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# Normal distribution problem: z-scores (from ck12.org)

## Video transcript

here's the second problem from ck-12 dot orgs ap statistics flex book it's an open source a textbook essentially on I'm using it essentially to get some practice on some statistics problem so here number two the grades on a statistics midterm for a high school are normally distributed with a mean of 81 and a standard deviation of 6.3 all right calculate the Z scores for each of the following exam grades draw and label draw and label a sketch for each example we could probably do it all on the same example but you know the first thing we'd have to do is just remember what is a z-score what is a z-score z-score is literally just measuring how many standard deviations away how many standard deviations away from the mean from the mean from the mean just like that so we learned just have to calculate how many standard deviations each of these guys are from the mean and that's their z-scores so let me do number or part a so we have 65 so first we can just figure out how far is 65 from the mean let me just draw one chart here that we could use the entire time so if this is our distribution see we have a mean of 81 so we have a mean of 81 that's our mean and then a standard deviation of six point three so our distribution say saying they're telling us that it's normally distributed so I can draw a nice bell curve here they're saying that it's normally distributed so that's as good of a bell curve as I am capable of drawing this is the mean right there at 81 and the standard deviation is six point three so one standard deviation above and below is going to be six point three away from that mean so if we go six point three in the positive direction that value right there is going to be eighty seven point three if we go six point three in the negative direction where does that get us that is close what seventy four point seven seventy four point seven right if we had six it will get to 80 point seven and then point three will get us to 81 so that's one standard deviation below and above the mean and then you would add another six point three two goes to standard deviations so on and so forth so that's the at least a drawing of the distribution itself so let's figure out the Z scores for each of these for each of these scores so 65 or each of these grades 65 is how far 65 is you know it's maybe going to be here someplace so we first want to say well how far is it just from our mean from our mean so the distance is we just want a positive number here well actually you want a negative number because you want your z-score to be positive or negative negative would mean to the left of the mean and positive would mean to the right of the mean so we say 65 minus 81 so that's literally how far away we are but we want that in terms of standard deviations so we divide that by the length or the magnitude of our standard deviation so 65 minus 81 let's see 81-65 is what it is 5 plus 11 it's 16 so this is going to be minus 16 over 6.3 and take our calculator out and let's see if we have minus 16 divided by 6.3 you get minus 2 point well looks like 5/4 so approximately equal to minus 2.5 4 that's the z-score for a grade of 65 pretty straightforward let's do a couple more let's do all of them 83 so how far it is away from the mean what's 83 minus 81 it's 2 grades above the mean but we want in terms of standard deviations how many standard deviations so this was part a a was right here this was a we were 2.5 standard deviations below the mean so this was Part A one two and then point five so this was a right there 65 and then Part B 83 83 is going to be right here a little bit higher where right here and the z-score here 83 minus 81 / 6.3 will get us let's see clear the calculator so we have well 83 minus 81 is 2 / 6.3 0.32 roughly so here we get 0.32 so 83 is 0.32 standard deviations above above the mean and if so it'd be roughly one-third of the standard deviation along the way right because this was one whole standard deviation so we're 0.3 of a standard deviation above the mean choice number see we're not choice Part C I guess I should call it 93 well we do the same exercise 93 is is how much above the mean well it's it's 93 minus 81 is 12 but we want it in terms of standard deviation so 12 is how many standard deviations above the mean well it's going to be almost 2 let's take the calculator out so we forget 12 divided by 6 point 3 it's 1 point 9 standard deviations its z-score it's z-scores 1.9 which means it's 1 point 9 standard deviations above the mean so the mean is 81 we go one whole standard deviation and then point 9 standard deviations and that's where score of 93 would lie right there it's z-score is 1 point 9 that all that means is 1 point 9 standard deviations above let's do the last one I'll do it in magenta D Part D score of 100 a score of 100 we don't even need the problem anymore a score of 100 well same thing we figure out how far is 100 above the mean or the mean was 81 and we divide that by the length or the size or the magnitude of our standard deviation so 100 minus 81 is equal to 19 over six point three so it's going to be a little over three standard deviations and well in the next problem we'll see what does that imply in terms of the probability of that actually occurring but if we just want to figure out the z-score 19 divided by six point three is equal to three point zero one so it's very close 3.0 to really if I were to round so it's very close to 23.0 - its z-scores 3.0 - or a grade of 100 is 3.0 - standard deviations above the mean so remember this was the mean right here right here at 81 we go one standard deviation above the mean two standard deviations above the mean the third standard iation above the mean is right there so we're sitting right there on our chart a little bit above that 3.0 two standard deviations above the mean that's where a score of a hundred would be and you can see the probability the height of this that's what the chart tells it's actually a very low probability and actually not just a very low property of getting something higher than that because we learned before and a probability density function the probability if you if you if this is a continuous not a discrete but probability of getting exactly that is zero if this wasn't discrete but since this is a scores on a test we know that it's it's actually a discrete probability function but the probability is is low of getting higher than that because you can see where we sit in the bell curve well anyway hopefully this at least clarified how does how to solve for Z scores which is pretty straightforward mathematically and in the next video will interpret z-scores and probabilities a little bit more