1 00:00:00,120 --> 00:00:07,020 So we left off talking about the standard normal distribution and this formula here for Z, where we 2 00:00:07,020 --> 00:00:14,550 say that Z is equal to X minus MU, divided by sigma, where X represents the random variable that we 3 00:00:14,550 --> 00:00:20,130 started with Z is the random variable specifically associated with the standard normal distribution 4 00:00:20,130 --> 00:00:23,040 MU is the mean and sigma is the standard deviation. 5 00:00:23,040 --> 00:00:29,610 And the point that we want to make now is specifically that if our random variable x is normally distributed, 6 00:00:29,610 --> 00:00:36,630 we can always use this formula to convert that normal distribution into the standard normal distribution 7 00:00:36,630 --> 00:00:38,070 associated with Z. 8 00:00:38,070 --> 00:00:43,980 So if we pull back up just a normal distribution, we said that sometimes we have normal distributions 9 00:00:43,980 --> 00:00:49,140 like this with this bell shape, but we can have normal distributions that are a little shorter and 10 00:00:49,140 --> 00:00:49,770 wider. 11 00:00:49,770 --> 00:00:54,270 So the distribution might be shorter than this one, but we can also have normal distributions that 12 00:00:54,270 --> 00:00:57,480 are taller and narrower than this one. 13 00:00:57,480 --> 00:01:02,880 They're all normally distributed, but they just have different measures of spread, they have different 14 00:01:02,880 --> 00:01:03,990 standard deviations. 15 00:01:03,990 --> 00:01:09,360 They're still normal, but some are more spread out and some are less spread out, whereas the standard 16 00:01:09,360 --> 00:01:15,480 normal distribution Z always has that same mean of zero and standard deviation of one. 17 00:01:15,480 --> 00:01:20,310 So that measure of spread for the standard, normal distribution is always the same. 18 00:01:20,310 --> 00:01:25,920 Standard deviation is one, whereas we can have infinitely many normal distributions represented by 19 00:01:25,920 --> 00:01:29,910 X with different measures of spread with different standard deviations. 20 00:01:29,910 --> 00:01:36,240 The point here is that regardless of the normal distribution that we have, regardless of the distribution 21 00:01:36,240 --> 00:01:42,960 of X with its own mean and standard deviation, we want to use this formula, which, if you remember 22 00:01:42,960 --> 00:01:51,030 from the last lecture, uses a shift of MU and a scaling of sigma to change the random variable x into 23 00:01:51,030 --> 00:01:52,410 the random variable z. 24 00:01:52,410 --> 00:01:57,270 We always want to use this formula to convert all of the values that make up the data. 25 00:01:57,270 --> 00:02:05,130 Set X into their corresponding values in the data set Z, and we can see this formula working in action. 26 00:02:05,130 --> 00:02:11,070 Remember that throughout the last few lectures we've been looking at a normal distribution that has 27 00:02:11,070 --> 00:02:15,870 a mean of five and a standard deviation of three. 28 00:02:15,870 --> 00:02:21,450 This normal distribution is obviously not the standard normal distribution, because the standard normal 29 00:02:21,450 --> 00:02:24,570 distribution has a mean of zero and a standard deviation of one. 30 00:02:24,570 --> 00:02:28,620 And this normal distribution has a mean of five and a standard deviation of three. 31 00:02:28,650 --> 00:02:33,000 So we're dealing with the normal distribution, but it is not the standard, normal distribution. 32 00:02:33,090 --> 00:02:40,290 So these two pieces of information here come from our random variable x, which is normally distributed, 33 00:02:40,380 --> 00:02:43,560 and we want to convert X into Z. 34 00:02:43,560 --> 00:02:47,490 We want to change this distribution into the standard normal distribution Z. 35 00:02:47,490 --> 00:02:53,280 And you can see that if we just take a few test values here, let's say that the random variable X takes 36 00:02:53,280 --> 00:02:55,050 on a value of five. 37 00:02:55,050 --> 00:03:04,170 Well, if we plug in five for x five from you and three for Sigma, what we get is Z equal to five minus 38 00:03:04,170 --> 00:03:07,200 five divided by three or zero. 39 00:03:07,230 --> 00:03:11,910 So the Z score associated with x equals five. 40 00:03:11,910 --> 00:03:15,960 We'll put a little five subscript here to indicate that x was equal to five. 41 00:03:15,990 --> 00:03:19,290 The z score associated with x equals five is zero. 42 00:03:19,290 --> 00:03:22,620 Let's look at what happens if we set x equal to eight. 