WEBVTT 00:02.290 --> 00:04.720 In this session, we will discuss a potential. 00:06.130 --> 00:10.160 So, first of all, let us discuss about the frequency distribution. 00:10.510 --> 00:12.610 What is the probability distribution? 00:13.650 --> 00:22.500 So the frequency distribution gives the exact frequency or the number of times a data point occurs, 00:23.160 --> 00:30.780 while the probability distribution gives the probability of occurrence of the given data point, when 00:30.780 --> 00:38.490 the first number of the best cases are large, the frequency distribution and probability distribution 00:38.610 --> 00:40.350 are similar in shape. 00:42.390 --> 00:47.230 So let us see what is the difference between frequency distribution and probability distribution. 00:48.000 --> 00:50.320 So let's say we have these data points. 00:51.000 --> 00:54.390 Now, these data points range from 10 to AP. 00:55.660 --> 01:07.100 And the frequency of these data point is two for them, three elevens, five to six turbines for four 01:07.300 --> 01:16.550 beings, zettl, 15 values, 14, 16 values, then 17 values and six 18 values. 01:17.410 --> 01:23.560 So if we want to find out and the total number of these values is 50. 01:24.500 --> 01:32.280 So there are in total 15 values and this is the number of times this particular value occurs. 01:32.840 --> 01:36.500 So 14 times 16 is occurring in the data. 01:36.620 --> 01:39.140 So 16 is the mode of the data. 01:40.010 --> 01:45.620 Similarly, we can find out the values and how often they occur and calculate the mean and median. 01:47.090 --> 01:55.670 When you want to find out the relative frequency related frequency, you will give the probability of 01:56.090 --> 01:57.290 the number occurring. 01:58.040 --> 01:59.450 So let us see. 01:59.600 --> 02:03.070 We want to find out the relative frequency of then occurring. 02:03.590 --> 02:08.660 So the relative frequency of then occurring will be two out of 50. 02:09.580 --> 02:17.770 So two times out of 50 then will be happening, three out of 50 times 11 will be a. 02:18.920 --> 02:28.280 Five times out of 50 will be the opponents of 12, similarly, 14 times out of the 16 will be occurring. 02:28.790 --> 02:33.440 So these are the frequencies it related to each other. 02:33.500 --> 02:40.430 So these are the number of frequencies where these data will be happening and the complete some of these 02:40.430 --> 02:43.250 relative frequencies comes out to be one. 02:43.900 --> 02:51.070 And when we find out when we divide these values, we actually get the probability of the value occurring. 02:51.860 --> 02:59.150 So the frequency actually helps us in driving the probability of the data occurring. 03:00.610 --> 03:09.940 So the probability of occurring is zero point zero, which means that there is a four percent chance 03:10.240 --> 03:11.800 of then occurring. 03:12.740 --> 03:21.080 When we randomly pick out the value from this particular population, similarly, there is a 12 percent 03:21.080 --> 03:23.870 chance of 13 occurring. 03:24.910 --> 03:33.720 Here there is a 28 percent chance of 16 occurring when we take a sample out of this data. 03:35.140 --> 03:42.880 So this is what the probability is and this is what the frequency is, and we can plot these together, 03:42.880 --> 03:46.460 frequency distribution and the probability distribution. 03:46.660 --> 03:53.960 So when we plot the frequency distribution, the area and the frequency distribution actually provides 03:53.960 --> 03:56.040 the probability of the value occurring. 04:03.020 --> 04:11.150 So here you can see this is the frequency distribution plot where the frequency of every value is given 04:11.870 --> 04:14.540 here, the frequency is almost Siedel here. 04:14.540 --> 04:19.460 The frequency used in frequency is 20 year, frequency is almost 22. 04:20.000 --> 04:26.480 And out of this, the probability distribution plot is created, which is this line of code which is 04:26.480 --> 04:26.800 given. 04:27.110 --> 04:30.750 And this will give the probability of a particular value of three. 04:30.950 --> 04:38.180 So if you want to find out the probability of one occurring, then we can find out the area below this 04:38.180 --> 04:46.030 particular area, below this particular area, and it will give the probability of one knackering. 04:46.250 --> 04:51.580 And similarly, we want to find out the probability of value occurring below 1.0. 04:51.860 --> 04:57.230 Then we can find out the area occurring below this particular area of Dakhil. 