# Properties of the z Score Normal Distribution A free video tutorial from Quantitative Specialists
Specializing in Statistics, Research Design, and Measurement
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In this video lecture, we take a look at the properties of the z score normal distribution, including (1) that it is symmetrical, (2) that the mean, median, and mode are all equal to zero, and (3) that the standard deviation is equal to 1.

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### Introduction to Statistics

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02:39:52 of on-demand video • Updated March 2018

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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 100 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. 100 percent of the values fall within this curve. Now the fact that it's symmetrical means that if we drew a line down the center of this distribution where it ends is 0 which is equal to the mean of the Z square distribution. I mean if 0 then 50 percent or point five zero of the values occur to the left of the mean and 50 percent or point 5 0 the values occur to the right of the ME. So in other words that line splits the distribution exactly in half because it's symmetrical. Next we can see that the mean of disease distribution is equal to zero which I just said a minute ago. So the mean is equal to zero. And the standard deviation is equal to 1. Now here on your screen we're using the population symbols the Greek symbols for the mean. This is called myu is equal to zero and sigma or the standard deviation is equal to 1. 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 zero then that tells us also that the median and the mode are also equal to zero. 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 50 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 core values which we'll talk about in other videos. One last thing before we close here. What is a z score. What disease 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 as the score of negative two as another example is two standard deviations below the mean and it's below because it's negative as the score of 0 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 as score of zero is just equal to the mean OK that's about it. For the properties of the z score a normal distribution. Thanks for watching.