43 00:03:22,620 --> 00:03:31,350 So we say z for the x value eight is equal to we would get eight minus five divided by three or three 44 00:03:31,350 --> 00:03:33,420 divided by three is equal to one. 45 00:03:33,420 --> 00:03:37,290 If we pick, let's say x equal to two. 46 00:03:37,950 --> 00:03:42,690 We get two minus five divided by three is equal to negative one. 47 00:03:42,690 --> 00:03:47,310 And if we pick maybe x equals 6.5. 48 00:03:48,370 --> 00:03:59,800 We get 6.5 minus five divided by three is equal to 1.5 divided by three or one half or 0.5. 49 00:03:59,830 --> 00:04:02,880 Now, here's what these values are actually telling us. 50 00:04:02,890 --> 00:04:10,420 The value we get for Z is giving us the distance of whatever X value we picked five 8 to 6.5. 51 00:04:10,570 --> 00:04:16,899 It's giving us the distance of that particular value of the random variable x away from the mean in 52 00:04:16,899 --> 00:04:18,970 terms of standard deviations. 53 00:04:18,970 --> 00:04:24,550 So when we picked x is equal to five, the mean of the random variable x is five. 54 00:04:24,550 --> 00:04:29,640 So the distance between the x value five and its mean is of course zero. 55 00:04:29,650 --> 00:04:35,050 So the z score we get is zero because the distance between the x value we picked and the mean is zero. 56 00:04:35,050 --> 00:04:41,500 So we get a Z score of zero, but if we pick X is equal to eight, the distance between eight and the 57 00:04:41,500 --> 00:04:42,820 mean five is three. 58 00:04:42,820 --> 00:04:46,000 We see that here, eight minus five in the numerator is three. 59 00:04:46,000 --> 00:04:50,350 So we could say that the distance between x equals eight and its mean is three. 60 00:04:50,350 --> 00:04:55,960 But we always want to give the distance in terms of standard deviations and that's why we divide by 61 00:04:55,960 --> 00:04:57,610 the standard deviation sigma. 62 00:04:57,610 --> 00:05:01,330 So when we take three divided by three, the result is one. 63 00:05:01,330 --> 00:05:08,830 And so this value tells us that the distance of eight away from the mean five in terms of standard deviations 64 00:05:08,830 --> 00:05:09,790 is one. 65 00:05:09,790 --> 00:05:16,120 In other words, X equals eight is one standard deviation away from the mean, more specifically one 66 00:05:16,120 --> 00:05:20,770 standard deviation above the mean because we got a value of positive one. 67 00:05:21,010 --> 00:05:26,830 But when we pick x is equal to two and we calculate the associated Z score, we say that the distance 68 00:05:26,830 --> 00:05:32,710 between x equals two and its mean x equals five is two minus five or negative three. 69 00:05:32,710 --> 00:05:39,640 So the distance between the specific value of x that we chose and its own mean is negative three that 70 00:05:39,640 --> 00:05:40,750 negative three value. 71 00:05:40,750 --> 00:05:45,520 The fact that it's negative tells us that the specific value of x that we picked is below the mean. 72 00:05:45,520 --> 00:05:51,250 It is less than the mean, which of course we see x equals two is less than x equals five the mean. 73 00:05:51,250 --> 00:05:56,710 So we get a distance of negative three and then when we divide by the standard deviation of three, 74 00:05:56,710 --> 00:05:58,990 the result we get is negative one. 75 00:05:59,170 --> 00:06:05,170 And this result tells us that x equals two is one standard deviation below the mean. 76 00:06:05,170 --> 00:06:07,420 So it's one standard deviation away from the mean. 77 00:06:07,420 --> 00:06:13,420 The negative sign tells us one standard deviation below the mean, whereas over here the positive one 78 00:06:13,420 --> 00:06:18,460 tells us one standard deviation away from the mean and specifically one standard deviation above the 79 00:06:18,460 --> 00:06:18,970 mean. 80 00:06:18,970 --> 00:06:22,120 And then here we see x equals 6.5. 