1 00:00:00,120 --> 00:00:06,060 Now that we understand how to calculate, mean variance and standard deviation for a normal data set, 2 00:00:06,060 --> 00:00:11,100 we want to talk about the equation of the probability density function of the normal distribution. 3 00:00:11,340 --> 00:00:19,510 So this function F of x is the standard form of the probability density function for a normal distribution. 4 00:00:19,530 --> 00:00:23,820 Notice that it relies upon the standard deviation given here by Sigma. 5 00:00:23,820 --> 00:00:28,250 We also see it here and the mean MU, which we see here. 6 00:00:28,260 --> 00:00:33,870 So in order to build the equation for the probability density function, we need to have the mean and 7 00:00:33,870 --> 00:00:36,740 standard deviation that we learned to calculate earlier. 8 00:00:36,750 --> 00:00:45,900 So let's say, for instance, that we had from our dataset calculated a mean of five and a standard 9 00:00:45,900 --> 00:00:52,950 deviation of three, then we could plug these values into the probability density function and we would 10 00:00:52,950 --> 00:00:58,380 get one over three times the square root of two pi. 11 00:00:58,380 --> 00:01:07,170 And then Euler's number here raised to the negative one half power multiplied by X minus our mean of 12 00:01:07,170 --> 00:01:13,590 five, so x minus five, all divided by standard deviation, so three quantity squared. 13 00:01:13,590 --> 00:01:20,400 And now we have here the equation for the probability density function of the normal distribution that 14 00:01:20,400 --> 00:01:23,470 has a mean of five and a standard deviation of three. 15 00:01:23,490 --> 00:01:28,550 Now we talked about earlier how a normal distribution was a bell shaped curve. 16 00:01:28,560 --> 00:01:34,050 Well, here are some examples of these probability density functions we've been talking about. 17 00:01:34,050 --> 00:01:40,890 In fact, this distribution in green here is the distribution for the normally distributed data with 18 00:01:40,890 --> 00:01:43,140 mean five and standard deviation three. 19 00:01:43,140 --> 00:01:47,250 In red, we have a mean of five and a standard deviation of two. 20 00:01:47,250 --> 00:01:51,540 And in blue we have a mean of five and a standard deviation of one. 21 00:01:51,570 --> 00:01:59,010 We can see that the mean is five for all three distributions because we see here that all three distributions 22 00:01:59,010 --> 00:02:00,750 are centered at five. 23 00:02:00,750 --> 00:02:06,630 Remember that for any normal distribution, the mean is right here at the center of the curve and for 24 00:02:06,630 --> 00:02:09,900 any normal distribution, this is also where we'll find the median. 25 00:02:09,900 --> 00:02:14,190 In other words, in a normal distribution, the mean and median are always equivalent. 26 00:02:14,220 --> 00:02:20,550 Now what we notice is that the curve, the graph of the probability density function is of course just 27 00:02:20,550 --> 00:02:25,830 dictated by these two values, the mean and the standard deviation, because those are the only two 28 00:02:25,830 --> 00:02:30,720 values we need in order to plug into this formula for the probability density function. 29 00:02:30,720 --> 00:02:38,640 And that's why we can express any normal distribution as just its mean and standard deviation, or what 30 00:02:38,640 --> 00:02:44,160 we actually use for our standard notation, which is variance of course, just the square of standard 31 00:02:44,160 --> 00:02:44,910 deviation. 32 00:02:44,910 --> 00:02:50,610 So whenever we're dealing with any normal distribution, we will often just express it this way the 33 00:02:50,610 --> 00:02:53,940 normal distribution of the mean comma, the variance. 34 00:02:53,940 --> 00:03:00,300 And so instead of having to express this entire probability density function or give extra information 35 00:03:00,300 --> 00:03:05,160 or a sketch of the curve, all we have to do is say we're dealing with the normal distribution. 