1 00:00:00,150 --> 00:00:05,460 All we want to do here is learn how to convert from the normal distribution to what's called the standard 2 00:00:05,460 --> 00:00:06,620 normal distribution. 3 00:00:06,630 --> 00:00:13,560 So previously we looked at a normal distribution and we said that the normal distribution was this symmetric 4 00:00:13,560 --> 00:00:17,430 bell shaped curve that had its center, both its mean and its median. 5 00:00:17,430 --> 00:00:21,360 At the center of the distribution here where the mean is mu. 6 00:00:21,390 --> 00:00:27,060 We said also that the normal distribution followed the empirical rule which told us the percentage of 7 00:00:27,060 --> 00:00:32,220 area that we could expect to find under each section of the curve where these sections here are broken 8 00:00:32,220 --> 00:00:35,730 up based on the standard deviation of the data. 9 00:00:35,730 --> 00:00:40,500 So this blue area here represents one standard deviation around the mean. 10 00:00:40,500 --> 00:00:46,020 So from the mean minus one standard deviation on the left to the mean plus one standard deviation on 11 00:00:46,020 --> 00:00:46,650 the right. 12 00:00:46,650 --> 00:00:53,280 If we add on to that and include this red area that represents the interval from the mean minus two 13 00:00:53,280 --> 00:00:57,990 standard deviations on the left all the way over here to the mean plus two standard deviations on the 14 00:00:57,990 --> 00:00:58,440 right. 15 00:00:58,440 --> 00:01:02,190 And then we have our interval for three standard deviations around the mean as well. 16 00:01:02,190 --> 00:01:08,820 But with a normal distribution as we saw before, the mean and the standard deviation can take on any 17 00:01:08,820 --> 00:01:09,540 values. 18 00:01:09,540 --> 00:01:15,270 In fact, we said that the normal distribution could always be identified as capital N and then mean 19 00:01:15,270 --> 00:01:17,370 and variance. 20 00:01:17,370 --> 00:01:22,140 So for instance, if we had a mean of five. 21 00:01:22,840 --> 00:01:25,210 And a standard deviation of three. 22 00:01:25,210 --> 00:01:29,540 We could indicate that normal distribution as five and then nine. 23 00:01:29,560 --> 00:01:32,710 Because standard deviation is sigma, variance is sigma squared. 24 00:01:32,710 --> 00:01:36,970 So if our standard deviation is three, then our variance is three squared or nine. 25 00:01:36,970 --> 00:01:40,180 And in fact, we looked at that exact normal distribution. 26 00:01:40,180 --> 00:01:48,190 But the standard normal distribution is the special normal distribution where the mean is always zero 27 00:01:48,400 --> 00:01:50,920 and the standard deviation is always one. 28 00:01:50,920 --> 00:01:57,100 And we indicate the standard normal distribution with the capital letter Z instead of just any normal 29 00:01:57,100 --> 00:01:59,200 distribution with the capital letter N. 30 00:01:59,200 --> 00:02:03,250 So if we see capital Z, that's indicating the standard normal distribution. 31 00:02:03,250 --> 00:02:11,290 And of course if we were writing this as a normal distribution, we could write it as capital N of zero 32 00:02:11,290 --> 00:02:16,240 comma one, because in the standard normal distribution, the mean is zero, the standard deviation 33 00:02:16,240 --> 00:02:19,270 is one, therefore the variance is one squared or one. 34 00:02:19,270 --> 00:02:25,090 So the standard normal distribution is just equal to the normal distribution that we've looked at before. 35 00:02:25,090 --> 00:02:30,940 But with this particular mean and standard deviation, which means that if we're sketching the standard 36 00:02:30,940 --> 00:02:35,350 normal distribution here, we have to indicate that the mean is at zero. 37 00:02:35,350 --> 00:02:41,530 And then by definition, the lower boundary of one standard deviation around the mean is negative one, 38 00:02:41,530 --> 00:02:48,550 the upper boundary is positive one, and then we have negative two and negative three and positive two 39 00:02:48,550 --> 00:02:54,340 and positive three for the boundaries around the two standard deviations around the mean interval and 40 00:02:54,340 --> 00:02:56,560 three standard deviations around the mean interval. 41 00:02:56,560 --> 00:03:03,340 So when we have a standard normal distribution, then these boundaries are always negative three, negative 42 00:03:03,370 --> 00:03:05,920 two, negative one 0123. 43 00:03:05,920 --> 00:03:12,250 Whereas in this example here for the normal distribution with mean five and standard deviation three, 44 00:03:12,250 --> 00:03:15,310 those values would have been a mean of five. 45 00:03:15,310 --> 00:03:22,900 And then with a standard deviation of three, we would have had eight, 11 and 14, and then here two 46 00:03:22,900 --> 00:03:25,270 negative one and negative four. 47 00:03:25,300 --> 00:03:31,450 Now what we want to understand at this point is how we can take any normal distribution and convert 48 00:03:31,450 --> 00:03:33,640 it into the standard normal distribution. 49 00:03:33,640 --> 00:03:35,410 In other words, to standardize it. 50 00:03:35,410 --> 00:03:42,820 If we could have a way to convert any normal distribution into this super standardized, normal distribution 51 00:03:42,820 --> 00:03:48,550 with this mean of zero and standard deviation of one, that would certainly make comparing normal distributions 52 00:03:48,550 --> 00:03:49,270 easier. 53 00:03:49,270 --> 00:03:53,200 It would make identifying values more clear. 54 00:03:53,200 --> 00:03:55,420 We could make our reporting more clear. 