hey guys it's mark from ace tutors and in this video i'm going to explain one of the most common probability distributions covered in stat the normal distribution first i'll go over this concept at a high level then i'll explain how this distribution is used to find probabilities and finally we'll finish things off with an example and just quickly before i begin if you would please consider tapping the subscribe and like buttons below to help us keep making videos like this for students like you so now let's dive right in first of all a normal distribution looks like this symmetric bell looking thing which is why it's often called a bell curve now this curve is centered around the distribution's mean and spreads out with decreasing probability as you move away in either direction although the normal distribution technically continues indefinitely in either direction it only really has significant probability values between three standard deviations below and above the mean in fact as a general rule of thumb for a normal distribution between plus or minus one standard deviation there is 68 of the distribution's data between plus or minus 2 standard deviations there is 95 percent of the data and finally between plus or minus 3 standard deviations there's 99.7 percent of the data now the reason i called this distribution one of the most popular ones is because it can be used for so many types of real world applications for example certain things like people's heights weights and even iq follow a normal distribution pretty closely however because the normal distribution is so widely applicable and each application comes with its own mean standard deviation and most importantly units this causes some problems to solve this issue instead of keeping the data in its original units we get rid of the units by doing what's called standardizing the data essentially we take our original distribution and transfer it to have a mean of 0 and standard deviation of 1. as a result any normal distribution of any magnitude or unit type can be converted to be centered around 0 and extend more or less from negative three to positive three okay that's cool but how do we actually do that well in order to standardize the data we need something you may have already heard about in your class the infamous z-score which has a formula of z equals the data point x minus the mean mu divided by the standard deviation sigma essentially what the z-score tells you is how many standard deviations a certain data point is away from the mean so for example if the heights of people in your stat class were normally distributed with a mean of 66 inches and standard deviation of 2 inches your unstandardized distribution would look like this centered at 66 inches and spreading out by increments of the standard deviation like this if you happen to have a height of 67 inches you would fall right here in the distribution slightly above the mean if we were to standardize your data point we would find that your z-score equals 67 minus 66 divided by 2 which is 1 over 2 or 0.5 this z score of 0.5 means that your data point is half of a standard deviation above the mean and located here the same exact location in the distribution as before all right now that we covered how and why we standardize normal distributions let's move on to the most important part of all probability distributions and that's well probability now unlike the other distributions we covered there isn't a nice and pretty formula for figuring out probabilities for the normal distribution instead to calculate probabilities for this distribution we have to rely on using charts which might look like these where numbers on the leftmost column and topmost row correspond to your z-score and the interior numbers correspond to those z-scores associated probability values so for example if your z-score was 0.63 you'd go down to the 0.6 row corresponding to the 6 in the z-score's tenths place and over to the 0.03 column corresponding to the 3 in the z-score's hundredths place to get the associated probability value of 0.7357 okay but what does that value actually mean well in order to know that it's very important to understand what values the z chart you're looking at gives you for these z charts it happens to give you the area or probability of being to the left of your z score so for our previous example based on the value found in the chart there's a 73.57 percent chance of having a z-score less than 0.63 and one last quick note it's also important to make sure you are using the correct chart based on whether you have a negative or positive z-score the one on the left is for negative z-scores or values below the mean and the one on the right is for positive z-scores or values above the mean all right i know i've been throwing a ton of info at you so let's solidify all of this with a quick example so you love pizza but your local pizza shops pizzas have been looking smaller and smaller lately they claim that their large is at least 16 inches or it's free over your expansive pizza eating career you've found that their pizzas are normally distributed in size with a mean of 16.3 inches and standard deviation of 0.2 inches now you want to know what the probability is of getting a free pizza also what's the probability of getting lucky with a pizza that's over 16.5 inches and just because you love stat and love eating pizza what's the probability of getting a pizza that's between 15.95 inches and 16.63 inches okay first let's draw our normal distribution centered at the mean of 16.3 inches and we want to find the probability of getting a pizza that is less than 16 inches which looks like this now our first step for these problems is always going to be standardize our distribution using our z-score equation and plugging in our data point of 16 we get 16 minus 16.3 divided by 0.2 or negative 0.3 over 0.2 which gives us a z-score of negative 1.5 so instead of finding the probability of a pizza x being less than 16 inches we are now finding the probability of a pizza having a z score less than negative 1.5 that we have a z-score we can use our z-chart to find its associated probability looking at row negative 1.5 and column .00 we find a probability of 0.0668 since we want to find the probability of being less than our z score and not being exactly what the chart gives us we're done we have a 6.68 chance of getting a free pizza next we want to find the probability of a pizza being over 16.5 inches like before our first step is to find this data point's z-score which equals 16.5 minus 16.3 divided by 0.2 or 0.2 over 0.2 or just 1. so this time we want to find the probability of having a z-score higher than 1. looking at our chart for our z-score we go to row 1.0 and column .00 to get a probability of 0.8413 but keep in mind this chart always gives you the probability to the left of your z-score since the area under the entire distribution is always equal to 1 and the area to the left is 0.8413 we can get the area to the right by subtracting 1 minus 0.8413 to get our final answer of 0.1587 or a 15.87 percent chance of getting a pizza larger than 16.5 inches not too shabby finally we want to find the probability of getting a pizza between 15.95 and 16.63 inches first as always let's find some z-scores for 15.95 we get negative 0.35 divided by 0.2 or negative 1.75 and for 16.63 we get 0.33 divided by 0.2 or 1.65 all right we have our z scores now but how do we use them to find the area between them well one way to do it would be first look in the chart for the larger z score this would get you this area here to the left of 1.65 then you can find the area to the left of the smaller z-score which would get you this area in the tail here once you have those you can subtract the smaller area from the larger area to leave you with the area in between them looking in our positive z-score chart we find the area below a z-score of 1.65 is 0.9505 and then looking in our negative z-score chart we find the area below a z-score of negative 1.75 is .0401 subtracting these values we get our answer of 0.9104 or a 91.04 chance of getting a pizza between 15.95 and 16.63 inches so that right there is the normal distribution if you found this video helpful please hit that subscribe button down below to support us making more videos for you if you didn't please leave us a comment down below to let us know what we can do better thanks again for watching and remember you have big dreams don't let a class get in the way