 live I can live with that I won't die at least I won't die if I do that I probably won't die if I get below that anyways but I might there's a chance that it'll kill me so you could sum those up but again it won't be exact because because it's the area so we can be more exact like this we can write this a couple different ways p of x I like is is is so now you have x is greater than or equal to 70 and less than or equal to 80 I kind of like writing it this way which is probably not the most efficient way because I like to have the X on the left so X is greater than or equal to 75 and less than or equal to the 90 but in any case the formula would look like this it's a bit uglier because we have to use that cumulative thing as we saw in the Poisson distributions we want to take the cumulative on the high side so if I look at my picture over here and the pictures are nice to use we're going to show how to make these in excel so if I'm if I'm trying to find you know the the if I'm trying to find like this side then I could take the whole thing I could take the whole thing minus so if I'm trying to find the blue area I could take the whole thing minus the cumulative minus the orange area will give us the blue area 100% minus the orange area if I'm trying to find something in the middle then I can take the cumulative up to the top point and then take the cumulative up to would be would be the bottom point and you would be left with the middle right and so that's what we would do here so so that's what this formula is doing norm dot dist of the higher x the 90 the mean standard deviation it needs to be cumulative that's what the one is minus the norm dot dist of the lower bit the 75 will give us the bit in the middle 42.93% remembering this does not mean that we're talking about you're getting going to get a 43% on the test it means you have a the of all the people taking the tests we expect around 43% to have scores between 75 and 90 so or equal to include a 75 and 90 we could do that with the Z scores as well so if I was to calculate the 75 Z score so I can say I could say okay what are we doing with the Z scores the Z score if I'm talking the 75 75 minus the minus the 74 minus 74.94 that's the distance from the middle point divided by the standard deviation 1010.09 is going to give us that that point 00 that didn't it didn't work it's 75 I think I said 70 75 minus 74.92 divided by 10.09 gives us that around 0.008 if I move the decimal two places over and this Z score is 90 minus the 74.92 divided by the standard deviation of 10.09 and that gives us the 1.494 so I can do the same concept but instead of using X's now using the Z's which would be norm dot s dot dist then all I need is the Z because I already have kind of the X I'm sorry the mean and the standard deviation the X the mean the standard are all compacted into that Z cumulative that's for the higher bit the 1.494 minus the norm dot s dot dist for the lower Z and that'll give us the same 42.93 the part in the middle so there's our there's our Z's and we can actually plot this over here so if we want to make our graph notice that I made these with bar graphs and this is an actual area graph now so we did this with the area and then the question is well how can I draw these graphs which is quite useful especially if you're like me and you're really not good at drawing these grasses kind of really hold held me back learning math in school because I would get all messed up on how many numbers should be on the X and the Y and then and so if you can make these in Excel then it's great but it's still a little bit complicated to do so this so we do this in Excel so you can you can plot this out we're doing this with an if logical test so if we're saying that if this X here if this X is greater so we're right here we're looking for less than or equal to 90 X is greater than or equal to 90 so we're taking this X and we're saying if it is less than or equal to I believe the 90 is what we're picking up in our formula over here and we'll do this I highly recommend taking a look at it in Excel if you're interested in this there's the 90 and then we're saying that if that is true then we want you to give me the result which is this P of X and if it's not true we want you to just give me a blank cell which is represented by the double quotes because whenever you have text it's with the quotes so you can see it's got stuff in it up to here which stops at the 90 and now you have blank stuff so if I graph this on top of my other chart here then that's where that's where we get this line which gives us that that nice representation so we can try to understand a little bit more pictorially what is happening notice that this graph is really pretty neat as well in that we also put the X's here as well as the Z scores down here so now we can look at two X axes because remember that we talked about the idea that we can represent a lot of this stuff in terms of talking about it in X's which are the grades right and talking about it in the Z scores which is the distance from that middle point so we can actually put two of these in here which is a little fancy tricky thing to do in Excel so you want to check that out in Excel so you can you can utilize that tool as well also note that you might be able to get away with like one of these graphs because remember there's that there's basically three questions we ask oftentimes there's more than three but one more of the common ones would be what is above so we asked what is everything above a certain point like the 90 so that would be the area of the blue side and then we ask well what's everything below a certain thing and notice that if you use one graph you can kind of ask both those questions because if I'm asking what is above a certain point that's the blue side that also means that because it adds up to 100% the orange represents the question of what if something is summed up up to and including everything below a certain point and then the next question often asked is the one that's in the middle and that's the one you can't you might have to do a little bit more fancy graph so we have a whole bunch of different ways that we can make these graphs that will demonstrate in Excel and try to explain them a little bit here in the one notes also note with this graph we when we graphed this this is the actual bell curve in a bar chart in our actual data this is the this is the bell curve in the actual data and when we graph those we graph this one on top of the percent of total column to get this one so those are the that's the general idea again I highly recommend checking the staff out in Excel