36 00:03:05,160 --> 00:03:10,110 Five one and that indicates a mean of five and a variance of one. 37 00:03:10,110 --> 00:03:15,000 Here we have a normal distribution with a mean of five and a variance of four, and here we have a normal 38 00:03:15,000 --> 00:03:18,030 distribution with a mean of five and a variance of nine. 39 00:03:18,030 --> 00:03:23,310 Now we always give the variance, but of course we know that standard deviation is just the square root 40 00:03:23,310 --> 00:03:24,300 of variance. 41 00:03:24,300 --> 00:03:30,030 So based on these expressions of these three normal curves which correspond to these three curves that 42 00:03:30,030 --> 00:03:38,790 we already sketched, we know that the standard deviation here is one, two and three, the square root 43 00:03:38,790 --> 00:03:41,070 of one, four and nine respectively. 44 00:03:41,070 --> 00:03:46,620 So just from this notation here, we can immediately see mean variance. 45 00:03:46,620 --> 00:03:52,350 We can take the square root of variance to get standard deviation, and then we can use this information 46 00:03:52,350 --> 00:03:58,590 to plug into the equation of the probability density function for a normal curve and use that equation 47 00:03:58,590 --> 00:04:00,450 then to sketch the curve. 48 00:04:00,450 --> 00:04:06,990 If we graph this equation after we've plugged in the mean and the standard deviation, we will get this 49 00:04:06,990 --> 00:04:09,180 picture of the normal curve. 50 00:04:09,180 --> 00:04:14,610 Let's notice again here too, that as the variance and standard deviation increase, the normal curve 51 00:04:14,640 --> 00:04:16,950 gets flatter and wider. 52 00:04:16,950 --> 00:04:21,180 And that's because variance and standard deviation are measures of spread. 53 00:04:21,180 --> 00:04:26,430 And so when we see a higher value for variance and therefore a higher value for standard deviation, 54 00:04:26,460 --> 00:04:29,610 that means the data is spread out further from the mean. 55 00:04:29,610 --> 00:04:36,300 The data gets pushed out further away from the mean to both the right and left, as opposed to when 56 00:04:36,300 --> 00:04:38,730 variance and standard deviation are smaller. 57 00:04:38,730 --> 00:04:45,330 And we see more of the area under the curve, more of the data pushed closer to the mean on the right 58 00:04:45,330 --> 00:04:46,350 and the left hand side. 59 00:04:46,350 --> 00:04:48,480 That data is pulled in toward the mean. 60 00:04:48,480 --> 00:04:51,720 And so we see the curve rise up and become taller. 61 00:04:51,720 --> 00:04:53,790 That normal curve is taller. 62 00:04:53,790 --> 00:04:56,700 The data is more tightly clustered around the mean. 63 00:04:56,700 --> 00:04:59,910 The spread is smaller, but as long as our data. 64 00:05:00,070 --> 00:05:05,680 Creates this symmetric bell shape following this form of a probability density function. 65 00:05:05,680 --> 00:05:12,310 We know that we have normally distributed data regardless of whether the curve is taller and more narrow, 66 00:05:12,310 --> 00:05:16,900 like this blue curve or shorter and wider like this green curve. 67 00:05:16,930 --> 00:05:23,260 Now, that being said, let's talk about one of the most important conclusions we can draw from a normal 68 00:05:23,260 --> 00:05:30,970 distribution, and that is breaking up the area under the curve into one, two and three standard deviations 69 00:05:30,970 --> 00:05:32,020 around the mean. 70 00:05:32,350 --> 00:05:38,920 So here we have a normal curve, we have the mean here at MU, and we've separated the area under the 71 00:05:38,920 --> 00:05:41,440 curve based on standard deviation. 72 00:05:41,440 --> 00:05:49,780 So this value right here, Mu plus sigma means the mean plus one standard deviation, whereas mu plus 73 00:05:49,780 --> 00:05:53,920 two sigma here means the mean plus two standard deviations. 74 00:05:53,920 --> 00:06:00,430 So here for this green curve, we said that the mean was here at five and that the standard deviation 75 00:06:00,430 --> 00:06:01,540 was three. 