04:57.530 --> 04:59.750 Then it will give the probability of one point to work. 05:00.260 --> 05:05.510 Similarly, if you want to find out the probability of value occurring between and B, then we can find 05:05.510 --> 05:12.260 out the area below the Gulf between A and B, and it will give the probability of values B this between 05:12.260 --> 05:13.370 this and B. 05:14.560 --> 05:18.400 So this is what the distribution gulf will provide us. 05:19.760 --> 05:27.500 Now, what is the distribution of sample means now we are getting close to the central limit. 05:27.860 --> 05:30.890 Now we know what the frequency distribution is. 05:31.130 --> 05:33.580 We know what probability distribution is. 05:33.740 --> 05:37.400 So let's see what is the distribution of sample means. 05:38.030 --> 05:47.420 So the distribution of sample means is the collection of sample means for all the possible random samples 05:47.570 --> 05:49.400 of a particular size and. 05:50.930 --> 05:58.820 So what we are doing is we are creating a collection of some means, so we are collecting. 05:58.910 --> 06:00.970 We are creating a lot of samples. 06:01.160 --> 06:06.620 Let us say we create hundred samples from those hundred samples. 06:06.620 --> 06:09.200 We calculate different means. 06:09.440 --> 06:15.220 We take out different means, and then we block these means together. 06:16.070 --> 06:23.440 So we collect the sample means for all the possible random samples of a particular size. 06:23.450 --> 06:28.400 And so the samples which we will be taking from the population will have a common size. 06:28.580 --> 06:34.730 So the size will be seen and because the samples will be taken randomly, then the values in the samples 06:34.730 --> 06:35.480 will be different. 06:36.680 --> 06:43.010 So because the values of the samples will be different, the mean value will also be different for all 06:43.010 --> 06:43.880 the samples. 06:44.950 --> 06:51.730 So we can obtain these samples and the means from the population and then we applaud them together. 06:52.250 --> 07:00.400 Now it is not a distribution of scores, but the distribution of statistics because they are not taking 07:00.400 --> 07:02.540 the mean of the entire population. 07:02.770 --> 07:06.240 We are not taking values from the entire population. 07:06.460 --> 07:14.980 We are only plotting the needs of all this data means of different samples. 07:16.440 --> 07:24.420 Now, when we do this, what will happen is the distribution will be kind of in a normal form, the 07:24.430 --> 07:33.570 distribution will be normal in nature, life will be normal in nature, because when we take out samples, 07:33.750 --> 07:39.900 the samples will be ranging from, let's say we have a population and the values of the population range 07:39.910 --> 07:41.130 from one two hundred. 07:42.080 --> 07:49.780 So now when we take all the samples, the sample values will be randomly distributed from one to hundred. 07:51.070 --> 08:00.040 And they are taking the mean of these samples now, the mean of the sample will never be always the 08:00.040 --> 08:02.500 mean of the samples and things would be fafi. 08:02.500 --> 08:05.470 Sometimes it will be 51, sometimes it will be 52. 08:05.470 --> 08:06.980 Sometimes it will be 48. 08:07.000 --> 08:08.590 Sometimes it will be 42. 08:08.590 --> 08:16.330 To now, based on the sample and how the random sample is generated, those values will be more number 08:16.330 --> 08:24.700 of samples will have values closer to the center values, which is 50, and they will be very few samples 08:24.910 --> 08:30.250 for which somehow the mean value will be near the edges. 08:31.060 --> 08:38.280 So this is the reason why the gold which will be created from these machines will be in normal nature. 08:38.680 --> 08:46.330 So it will be almost perfectly normal if either the population from which the sample is drawn is normal 08:46.600 --> 08:50.170 or the number of sample is relatively large. 08:51.250 --> 08:58.030 That is be almost basically the size of the sample is large, so if the size of the sample is large, 08:58.240 --> 09:02.170 then the distribution has a mean that is equal to the population. 09:04.400 --> 09:08.620 So let's look at some properties of normal, Goebels. 09:09.140 --> 09:11.450 So what is the properties of the namiko? 09:11.810 --> 09:19.520 A normal distribution of esveld sheep, the mean median and mode will be equal and will be located in 09:19.520 --> 09:20.950 the center of the distribution. 