81 00:06:22,150 --> 00:06:26,110 The distance away from the mean five is one and a half, right? 82 00:06:26,110 --> 00:06:29,890 6.5 minus five is 1.5 or one and one half. 83 00:06:30,070 --> 00:06:34,570 So our distance away from the mean for this particular value of x is 1.5. 84 00:06:34,570 --> 00:06:37,570 But what is that distance in terms of standard deviations? 85 00:06:37,570 --> 00:06:42,130 Well, it's half of a standard deviation because one standard deviation is three. 86 00:06:42,130 --> 00:06:44,890 So what is 1.5 of three? 87 00:06:44,890 --> 00:06:47,020 1.5 is half of three. 88 00:06:47,050 --> 00:06:50,320 And so the result that we get is 0.5 or one half. 89 00:06:50,320 --> 00:06:55,690 6.5 is half of a standard deviation away from the mean. 90 00:06:55,690 --> 00:07:02,470 And so this standardized Z score is 0.5 or half of a standard deviation above the mean. 91 00:07:02,470 --> 00:07:07,630 That is what we're calculating These values here that we're calculating are Z scores. 92 00:07:07,630 --> 00:07:13,870 And the Z score always tells us how many standard deviations we are away from the mean when we choose 93 00:07:13,870 --> 00:07:16,030 a particular value of our random variable. 94 00:07:16,210 --> 00:07:21,940 If we get a positive value for the Z score, it means the particular value we chose is above the mean. 95 00:07:22,030 --> 00:07:26,290 If we get a negative value, it means the value we chose is below the mean. 96 00:07:26,320 --> 00:07:31,990 If we get zero, it means the value we chose is exactly at the mean, it is the mean itself. 97 00:07:31,990 --> 00:07:37,030 Now we don't call the distribution of Z the standard normal distribution for nothing. 98 00:07:37,030 --> 00:07:44,830 It really is standardized, so much so that we actually have a full table of values for every different 99 00:07:44,830 --> 00:07:47,680 Z score that we can possibly calculate. 100 00:07:47,680 --> 00:07:50,470 And here's what that big table looks like. 101 00:07:50,470 --> 00:07:55,570 Now, there's a lot going on here, but it's actually pretty simple when we break it down, What we 102 00:07:55,570 --> 00:08:00,700 see in this whole left side of the table here is this first column of Z scores. 103 00:08:00,700 --> 00:08:03,040 And these values are all negative. 104 00:08:03,040 --> 00:08:04,330 You see that in the first row. 105 00:08:04,330 --> 00:08:11,680 We start with -3.4 and we go all the way down here to the bottom where we see -0.1 and then 0.0. 106 00:08:11,920 --> 00:08:18,370 So these are all negative Z scores, which we already said we find when we pick a particular value that 107 00:08:18,370 --> 00:08:19,660 is below the mean. 108 00:08:19,660 --> 00:08:25,390 We saw that here when we calculated the Z score for X equals two, whereas this whole right hand side 109 00:08:25,390 --> 00:08:29,500 of the table we see here in this column, the Z scores are all positive. 110 00:08:29,500 --> 00:08:33,490 They start at 0.0 and move all the way up to 3.4. 111 00:08:33,760 --> 00:08:36,640 So you can almost think about the normal distribution. 112 00:08:36,640 --> 00:08:42,730 And if we're starting at the left edge of the normal distribution, we are starting here with this most 113 00:08:42,730 --> 00:08:45,280 negative value, -3.4. 114 00:08:45,370 --> 00:08:47,950 And as we move from left to right along the. 115 00:08:48,000 --> 00:08:49,020 Normal distribution. 116 00:08:49,020 --> 00:08:56,670 We move this way in our Z table until we get all the way down here to the bottom at 0.0, where we reach 117 00:08:56,670 --> 00:09:01,170 the halfway point of the normal distribution or the mean of the normal distribution. 118 00:09:01,170 --> 00:09:06,180 Because remember, a Z score of zero is associated with the mean. 119 00:09:06,360 --> 00:09:13,500 We saw that here and we see that this Z score of zero is giving us exactly 0.5. 