76 00:06:01,540 --> 00:06:06,880 So one standard deviation above the mean has to be five plus three or eight. 77 00:06:06,880 --> 00:06:13,630 So we see that here at eight and then one standard deviation below, the mean has to be five minus three 78 00:06:13,630 --> 00:06:15,730 or over here at two. 79 00:06:15,730 --> 00:06:23,380 And so we can see for this green curve that we have the mean here and then this is one standard deviation 80 00:06:23,380 --> 00:06:26,740 below the mean and one standard deviation above the mean. 81 00:06:26,740 --> 00:06:35,590 Contrast that with this blue curve here where we see the mean at the same value of five. 82 00:06:35,590 --> 00:06:41,230 And then we said that standard deviation was equal to one, which means one standard deviation above 83 00:06:41,230 --> 00:06:46,540 the mean of five has to be at six here and one standard deviation below the mean. 84 00:06:46,540 --> 00:06:50,260 It has to be at five minus one or four right here. 85 00:06:50,290 --> 00:06:57,910 So we have four and six and we can say that for the blue curve, an interval of one standard deviation 86 00:06:57,910 --> 00:07:00,610 around the mean ranges from 4 to 6. 87 00:07:00,610 --> 00:07:06,520 Whereas for the green curve, an interval of one standard deviation around the mean ranges from 2 to 88 00:07:06,520 --> 00:07:12,610 8, a much wider interval for one standard deviation around the mean, which of course makes sense because 89 00:07:12,610 --> 00:07:18,250 we know that for the green curve the data is more spread out, variance and standard deviation are larger. 90 00:07:18,250 --> 00:07:23,560 That measure of spread is larger, the data is more spread out away from the mean, whereas for the 91 00:07:23,560 --> 00:07:26,500 blue curve, the data is more tightly clustered around the mean. 92 00:07:26,500 --> 00:07:28,720 We see that smaller standard deviation. 93 00:07:28,720 --> 00:07:34,720 So for the normal distribution, we're particularly interested in these intervals that define one standard 94 00:07:34,720 --> 00:07:40,630 deviation around the mean between mu minus sigma and mu plus sigma, two standard deviations around 95 00:07:40,630 --> 00:07:46,810 the mean between mu minus two sigma and mu plus two sigma and three standard deviations around the mean 96 00:07:46,810 --> 00:07:50,230 between U minus three sigma and U plus three sigma. 97 00:07:50,260 --> 00:07:59,560 Because what we know to always be true for normally distributed data is that 68% of the data will always 98 00:07:59,560 --> 00:08:02,500 fall within one standard deviation of the mean. 99 00:08:02,500 --> 00:08:09,430 And because the normal distribution is always symmetric around the mean, that means 34% of the data. 100 00:08:09,430 --> 00:08:17,380 Or we could also say 34% of the area under the probability density function under this curve, or 34% 101 00:08:17,380 --> 00:08:23,950 of the probability has to fall between one standard deviation below the mean and the mean, And 34% 102 00:08:23,950 --> 00:08:29,110 of the probability or the area under the curve has to fall between the mean and one standard deviation 103 00:08:29,110 --> 00:08:30,040 above the mean. 104 00:08:30,040 --> 00:08:36,970 We also know that for every normal distribution, about 95% of the area under the curve will fall within 105 00:08:36,970 --> 00:08:45,160 two standard deviations of the mean, and that about 99.7% of the area under the curve or the probability 106 00:08:45,160 --> 00:08:48,040 will fall within three standard deviations of the mean. 107 00:08:48,070 --> 00:08:54,790 These numbers, this idea, these percentages are called the empirical rule or more blatantly, the 108 00:08:54,790 --> 00:09:00,670 6895, 99.7 rule given by these values 68, 95, 99.7. 