Using The Empirical Rule to Determine Percentiles

Hi, everyone. I wanted to do a problem for you guys from section 2.5. In the past few semesters, this has been a problem that people have emailed me about a lot. So I figured I would do a video for you. I've only done these videos two or three times, so please forgive me if it's a little messy. So the problem we're going to do is, let's say, the mean amount of time spent shovelling in the blizzard of 2013 is 16 hours, the standard deviation of 3. OK. I know that that-- 16 was not our time, we were definitely about that. But let's just say that was our mean. And let's also assume-- I should have added this in-- let's assume that this is normal. OK. Which means, as soon as we assume that, we know we're thinking we're going to use empirical rule. OK. I promise I have better handwriting in person. OK. So let's say we wanted to figure out what percentile does a person who shovels 19 hours represent. The first thing we need to do is translate this into number of standard deviations. OK? 19 doesn't mean anything to me. What means something to me is, how many standard deviations is that from 16? All right. So that's our z-score. So first thing we do is find our z. So we do our 19 minus our mean of 16, all over our standard deviation. Which comes out to be 1. So that means we're one standard deviation from the mean. That's what we want. That's what tells us, really, if this is unusual or not. So next thing I always tell people do, especially when I have them on ground for a class, is you want to draw the normal curve. Any time you're using it, draw it so you can picture what you're trying to find. So this is a mean of 16, and we're up here at 19. So what I'm going to do is I'm going to draw the other side of this. I'm going to make it symmetric, and let's say that was 13. OK. So from the empirical rule, we know that this shaded piece in here, the middle from one standard deviation on either side, is 68% of our data. Since its normal. OK. So that's not our percentile. Percentile, remember, is area to the left. Always. So the percentile is we want all of this whole piece, not just the middle. But 68% is going to get us started. If I take the whole curve, which is 100%, and I take off 68%, I'm left with 32% as a whole for these two pieces on the side that are not shaded. Which means in here is 16%. And in here is 16%. OK. So now my percentiles all of this shaded piece from 19 all the way down to the end. OK. So if I add up those two values, 68% plus 16, that's going to get me my percentile. That's going to get me my percentile of 84. So the person that shovels 19 hours represents the 84th percentile. OK? So you're going to use that same process for any one of those. It's a matter of drawing a normal curve, figuring out what the empirical rule will tell you, and then filling in the other percentiles and seeing what you need to add up. OK. So I have one more. So now let's say what percentile that a person who shovels 10 hours represent. So same thing. Let's see how many standard deviations far away this is. So this is below the mean. So it's going to come out negative. So we've got a person that shovels 10 against a mean of 16 over 3. So we end up getting a negative 2. So that means we're two standard deviations to the left of the mean. So let's draw it and see where we're at. OK. So here's our mean of 16. We're down here at 10, two standard deviations. So we could just kind of fill in the other side. We don't even need that value. But empirical rules gives me the middle piece. OK. And empirical rule says that 95% of my data falls within two standard deviations. OK. So that's where we're at. Again, that's not my percentile. I want to know what is to the left of that. So in this case, my percentile, since I'm the person shoveling 10, my percentile's over here. So 100 minus that 95 gives me 5% for the rest of my whole graph here. Whatever all the way to the left, all the way to the right. OK. Cut that in half, which means this gives me 2.5%, and this gives me 2.5%. Which means the percentile for a person that shovels 10, that person is in the 2.5% percentile. It comes out really low. If you think about that makes sense, it's a really low percentile because they're quite a bit below the mean. Whereas the last one that we did, it came out to be I think it was the 84th percentile, and they were above the mean. So it should be above the 50th percentile. If we were looking for, say, this value over here, the percentile for that, we would have added up the 95 and the 2.5. OK. But this is down here below the mean. Hopefully that helps you with some of those homework problems in 2.5. If you have any other questions, just shoot me an email. Enjoy your snow days. Bye.