120 00:09:13,500 --> 00:09:20,280 And that's because 50% of the area under the curve of the normal distribution is to the left of this 121 00:09:20,310 --> 00:09:20,910 Z score. 122 00:09:20,910 --> 00:09:22,800 And we'll talk more about that in a second. 123 00:09:22,800 --> 00:09:27,480 But we move all the way down through the table here, all the way through these negative Z scores, 124 00:09:27,480 --> 00:09:33,090 from the most negative Z score to a just slightly negative Z score of -0.1. 125 00:09:33,090 --> 00:09:36,900 We hit this 0.0, which is right at the mean or right at the middle. 126 00:09:36,990 --> 00:09:42,690 And then we continue on through this right side of the table starting at 0.0. 127 00:09:42,720 --> 00:09:43,110 The mean. 128 00:09:43,110 --> 00:09:51,690 So you can see how these two values match between the tables because they're both at 0.0. 129 00:09:51,780 --> 00:09:58,350 And then we continue on through positive Z scores, starting with a slightly positive Z score of 0.1 130 00:09:58,350 --> 00:10:02,940 until we get to an extremely positive Z score of 3.4. 131 00:10:02,940 --> 00:10:07,740 And of course, up here we're at the far right edge of the standard normal distribution. 132 00:10:07,740 --> 00:10:14,790 The idea here with these Z tables, the value that we're getting out of the body of the Z table is the 133 00:10:14,790 --> 00:10:21,720 percentage of area underneath the curve that falls to the left of the Z score that we look up. 134 00:10:21,990 --> 00:10:28,500 So we already said here that for a Z score equal to zero, we could look in either side of the table 135 00:10:28,500 --> 00:10:30,900 and we see this 0.0 value. 136 00:10:30,900 --> 00:10:35,790 So we go to 0.0 that takes us out to the first decimal place. 137 00:10:35,790 --> 00:10:42,840 But then we have to look across the top row of the table to get the second decimal place for a Z score 138 00:10:42,840 --> 00:10:44,040 of exactly zero. 139 00:10:44,040 --> 00:10:49,680 We want to stay in this first column here to get a true Z score of 0.00. 140 00:10:49,680 --> 00:10:54,390 And so we go to the last row, this first column, and we see here 0.5. 141 00:10:54,390 --> 00:11:01,440 That means that 50% of the data, 50% of the area under the standard normal distribution lies to the 142 00:11:01,440 --> 00:11:03,750 left of Z equals zero. 143 00:11:03,750 --> 00:11:08,400 If we want to look up a Z score of one, well, this is a positive one here. 144 00:11:08,400 --> 00:11:10,440 So we're going to be in this right hand side of the table. 145 00:11:10,440 --> 00:11:12,240 We're looking for a positive one. 146 00:11:12,240 --> 00:11:16,680 Well, we find that at 1.00. 147 00:11:16,680 --> 00:11:22,860 And so we see this value right here, which is 0.8413. 148 00:11:22,860 --> 00:11:30,960 That tells us that about 84% of the data or 84.13% of the area under the standard normal distribution 149 00:11:30,960 --> 00:11:33,960 lies to the left of Z equals one. 150 00:11:33,960 --> 00:11:39,600 Now, if we look for Z equals negative one, because we have a negative Z score, where in this left 151 00:11:39,600 --> 00:11:52,140 side table here, if we go to Z equals -1.00, that puts us here in this first column at 0.1587. 152 00:11:52,140 --> 00:11:58,380 This tells us that the area under the standard normal distribution to the left of negative one is about 153 00:11:58,380 --> 00:12:01,950 15.87% of the total area. 154 00:12:01,980 --> 00:12:05,580 Now, here's an interesting point to make note of. 155 00:12:05,580 --> 00:12:11,970 Now that we have the Z score for both positive one and negative one, if we just sketch a super simple, 156 00:12:12,000 --> 00:12:17,220 normal distribution, we'll say that it looks like this doesn't need to be perfect, but we have here 157 00:12:17,400 --> 00:12:18,240 our. 158 00:12:18,970 --> 00:12:26,120 Mean, and we just found the percentage of area below Z equals one and Z equals negative one. 159 00:12:26,140 --> 00:12:30,610 So let's say that this right here is one standard deviation above the mean. 160 00:12:30,610 --> 00:12:33,700 We'll say that's one and that then symmetrically. 