109 00:09:00,670 --> 00:09:06,850 But the empirical rule is what tells us that for every normal distribution, for every set of normally 110 00:09:06,850 --> 00:09:12,550 distributed data, for every probability density function, representing a normal curve, any time we 111 00:09:12,550 --> 00:09:18,430 have this bell shaped curve that is normally distributed, then we can always break up the area under 112 00:09:18,430 --> 00:09:24,490 the curve based on these percentages, which allows us to answer all kinds of probability questions. 113 00:09:24,490 --> 00:09:30,430 For instance, with this information alone, if we know that our normally distributed data has a mean 114 00:09:30,430 --> 00:09:35,440 of five and a standard deviation of three, going back to this curve that we started with here and we 115 00:09:35,440 --> 00:09:40,720 want to answer the question, what is the probability that a randomly selected value in our data set 116 00:09:40,720 --> 00:09:46,210 will fall between five and eight or we'll have a value between five and eight? 117 00:09:46,210 --> 00:09:52,630 Well, we know that the mean is five, and that the boundary here of one standard deviation above the 118 00:09:52,630 --> 00:09:54,010 mean is eight. 119 00:09:54,010 --> 00:09:59,080 So the probability that one randomly selected data point is going to fall between five and eight. 120 00:09:59,110 --> 00:09:59,890 That's just. 121 00:09:59,970 --> 00:10:04,590 This particular section of the normal distribution. 122 00:10:04,590 --> 00:10:11,460 And we know that 34% of the area under the curve falls within this interval. 123 00:10:11,460 --> 00:10:17,100 And so we know right away that there's a 34% chance that a randomly selected value from our data set 124 00:10:17,100 --> 00:10:19,640 will take on a value between five and eight. 125 00:10:19,650 --> 00:10:23,850 And of course, we can use these percentages in an infinite number of other ways. 126 00:10:23,850 --> 00:10:26,030 So take this same data set. 127 00:10:26,040 --> 00:10:29,430 What's the probability that we get a value less than five? 128 00:10:29,430 --> 00:10:30,960 Well, five is the mean. 129 00:10:30,960 --> 00:10:36,450 And in a normally distributed data set, 50% of the data is below the mean, 50% of the data is above 130 00:10:36,450 --> 00:10:36,780 the mean. 131 00:10:36,780 --> 00:10:40,890 So the probability of getting a value less than five is 50%. 132 00:10:40,890 --> 00:10:44,670 The probability of getting a value greater than five is also 50%. 133 00:10:44,670 --> 00:10:48,240 The probability of getting a value less than 11. 134 00:10:48,240 --> 00:10:51,840 For this normal curve here the mean is five. 135 00:10:51,840 --> 00:10:53,640 The standard deviation is three. 136 00:10:53,760 --> 00:10:58,320 So one standard deviation above the mean is five plus three equals eight. 137 00:10:58,320 --> 00:11:02,130 Two standard deviations above the mean is eight plus three equals 11. 138 00:11:02,130 --> 00:11:04,920 So 11 is two standard deviations above the mean. 139 00:11:04,950 --> 00:11:09,300 11 puts us right here at two standard deviations above the mean. 140 00:11:09,300 --> 00:11:14,430 And so to find the probability that we get a value less than 11 would just mean that we add up all of 141 00:11:14,430 --> 00:11:21,810 these probabilities, starting with 13.5 and then 3430 for 13.5 and 2.35. 142 00:11:21,810 --> 00:11:27,270 Or how about the probability that we get a value outside three standard deviations around the mean? 143 00:11:27,360 --> 00:11:34,290 Well, we know that within three standard deviations of the mean, we have 99.7% of the area under the 144 00:11:34,290 --> 00:11:40,830 curve, which means the area outside of three standard deviations is 0.3%. 145 00:11:41,070 --> 00:11:47,430 If we divide that by two to put half of that area into each tail, we see these values right here. 146 00:11:47,430 --> 00:11:54,060 So instead of the total area in these two tails being 0.3% in this upper tail here we have 0.15%. 147 00:11:54,060 --> 00:12:00,630 And in the lower tail here, we have 0.15% of the area outside of three standard deviations away from 148 00:12:00,630 --> 00:12:01,170 the mean. 149 00:12:01,170 --> 00:12:07,080 And that 0.15% captures all of the area under the curve infinitely. 