exists with us, how to read it.

Let's see it with an example once.

Suppose, I have one example where I have Z’s value as 0.68.

What is the meaning of Z’s value being 0.68? That 0.68 standard deviation away, my value lies from the mean.

In this case, I have to find what that would be my probability.

So, in which ways we will use it, we will use the Z table, Z table we can get even on Google.

So, you can refer to it from there.

Now if I see the Z score, Z equals to 0.68, which means that...how will we read 0.68? First of all, we will see in the row, where 0.6 lies? We will mark over there, then the value comes to us 0.08 because 0.6 we have already calculated.

So, we have seen in the above section, which means we have seen in the column that what is the value of 0.08.

Wherever these two intersect that point gives me the value of cumulative probability.

Like in this case, 0.7517 is the intersection of both these points.

So, in this way we can find cumulative probability on the basis of Z score or we can find from Z score table.

So, we have now seen what is Z score and how it is useful to find the probabilities and to tell us how many standard deviations away is my any variable from the mean.

It’s very important application is given to me by standard normal distribution.

So, what is the standard normal distribution? If in our normal distribution, we put the mean’s value as 0 and standard deviation’s value as 1.

So, basically what we are doing is if we convert the normal distribution in Z scores, where in the normal distribution we were using x’s value.

Now instead of if we start using Z score in our formula, then that will become our standard normal distribution.

Now if you see this curve where my Z score is on the x axis, and my probability distribution is on the y axis.

The mean on the Z score is 0 and the standard deviation that we have is 1.

So, I will call this curve a standard normal distribution.

We also call it the Z distribution.

Why? Because we use Z scores in it.

Standard normal distribution, basically there are many applications of this, which we will see in the future chapters, in which way every normal distribution is useful for us to do all the future analysis.

Even if we see any algorithms of machine learning or we have to do the standardisation, normalisation, in all these processes, standard normal distribution is used.

Do not get worried about standardisation and normalisation.

What is it and what is the difference between these two? We will cover this in other chapters.

So, if we convert the normal distribution into standard normal distribution, then with this we can find two things very easily.

First, we can easily find the probability of one particular value.

How? On the basis of Z score.

Second, we can compare different means and standard deviation data sets and we can get to know which data set is giving me better results.

So, this standard normal distribution is called to be an extremely widely used distribution.

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