55 00:03:55,420 --> 00:03:58,360 It would just give us a good standard from which to start. 56 00:03:58,360 --> 00:04:04,210 Instead of starting with something like this, this normal distribution we looked at earlier with mean 57 00:04:04,210 --> 00:04:09,310 five and standard deviation three, where we have all of these different numbers and it's much harder 58 00:04:09,310 --> 00:04:11,950 to see that this is a normal distribution. 59 00:04:11,950 --> 00:04:17,829 And of course, as you may expect, we already have a way to convert from a normal distribution into 60 00:04:17,829 --> 00:04:19,480 a standard, normal distribution. 61 00:04:19,480 --> 00:04:25,270 And it goes back to what we looked at earlier when we learned about transforming random variables. 62 00:04:25,270 --> 00:04:31,660 Remember that we talked about how to apply a shift or a scale to a random variable, and that's all 63 00:04:31,660 --> 00:04:32,770 we're going to do here. 64 00:04:32,770 --> 00:04:35,290 We're going to start with our normal distribution. 65 00:04:35,290 --> 00:04:41,740 MN And we're going to apply a shift and then a scale in order to transform our normal distribution into 66 00:04:41,740 --> 00:04:43,330 the standard, normal distribution. 67 00:04:43,330 --> 00:04:49,630 So if we start with a normal distribution and instead of representing that random variable with N, 68 00:04:49,630 --> 00:04:55,960 let's say we're representing it with X, then the first thing we would do is apply a shift to X where 69 00:04:55,960 --> 00:04:57,790 we subtract out the mean. 70 00:04:57,790 --> 00:05:00,670 Now let's talk about what this looks like visually. 71 00:05:00,670 --> 00:05:07,960 So this is in fact a sketch of the normal distribution with mean five and standard deviation three. 72 00:05:07,960 --> 00:05:14,200 So what we're starting with here is the normal distribution of the random variable here with mean five 73 00:05:14,200 --> 00:05:17,470 and standard deviation three or variance nine. 74 00:05:17,470 --> 00:05:20,200 This is what it's probability distribution looks like. 75 00:05:20,200 --> 00:05:27,310 Now if we apply a shift by subtracting out the mean from every data point in our dataset that's making 76 00:05:27,310 --> 00:05:32,050 this distribution, it's going to move every data point to the left by five units. 77 00:05:32,050 --> 00:05:33,310 The mean is five. 78 00:05:33,310 --> 00:05:39,820 So when we subtract out the mean, the graph will shift and the curve moves over to the left. 79 00:05:39,820 --> 00:05:45,220 And now we have a distribution that is centered at zero that has a mean of zero. 80 00:05:45,220 --> 00:05:55,570 So we moved the mean from five here to zero here by applying this shift to all of the data points in 81 00:05:55,570 --> 00:05:56,290 our dataset. 82 00:05:56,290 --> 00:06:03,280 So now we have the mean of the standard normal distribution, which is that mean of zero, but we still 83 00:06:03,280 --> 00:06:05,980 have a standard deviation of three. 84 00:06:05,980 --> 00:06:08,410 We still have this wide stretched curve. 85 00:06:08,410 --> 00:06:14,350 So this curve we have right now, we started with the normally distributed random variable with mean 86 00:06:14,350 --> 00:06:15,790 five, standard deviation three. 87 00:06:15,790 --> 00:06:22,210 Now this curve that we're showing is the normally distributed variable with mean zero and standard deviation. 88 00:06:22,320 --> 00:06:22,720 Three. 89 00:06:22,770 --> 00:06:27,580 So we have the first piece of getting this over to a standard normal distribution. 90 00:06:27,600 --> 00:06:32,450 Now all we have to do is apply a scale, and here's what that scale looks like. 91 00:06:32,460 --> 00:06:37,760 We will divide every data point in the data set by the standard deviation. 92 00:06:37,770 --> 00:06:43,380 When we do that, when we divide by the standard deviation, that means every data point in the set 93 00:06:43,380 --> 00:06:46,470 is going to get pulled in closer to the mean in this case. 94 00:06:46,470 --> 00:06:50,190 And the resulting curve looks like this and this. 95 00:06:50,190 --> 00:06:56,580 Now, this final curve, after both the shift and the scale, is the normally distributed random variable 96 00:06:56,580 --> 00:07:05,370 with mean zero and standard deviation one, which of course we know to be equal to the standard normal 97 00:07:05,370 --> 00:07:06,270 distribution. 98 00:07:06,270 --> 00:07:12,930 So that's really all that we want to understand how we transform one normal distribution into the standard, 99 00:07:12,930 --> 00:07:17,070 normal distribution, regardless of the mean and standard deviation that we started with. 100 00:07:17,070 --> 00:07:24,540 And going forward, this formula here after we've applied the shift and the scale is going to be our 101 00:07:24,540 --> 00:07:25,920 next point of focus. 102 00:07:25,920 --> 00:07:32,070 So we want to imagine that we started with the random variable x, and after these two transformations 103 00:07:32,070 --> 00:07:35,370 we transformed it into the random variable Z. 104 00:07:35,370 --> 00:07:42,210 And now if we rework this formula a little bit, if we switch it around and consolidate this side here 105 00:07:42,210 --> 00:07:49,110 we get Z is equal to X minus the mean divided by the standard deviation. 106 00:07:49,290 --> 00:07:55,710 And it's this formula that we're going to be focusing on going forward as we talk about what we call 107 00:07:55,710 --> 00:08:00,120 the Z scores that are associated with the standard normal distribution.