161 00:12:33,730 --> 00:12:38,240 This is one standard deviation below the mean or Z equals negative one. 162 00:12:38,260 --> 00:12:46,540 What these values tell us is that 15.87% of the total area under this curve is going to fall to the 163 00:12:46,540 --> 00:12:49,900 left of Z equals negative one. 164 00:12:49,930 --> 00:12:56,410 So this area right here makes up 15.87% of the total area under the curve. 165 00:12:56,500 --> 00:13:03,820 Another way to put this is that the probability that we find a value below one standard deviation below 166 00:13:03,820 --> 00:13:06,550 the mean is 15.87%. 167 00:13:06,550 --> 00:13:12,010 Or in the case of this normal distribution here, this random variable x, remember we found a z score 168 00:13:12,010 --> 00:13:19,210 of negative one for this value x equals two, which means that the probability or the likelihood that 169 00:13:19,210 --> 00:13:25,530 we get a value of X in our data set that is less than two is about 15.87%. 170 00:13:25,570 --> 00:13:31,240 Now, remember, the standard normal distribution is symmetric, which means that if the likelihood 171 00:13:31,240 --> 00:13:38,830 of us getting a value below one standard deviation is 15.87%, that means that finding a value above 172 00:13:38,830 --> 00:13:40,030 one standard deviation. 173 00:13:40,030 --> 00:13:46,930 So over here, the area that is symmetric to the one that we already sketched over here, the probability 174 00:13:46,930 --> 00:13:53,400 of us finding a value above one standard deviation above the mean also has to be 15.87%. 175 00:13:53,410 --> 00:13:59,350 And of course that makes sense because when we looked here in the Z table for the value associated with 176 00:13:59,350 --> 00:14:03,780 a Z score of 1.00, we got 0.8413. 177 00:14:03,790 --> 00:14:11,020 So we're saying that the probability that we find a value for X that's less than one standard deviation 178 00:14:11,020 --> 00:14:14,050 above the mean is 84.13%. 179 00:14:14,050 --> 00:14:23,560 So this whole area here, everything below one standard deviation above the mean or to the left of a 180 00:14:23,770 --> 00:14:25,840 Z score of positive one. 181 00:14:25,840 --> 00:14:29,290 This whole area is 84.13%. 182 00:14:29,290 --> 00:14:33,190 So that is 84.13%. 183 00:14:33,190 --> 00:14:39,460 And we're saying this value in blue, which was symmetric to the blue area that we sketched on the left, 184 00:14:39,460 --> 00:14:42,060 that is 15.8, 7%. 185 00:14:42,070 --> 00:14:49,990 So if we add to that 15.87%, notice that what we get here. 186 00:14:51,250 --> 00:14:52,670 Is 100%. 187 00:14:52,690 --> 00:14:59,080 Which, of course makes sense because the area under the distribution always has to be equal to one 188 00:14:59,080 --> 00:15:00,240 or 100%. 189 00:15:00,250 --> 00:15:06,700 And so when we pick some Z score like Z equals one, and we look at all the area to the left of Z equals 190 00:15:06,700 --> 00:15:13,630 one, and all the area to the right of Z equals one, those two areas need to sum to one or those probabilities 191 00:15:13,630 --> 00:15:15,570 need to sum to 100%. 192 00:15:15,610 --> 00:15:23,140 They're complementary probabilities and we can see that with any two associated Z scores in this table. 193 00:15:23,140 --> 00:15:31,690 Let's say for instance, we pick 1.53 that would give us this value right here, 0.9370. 194 00:15:31,690 --> 00:15:40,270 And then if we go find -1.53, so we go over here to the negative side of the table, -1.53 gives us 195 00:15:40,270 --> 00:15:45,340 this cell here, which we see is 0.0630. 196 00:15:45,340 --> 00:15:51,820 And if we add 0.9370 to 0.0630, the result we get is one. 197 00:15:51,820 --> 00:15:58,990 So the values we get from the Z table for opposite Z scores are always going to add to one which makes 198 00:15:58,990 --> 00:16:05,890 sense given our understanding of the area under the probability distribution, always adding up to one. 199 00:16:05,890 --> 00:16:09,700 We can also think about the values in the Z table as percentiles. 