# Normal Distribution and z Scores Explained – Introductory Statistics

In this video we’ll take a look at the

z-score normal distribution including some of the properties of the z-score

normal distribution. And first of all the z score distribution is symmetrical. And

what that means is if we define the area under this curve as one hundred percent

or 1.00 if we’re speaking in terms of proportions, that means that all of the

values are underneath this curve or within this curve. One hundred percent of

the values fall within this curve. Now the fact is symmetrical means that if we

drew a line down the center of this distribution where it ends in 0 which is

equal to the mean of the z score distribution a mean of zero then 50% or .50 of the values

occur to the left of the mean and 50% or .50 the values occur to the

right of the mean. So in other words that line splits the

distribution exactly in half because it’s symmetrical. Next we can see that

the mean of a z-score distribution is equal to 0 which I just said a minute

ago. So the mean is equal to 0 and the standard deviation is equal to one. Now

here on your screen we’re using the population symbols the Greek symbols for

the mean this is called Mu is equal to 0 and sigma or the standard deviation is

equal to one. And then finally the mean the median and the mode are all equal in

a z-score distribution so if we know that the mean is equal to 0 then that

tells us also that the median and the mode are also equal to 0. Now this occurs

in a z-score normal distribution but it’s necessary that the

distribution is normal in order for the properties of the fifty percent on each

side to be true we could calculate these scores on any distribution but it’s

required that the distribution be normal in order for these properties to hold

here all of them and then for us to look at proportions in the back of our

introductory statistics textbook to pull out proportions for various z-score

values which we’ll talk about in other videos. One last thing before we close

here, what is a z-score? Well a z-score tells you how far away a value is

in terms of standard deviations from the mean. Let me show you what I mean there. So

if we have a z-score of 1 the value that occurs right here that indicates

it’s one standard deviation above the mean and it’s above the mean because

it’s positive. So a z-score of -2 as another example is two standard

deviations below the mean and it’s below because it’s negative. A z score of zero means that the value is

zero standard deviations away from the mean or no standard deviations away from

the mean so if you think about that if it’s zero standard deviations away that

actually means that it’s equal to the mean. So a z score of 0 is just equal to

the mean. OK that’s about it for the properties

of the z-score normal distribution. Thanks for watching.

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