150 00:12:07,080 --> 00:12:10,020 Out to the left here and then infinitely out to the right here. 151 00:12:10,020 --> 00:12:17,550 So within three standard deviations of the mean, we have 99.7% of the area, 99.7% of the probability. 152 00:12:17,550 --> 00:12:25,170 This 0.15% in each tail captures everything else As far as this tail goes out to the left or as far 153 00:12:25,170 --> 00:12:29,760 as this tail goes out to the right, everything outside of three standard deviations. 154 00:12:29,760 --> 00:12:35,670 So then, for instance, if we wanted to say what's the probability that we get a value less than 11 155 00:12:35,670 --> 00:12:41,250 going back to this normal curve here with a mean of five and a standard deviation of three? 156 00:12:41,340 --> 00:12:51,840 Well, if our mean is five and then one standard deviation above the mean is five plus three or eight, 157 00:12:51,870 --> 00:12:54,330 then two standard deviations. 158 00:12:55,150 --> 00:12:55,540 Above. 159 00:12:55,540 --> 00:13:01,300 The mean is five plus six or 11 or eight plus three equals 11. 160 00:13:01,300 --> 00:13:07,750 So we know that 11 for this curve puts us at two standard deviations above the mean, which is right 161 00:13:07,750 --> 00:13:08,230 here. 162 00:13:08,230 --> 00:13:11,480 So the probability that we get a value less than 11. 163 00:13:11,500 --> 00:13:20,270 All we have to do is add up all these percentages 0.15, 2.35, 13.5, 34, 34 and 13.5. 164 00:13:20,290 --> 00:13:30,430 Or to take the easier route since probability always sums to 100% or one, we can just take 100% and 165 00:13:30,430 --> 00:13:34,090 we can subtract from that this 2.35. 166 00:13:35,060 --> 00:13:48,470 Percent value and this 0.15% value and we get a result of 97.5%, which tells us that we have a 97.5% 167 00:13:48,470 --> 00:13:51,620 chance of finding a value less than 11. 168 00:13:51,650 --> 00:13:56,060 Given this normally distributed curve with these properties a mean of five, a standard deviation of 169 00:13:56,060 --> 00:14:02,540 three, it's this idea of the empirical rule and these standardized percentages, because the normal 170 00:14:02,540 --> 00:14:09,290 distribution is a standard curve that's going to allow us, as we continue on here, to answer all kinds 171 00:14:09,290 --> 00:14:15,200 of probability questions and then even start doing some statistical analysis on normally distributed 172 00:14:15,200 --> 00:14:22,430 data, which ultimately leads us to an idea about significance, being able to start with a hypothesis, 173 00:14:22,430 --> 00:14:29,420 run an experiment or collect data, and then determine the significance of the result of that data to 174 00:14:29,420 --> 00:14:33,160 determine how unusual it is to find the result that we did. 175 00:14:33,170 --> 00:14:36,320 We'll talk all about that when we get to hypothesis testing. 176 00:14:36,320 --> 00:14:43,160 But the idea here is that this normal distribution we see over and over and over again in real life 177 00:14:43,160 --> 00:14:49,880 and almost more importantly, even when we collect data and we determine that it's not normally distributed, 178 00:14:49,880 --> 00:14:57,080 we'll learn about a way that we can sort of change that non normal data into normally distributed data 179 00:14:57,080 --> 00:15:03,050 so that we're able to use this normal distribution and all of these probability figures that go along 180 00:15:03,050 --> 00:15:03,560 with it. 181 00:15:03,560 --> 00:15:09,710 And so because we'll be using this normal distribution over and over and over again for normally distributed 182 00:15:09,710 --> 00:15:17,180 data and even non normally distributed data, it's critical that we understand this idea of this symmetric 183 00:15:17,180 --> 00:15:23,330 bell shaped normal distribution, where the empirical rule can give us the likelihood that a particular 184 00:15:23,330 --> 00:15:28,670 data point falls within some certain standard deviation interval around the mean.