200 00:16:09,700 --> 00:16:16,270 So if the value that we get from the Z table is let's take this value here 0.9370, that tells us that 201 00:16:16,270 --> 00:16:24,940 the data point that was associated with a Z score of 1.53 is greater than 93.7% of the data points, 202 00:16:24,940 --> 00:16:29,910 or it's in the 93.7 percentile of all of our data. 203 00:16:29,920 --> 00:16:37,090 Or to say it another way, whatever value of the random variable x we used to arrive at a Z score of 204 00:16:37,090 --> 00:16:47,980 1.53 is a value of X larger than 93.7% of the other values of X and less than 6.3%, which we see right 205 00:16:47,980 --> 00:16:50,560 here of all of the other values of X. 206 00:16:50,560 --> 00:16:55,240 Now, keep in mind too, that we can also use this Z table to work backwards. 207 00:16:55,240 --> 00:17:02,860 So if we want to know, for instance, what the threshold is or the cutoff for the top 30% of our data, 208 00:17:02,860 --> 00:17:07,750 well, top 30% means that we have to be in that 70th percentile. 209 00:17:07,750 --> 00:17:10,690 100%, -30% is 70%. 210 00:17:10,690 --> 00:17:13,300 We have to beat that 70% cutoff. 211 00:17:13,300 --> 00:17:20,619 So if we come here to the positive side of the Z table and we locate the first value that is greater 212 00:17:20,619 --> 00:17:30,040 than 0.7, we see that it's right here, this cell right here, 0.7019. 213 00:17:30,040 --> 00:17:34,840 This is the first cell we see that is greater than 0.7. 214 00:17:34,840 --> 00:17:39,160 Just to the left of it, we see the first value that is less than 0.7. 215 00:17:39,160 --> 00:17:47,350 So we know that if we want to be within the top 30%, we'll need a Z score of at least 0.53 because 216 00:17:47,350 --> 00:17:55,600 we get the 0.5 from the row header over here, 0.5 and we get the 0.03 from the column header up here. 217 00:17:55,600 --> 00:18:06,040 So 0.5 and then 3.53 is the smallest Z score we can get that's associated with the top 30% of our data. 218 00:18:06,040 --> 00:18:12,460 So if we choose a value of X and we calculate its Z score and we get a Z score that is greater than 219 00:18:12,460 --> 00:18:19,030 or equal to 0.53, we know based on the table here that we're going to be in the top 30% of our data. 220 00:18:19,030 --> 00:18:25,750 So that's the idea of a Z score, the formula we use to calculate it, what it's actually telling us 221 00:18:25,870 --> 00:18:31,630 how to look up the Z scores that we find in a Z table, what the information in the Z table actually 222 00:18:31,630 --> 00:18:36,190 tells us, and how to work backwards from the Z table to the Z score. 223 00:18:36,280 --> 00:18:42,880 We will use these tables all the time and a quick search online for Z table will pull these up right 224 00:18:42,880 --> 00:18:43,180 away. 225 00:18:43,180 --> 00:18:46,390 They're so standard that you'll be able to find the Z table everywhere. 226 00:18:46,390 --> 00:18:50,020 But this is what we're going to be doing all the time going forward. 227 00:18:50,020 --> 00:18:55,060 We're going to have some random variable, we'll call it X, we're going to know its mean and it's standard 228 00:18:55,060 --> 00:18:55,870 deviation. 229 00:18:55,870 --> 00:19:02,260 But instead of using the normal distribution that's associated with X, which might be shorter and wider 230 00:19:02,260 --> 00:19:08,440 or taller and narrower than the standard normal distribution that represents Z, we're going to convert 231 00:19:08,440 --> 00:19:16,690 all values of X into values of Z to essentially find the associated value of Z for every value of X, 232 00:19:16,690 --> 00:19:21,160 We'll calculate here a Z score for any value of X that we need. 233 00:19:21,280 --> 00:19:27,190 That'll give us the value of Z that's associated with a particular value of X that we're interested 234 00:19:27,190 --> 00:19:27,460 in. 235 00:19:27,460 --> 00:19:32,800 In other words, it'll give us our Z score for that particular value of X, And once we have that Z 236 00:19:32,800 --> 00:19:40,210 score, then we have these Z tables available to us as a tool which will allow us to quickly look up 237 00:19:40,210 --> 00:19:47,680 all of these probability percentile threshold values that are associated with any specific Z score that 238 00:19:47,680 --> 00:19:48,820 we've calculated.