WEBVTT 00:01.880 --> 00:08.690 Now, the next measure in the descriptive statistics is the measures of variability. 00:09.950 --> 00:13.130 Which is also point the measure of spreading. 00:14.410 --> 00:20.680 Measure of spread sends us about how five the values are spread out. 00:22.040 --> 00:26.810 So here, if you see we have these values ranging from 1 to 100. 00:27.350 --> 00:36.230 So here to show that how these values are spreading, how these values are going from 1 to 100, if 00:36.230 --> 00:44.120 the values are more present toward between 1 to 12, I'm there's very few values between 12, 100. 00:44.360 --> 00:49.970 So this kind of data is shown by the measure of variability. 00:50.810 --> 00:54.550 This shows that how much the values are reading in between. 00:56.460 --> 00:56.880 So. 00:57.990 --> 01:06.300 We have 9 million more, which can provide information about the central points of the data, but they 01:06.300 --> 01:08.930 do not tell us about how the detail varies. 01:10.150 --> 01:18.250 So we use range in the four day range variance and standard deviation as the measures of. 01:19.900 --> 01:24.340 While 9 million and more are the measures of center attendance. 01:28.250 --> 01:29.840 Now let us discuss about. 01:31.090 --> 01:33.310 The quartiles and in the dating. 01:34.520 --> 01:35.870 So let us see. 01:35.870 --> 01:37.430 We have these values. 01:38.210 --> 01:41.780 We have data ranging from 3 to 19. 01:43.160 --> 01:48.960 So these values are ranging from 3 to 19 and then at a total of 15 values which represent. 01:50.100 --> 01:56.100 So quartiles are basically the data divided into four quarters. 01:57.240 --> 02:03.780 So the detail from 3 to 7 is the first quarter, but with just 25% of the values. 02:04.960 --> 02:09.760 Then 8 to 11 is the next 25% of the values. 02:10.860 --> 02:18.150 Then 12 to 15 is the next 25% of the value, and then 17 onwards is the next. 02:19.860 --> 02:28.770 25% of the values, the values from 3 to 11 mean that this is the 50% of the value. 02:29.010 --> 02:34.800 While 2217 mean that this contains 75% of the values. 02:35.340 --> 02:40.160 So each quarter versus 25% of the entire data. 02:41.280 --> 02:42.210 Which is shown here. 02:42.780 --> 02:45.810 So each quarter will have 25%, be they net. 02:46.910 --> 02:53.150 On the line, which is dividing this data is called coordinates. 02:54.050 --> 02:56.720 So this one becomes the Q one. 02:57.110 --> 03:01.430 This becomes Q, which is also the median of these values. 03:02.030 --> 03:07.790 And the third division between the 25, the 75 and 25 of the. 03:07.790 --> 03:10.370 This becomes the third quarter. 03:10.610 --> 03:11.570 But just three. 03:13.340 --> 03:20.540 Now interquartile range is the difference between Q3 and Q1. 03:21.910 --> 03:28.300 So the in the model range for this particular data would be 17 minus seven, which is day. 03:30.520 --> 03:34.870 Now to find out if a value is an outlier or not. 03:35.290 --> 03:37.630 We have the simple formula present. 03:38.050 --> 03:48.850 So the outlier is calculated by Q one, which is seven -1.5 times the in the quartile range. 03:49.930 --> 03:53.160 And so it will be something there below. 03:53.160 --> 03:53.650 Q What? 03:55.320 --> 04:05.120 I'm the next outlier, which is the above range would be Q three plus 1.5 times the in the quartile 04:05.130 --> 04:05.520 range. 04:05.760 --> 04:10.590 So 17 plus 115 would be 30. 04:10.920 --> 04:19.920 So any value above 32 will be an outlier on any value below minus eight would be an outlier. 04:21.270 --> 04:23.940 So this is what Cornell done in the ages. 04:24.690 --> 04:27.150 Now, here, let us see this data. 04:27.360 --> 04:28.850 And here we have a box. 04:28.860 --> 04:29.550 BLOCK three. 