09:21.560 --> 09:27.000 Then a normal distribution is unimportant, which is it will have a single peak. 09:27.380 --> 09:29.060 It will be symmetrical nature. 09:31.170 --> 09:37.880 Then the Gulf is above the mean, which is equivalent to saying that if this ship is same on the both 09:37.890 --> 09:38.680 side of the Gulf. 09:38.970 --> 09:44.970 So if we have this line passing, so the shape of the Gulf towards the left will be the same as the 09:45.000 --> 09:46.530 shape of the globe on the right. 09:47.590 --> 09:53.380 Then the goal was continuous, that is, the values will range from some value to another value, but 09:53.380 --> 09:55.390 there will not be any gaps in between. 09:57.170 --> 10:00.420 Then the Gulf will never touch the axis here. 10:00.440 --> 10:06.110 You can see this going endlessly beyond, but it is never touching the x axis here. 10:07.910 --> 10:14.670 So theoretically, no matter how far in either direction the globe extends, it never needs access. 10:16.330 --> 10:23.590 The total area under the normal distribution code is equal to one or two hundred, so the area and this 10:23.590 --> 10:29.320 entire gulf, which is the probability actually, so the probability of a value occurring under this 10:29.320 --> 10:30.040 Gulf will be. 10:31.510 --> 10:35.860 So it will be either it will range from zero to 100 or zero. 10:35.860 --> 10:42.250 The one will be the probability and the probability percentage will be zero to 100 percent. 10:42.730 --> 10:43.050 Right. 10:43.390 --> 10:47.680 So this is about the normal gulf and the area of the Gulf. 10:47.770 --> 10:50.320 Give the probability of the point falling under the sea. 10:50.950 --> 10:52.990 So if we have any point. 10:54.200 --> 11:00.540 Then the area under this particular point bin line will give the probability of that value occurring. 11:02.000 --> 11:08.240 Now, there are a few things about the Namiko, which is the area under the part of the normal Gulf 11:08.240 --> 11:10.400 that lies within one standard deviation. 11:10.910 --> 11:12.910 Now, this is the mean value. 11:12.950 --> 11:16.850 This is the mean median mode value, the central value. 11:17.910 --> 11:23.700 Now, this is one standard deviation away in the region and one standard deviation of it in the positive 11:23.700 --> 11:24.180 direction. 11:24.510 --> 11:29.790 So this is basically one standard deviation area of one standard deviation. 11:30.120 --> 11:36.270 This one standard deviation area contains approximately sixty eight percent of the. 11:38.090 --> 11:44.300 This one standard deviation bill contains approximately 60 percent of the data. 11:46.490 --> 11:49.070 Then the two standard deviation. 11:50.260 --> 11:57.850 This is the standard deviation area, it will contain almost 95 percent of the data and the three standard 11:57.850 --> 12:02.380 deviation will contain almost ninety nine point seven percent. 12:03.550 --> 12:12.660 This is the rule which is already defined for the normal golf or the baker, which is this is the percentage 12:12.670 --> 12:13.340 distribution. 12:13.810 --> 12:16.980 So how can we get help from this now? 12:17.050 --> 12:20.480 I think we have already discussed this example, but again, talking about. 12:21.040 --> 12:23.870 So we have this kind of distribution in the nonlegal. 12:24.040 --> 12:31.850 So what will happen is that let's say we want to find out the thought of the two percent people out 12:32.530 --> 12:39.340 of two point two percent, people who should be given a lease or should be given extra rewards for their 12:39.340 --> 12:39.820 hard work. 12:40.360 --> 12:41.630 So what will we do? 12:41.650 --> 12:44.880 We will find out later, see the dolphin things. 12:45.040 --> 12:50.800 And from the top ratings, we will find out the top two point, the weight person, people who are operating, 12:51.160 --> 12:54.170 and then we will be giving them a reward. 12:54.310 --> 12:57.610 So how we can do it, we can simply find out the mean value. 12:58.830 --> 13:01.020 Because we cannot really find this value, right? 13:01.040 --> 13:03.420 How will we know what is going to it? 13:04.110 --> 13:10.310 So what we can do is we can simplify the mean value and we can find out the standard deviation. 13:10.470 --> 13:18.060 And based on the mean value plus the standard deviation, we can declare that whoever is getting the 13:18.080 --> 13:20.580 rating above mean the standard deviation. 