 This is a video about the empirical rule which applies to data which are normally distributed. If a random variable is normally distributed, the empirical rule states that 68% of the data values will fall within one standard deviation of the mean, 95% will fall within two standard deviations of the mean, and about 99.7% of all data values fall within three standard deviations of the mean. So what this means is if I take my bell curve which is symmetric about the mean and I go one standard deviation above one standard deviation below that is all within one standard deviation of the mean, 68% of the data will fall there. Well if 68% has to be split equally among these two regions, that leaves 34%. In each region, more. If 95% of all data values fall within two standard deviations of the mean, that means two standard deviations below two standard deviations above should add up to 95%. These four regions should add up to 95%. Well out of that 95% we already have 68 that's accounted for. So you have to do 95 minus 68, 27. So I have to take 27% and split it up equally among the two regions, 27 divided by 2. Well that would be 13.5% for each of the regions. So 13.5% and 13.5%. All right then if 99.7% of all data values fall within three standard deviations, 99.7, we already have 95% accounted for. So try 95. That's going to give us how much? 4.7. I have to take 4.7% and split it equally among two regions. 4.7 divided by 2 is 2.35. That would be, I'll draw an arrow here, 2.35%, and 2.35%. In the area under this curve has to add up to 100% when you're talking percentages. It has to add up to 1 when you're talking decimals or non percentages. So if my entire area under the curve has to add up to 100% and I already have 99.7% accounted for, well that's 0.3% remaining in these little tiny tails off to the side. What is 0.3 divided by 2? It would be 0.15% in each of these little outer tails here. It's not much, but it is important to note that it's something. So we can use the empirical world to answer questions. It gives us really good approximations for the answers for questions. So the time to complete an exam is approximately normal or bell shape with a mean of 52 minutes and the standard deviation of 6 minutes. Find the probability a student takes between 40 and 58 minutes to complete the exam, then find the probability it takes a student more than 58 minutes to complete the exam. So I have here my bell curve with all of the empirical rule probabilities. Now I'm going to label my x values or x axis down here. Who is my mean? My mean is 52. If each standard deviation is 6 minutes, well 52 plus 6, that gives you one standard deviation above the mean, which is 58. Add another standard deviation. 58 plus 6 is 64. Add another standard deviation. 64 plus 6 is 70. Now let's go the other way. 52 minus 6, well that'd be 46. 46 minus 6, 40. Take away 6 again, 34. So now that we have my x axis labeled, to find the probability a student takes between 40 and 58 minutes to complete the exam, I find 40. I find 58. And I would have three regions here, region one, region two, region three. So add up the three percentages, 13.5 plus 34 plus 34. And that is going to give you a grand total of 81.5 percent or in decimal form 0.815. That is the probability that a student takes between 40 and 58 minutes to complete the exam. Now what about the probability it takes a student more than 58 minutes? So find 58 along your x axis and we include everything to the right of it because we want more than 58 minutes. So that's one, region two, and then our little tiny tail is our third region. So add up those percentages. You have 13.5 plus 2.35 plus 0.15. That's going to add up to give you a grand total of 16 percent or in decimal form 0.16. So that's the empirical rule and how you should use it. Thanks for watching.