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# Comparing with z-scores

AP.STATS:
VAR‑2 (EU)
,
VAR‑2.C (LO)
,
VAR‑2.C.1 (EK)
CCSS.Math:

## Video transcript

before applying to law school in the US students need to take an exam called the LSAT before applying to medical school students need to take an exam called the MCAT here are some summary statistics for each exam so the LSAT the mean score is 151 with a standard deviation of 10 and the MCAT the mean score is 25 point one with a standard deviation of 6.4 Juwan took both exams he scored 172 on the LSAT and 37 on the MCAT which exam did he do relatively better on so pause this video and see if you can figure it out so the way I would think about it is you can't just look at the absolute score because they are on different scales and they have different distributions but we can use this information if we assume it's the normal distribution or relatively close to a normal distribution with a meat centered at this mean we can think about well how many standard deviations from the mean did he score in each of these situations in both cases he scored above the mean but how many standard deviations above the mean so let's see if we can figure that out so in the LSAT let's see let me write this down on the LSAT he scored 172 so how many standard deviations is that going to be let's take 172 his score minus the mean so this is the absolute number that he scored above the mean and now let's divide that by the standard deviation so on the LSAT this is what this is going to be 21 divided by 10 so this is 2.1 standard deviations deviations above the mean above the mean you could view this as a z-score it's a z-score of 2.1 we are two point one above the mean in this situation now let's think about how he did on the MCAT on the MCAT he scored a 37 the mean is a 25 point one and there is a standard deviation of six point four so let's see 37 point one minus 25 would be 12 but now it's going to be eleven point nine 11 point nine divided by six point four so without even looking at this so this is this is going to be approximately well this is going to be a little bit less than two this is going to be less than two so based on this information and we could figure out the exact number here in fact let me get my calculator out so you get the calculator so if we do eleven point nine divided by six point four that's going to get us to one point I'll just say one point I would say approximately one point eight six so approximately one point eight six so relatively speaking he did slightly better on the LSAT there's more he did more standard deviations although this is close I would say they're comparable he did roughly two standard deviations if we were round to the nearest standard deviation but if you wanted to get precise he did a little bit better relatively speaking on the LSAT he did two point one standard deviations here while over here he did one point eight six or one point nine standard deviations but in everyday language you would probably say well this is this is comparable if this was three standard deviations and this is one standard deviation then you'd be like oh he definitely did better on the LSAT