13:20.790 --> 13:21.890 We will give them whatever. 13:23.640 --> 13:30.030 Or let's say someone who has everything to you mean minus two standard deviation, we will give them 13:30.030 --> 13:33.710 additional training so that they perform better next quarter. 13:34.770 --> 13:36.450 They actually can do something like the. 13:38.140 --> 13:40.740 So this is the population distribution. 13:40.870 --> 13:47.620 So this is the normal population distribution curve, and for this particular article, the mean is 13:47.800 --> 13:51.120 zero point one, the meanness zero point one. 13:51.940 --> 13:55.600 This is the distribution and the mean value is zero point one to. 13:56.840 --> 14:05.990 The standard deviation is twenty one 06, this is the standard deviation for this block, we have mean 14:05.990 --> 14:10.070 as zero point one two and standard deviation as twenty point zero six. 14:11.170 --> 14:17.370 Now, what we are doing is we are taking out several samples, so we will take, I would say, 50, 14:17.380 --> 14:23.760 100 or 200 or 500 samples out of this of this population data. 14:24.640 --> 14:28.010 Out of this entire data, we will take out different samples. 14:28.300 --> 14:35.190 Now we are taking out samples so the values will anyhow range from this minus two plus hundred. 14:35.710 --> 14:40.040 OK, but we need to find out some sample values out of it. 14:40.300 --> 14:44.020 So what we are doing is we are taking out a few samples. 14:44.350 --> 14:48.150 So the values of the samples, you can see the distributions are different. 14:49.850 --> 14:56.500 See this, there are different distributions for the sampas, they're not exactly the same samples. 14:57.290 --> 15:03.920 So what happens is the standard deviation is also slightly different for the samples here. 15:03.930 --> 15:06.470 The standard deviation is nineteen point something. 15:06.470 --> 15:09.700 Here, the standard deviation is eighteen point something and so on. 15:09.950 --> 15:13.580 So different samples will have different standard deviations. 15:14.730 --> 15:19.890 Similarly, these samples which we have generated will have some mean value. 15:20.910 --> 15:26.580 Now, this means that you will also be different here, the mean value is minus zero point one eight 15:26.580 --> 15:30.650 one here, the mean value is minus one point seventy seven. 15:30.930 --> 15:33.740 Here, the money will mean value is zero point three. 15:33.930 --> 15:36.390 Here, the mean values, minus zero point zero four. 15:36.660 --> 15:38.980 So these mean values are also different. 15:39.810 --> 15:48.240 So what happens is when we take five hundred or two hundred or any number of such samples, then we 15:48.240 --> 15:49.530 create the. 15:51.950 --> 15:59.680 We create the sampling distribution out of it, which is the distribution of the sample means so here 15:59.690 --> 16:04.910 what we have done is we have created a block of all the means. 16:05.950 --> 16:13.300 Of the samples that we have obtained, so here we have certain mean values, so we have only plotted 16:13.300 --> 16:17.230 these export values in this particular judge you. 16:18.570 --> 16:22.080 OK, so now when we have plotted this. 16:23.070 --> 16:24.880 Sample means here. 16:25.430 --> 16:29.100 Now, this is where central to them comes in. 16:30.820 --> 16:37.930 So what we have done enough, we had one population out of this population, we have taken out different 16:37.930 --> 16:44.700 samples, the sample size, we are expecting to be more than 30 or at least 30 so that it is a good 16:44.710 --> 16:46.540 representative of the population. 16:47.540 --> 16:54.350 And they have created several samples from these samples, we have calculated the mean value of each 16:54.350 --> 16:58.910 and every something so mean value of each sample comes out to the buttocks. 16:58.940 --> 17:02.660 But we have, let's say, one hundred or one, two, one hundred samples. 17:02.660 --> 17:05.090 And we have the economics blood from these samples. 17:05.780 --> 17:09.440 Now, from the samples, we have generated a lot. 17:10.250 --> 17:18.750 And this is actually a lot of the mean values, which is the X values from these samples. 17:19.100 --> 17:24.950 So this is a lot of different values from different samples of what we created. 17:24.960 --> 17:29.060 We needed a distribution means one 60 random samples. 17:29.060 --> 17:35.090 We have taken 160 random samples and taken out the mean value and plotted them here. 17:35.690 --> 17:38.890 Now each consists of one hundred and three observations. 17:39.140 --> 17:42.230 So the sample size here is one hundred and three. 17:42.530 --> 17:44.050 That is the end value. 17:44.180 --> 17:46.940 The end value here is one hundred and three. 17:47.510 --> 17:50.680 And the number of samples is 160. 17:51.230 --> 17:55.910 Now, from this sampling distribution, this distribution of means. 17:57.530 --> 18:00.890 The distribution of means is also already sampling distribution. 18:01.160 --> 18:08.290 So from this sampling distribution, the mean is zero point zero eight. 18:09.770 --> 18:16.040 The calculated mean of this sampling distribution is zero point zero eight letters compared with the 18:16.040 --> 18:17.050 population mean. 18:17.950 --> 18:20.890 Here, the mean is zero point one to. 18:22.680 --> 18:26.010 And the the standard deviation is twenty point zero six. 18:27.380 --> 18:31.360 Here, the standard error is one point seven four. 18:31.880 --> 18:32.580 Now let us see. 18:33.800 --> 18:34.250 So. 18:36.180 --> 18:38.970 The central limit theorem states that. 18:40.020 --> 18:42.780 The distribution of sample mean. 18:43.990 --> 18:46.910 The sampling distribution should be nearly normal. 18:47.860 --> 18:52.840 But just through the sampling distribution, which we have created, which is the distribution of sample 18:52.840 --> 18:55.000 means, is normal. 18:57.970 --> 19:04.240 And the mean of the sampling distribution should be approximately equal to the population. 19:05.110 --> 19:06.820 That is the mean of the sampling. 19:06.820 --> 19:11.500 Distribution should be approximately equal to the population me, which is zero point one. 19:11.720 --> 19:12.520 So what is it? 19:15.450 --> 19:19.010 The mean of the sampling distribution is zero point zero it. 19:20.290 --> 19:26.200 And the mean of the population is zero point one, which is almost similar. 19:26.350 --> 19:28.360 The mean is almost close to each other. 19:30.130 --> 19:36.970 Now, the standard error is the standard deviation of the sampling me, the standard error which has 19:36.970 --> 19:41.700 been calculated, is actually the standard, the standard deviation of the sampling distribution. 19:42.980 --> 19:46.790 Standard error is the standard deviation of the sampling distribution. 19:47.030 --> 19:48.480 So what does that signify? 19:48.860 --> 19:57.920 This standard error is actually equal to the standard deviation of the population divided by the square 19:57.920 --> 20:01.030 root of the size of the sample. 20:01.910 --> 20:05.690 So twenty point zero six, which is the. 20:07.440 --> 20:09.300 Standard deviation of the population. 20:11.080 --> 20:14.770 Twenty point zero six, Bob, standard deviation of the population. 20:18.510 --> 20:22.470 And here we have the square root of one zero three, which is the. 20:23.990 --> 20:27.380 One hundred and three observations, so Vera. 20:28.780 --> 20:29.650 110. 20:32.010 --> 20:35.430 Today is the end, which is the size of each sample. 20:36.840 --> 20:39.170 So this gives us one point nineteen. 20:41.160 --> 20:41.730 Which is. 20:42.750 --> 20:48.150 One point seven four, so this is also equivalent, so this is what we are doing from this. 20:49.230 --> 20:50.430 Send a to. 20:51.590 --> 20:55.460 So we can find out the characteristics of the population. 20:57.010 --> 21:02.950 From the sampling distribution, from the sampling distribution, we can create different sampling distribution 21:03.190 --> 21:05.810 and find out the characteristics of the population here. 21:06.100 --> 21:12.400 So this is what central imitative is, which is that the distribution of the sample mean should be nearly 21:12.400 --> 21:12.890 normal. 21:12.890 --> 21:18.790 In the mean of the sampling, distribution should be approximately equal to the population mean. 21:19.480 --> 21:25.960 That is the mean of the sampling distribution of this sampling distribution will be almost equal to 21:25.960 --> 21:27.910 the mean of the population distribution. 21:28.960 --> 21:36.550 And the standard error, which is the standard deviation of the sampling distribution, will be equivalent 21:36.550 --> 21:37.000 to the. 21:37.990 --> 21:46.300 Standard deviation of the population distribution divided by the square root of the sample size, the 21:46.300 --> 21:48.900 number of observations in each sample. 21:50.300 --> 21:53.840 So this is what we have learned from the central limit.