04:31.010 --> 04:33.500 This kind of diagram is called a box plot. 04:34.220 --> 04:36.890 So here, this again shows the same thing. 04:38.070 --> 04:39.690 So this law was good. 04:40.170 --> 04:43.470 Usually shows the law most value. 04:45.900 --> 04:53.970 And this second line, which is the beginning of the box, shows the Q one line. 04:55.040 --> 04:59.990 Which is which means that this much area contains 25% of the data. 05:01.440 --> 05:07.170 The next portion from the Q one line to the Q2 line. 05:08.110 --> 05:14.020 This is the line for the Q2 and this is called the median also. 05:14.230 --> 05:22.270 So this much area contains that 50% of the data and the area above this line is the next 50% of the 05:22.270 --> 05:27.520 data on this middle line or the end of the box line is the. 05:28.560 --> 05:31.370 Range from 50% to 75% of. 05:33.000 --> 05:34.350 I'm the line above. 05:34.650 --> 05:40.550 This end of the line is the values above 75% of the data. 05:41.470 --> 05:45.760 The above, the score shows the maximum value. 05:46.810 --> 05:49.990 In peace, there is any outlier present. 05:50.290 --> 05:57.550 Then this upper was this good and the lower good will not show the minimum and maximum value, but in 05:57.550 --> 05:59.740 dawn they will sure be. 06:01.270 --> 06:02.340 Outlier line. 06:03.130 --> 06:05.590 So this line would then go on. 06:05.590 --> 06:06.060 Endo. 06:06.580 --> 06:14.770 Q One -1.5 base IQ and any value above this line, this particular value which is depicted by this stuff, 06:15.040 --> 06:16.750 is the outlier value. 06:18.160 --> 06:21.160 So here the lower cost quartile value. 06:21.160 --> 06:27.100 Q one is seven, this is seven, which is Q1 and Q2 is 8.5. 06:28.100 --> 06:29.420 Which is also the median. 06:30.460 --> 06:33.140 Then Q3 is nine. 06:35.120 --> 06:40.850 Then the lower whisker is £1.5 in the port range, which is at fourth. 06:42.570 --> 06:49.530 Because that is data presented below what we are showing it as the outlier. 06:50.610 --> 06:57.750 Now the upper school would be calculated by people as 1.5 frames in the model range, which is nine 06:57.750 --> 06:58.320 plus three. 06:58.420 --> 07:01.530 Well, so this is the maximum one. 07:02.750 --> 07:08.900 If there is no datapoint that will then the highest point, less than 12 would be considered as the 07:09.140 --> 07:09.910 upper whisker. 07:10.760 --> 07:14.810 This means the 1.52 out of a school can be uneven in. 07:16.200 --> 07:22.680 So this whiskey would either be the maximum value or 1.5 things that up. 07:23.900 --> 07:27.050 And this one would be either the minimum value or. 07:27.500 --> 07:29.860 Q one -1.5 things like you are. 07:32.190 --> 07:39.810 Now that we have learned about the different measures of central tendency, let us go ahead and have 07:39.810 --> 07:45.240 a look at how we can get to know about the spread of data. 07:46.200 --> 07:47.610 Or you can also see this. 07:49.110 --> 07:58.950 Now the measures of central tendency let us know about what are the central values or the data which 07:58.950 --> 07:59.340 we have. 08:00.150 --> 08:04.650 But that does not give us an insight of how the day district. 08:05.550 --> 08:06.730 So that is what we will get 08:10.680 --> 08:11.430 for that. 08:11.970 --> 08:15.240 Let us first have a look at what are the values. 08:16.080 --> 08:21.300 The first one is variants, and the second one is standard deviation. 08:22.260 --> 08:33.580 Standard deviation is calculated by one upon N multiplied by v. 08:34.980 --> 08:39.240 Thus the square of difference of the mean. 08:39.240 --> 08:39.570 And 08:42.690 --> 08:48.330 these are some together and divided by in to find the same condition. 08:48.960 --> 09:00.750 Now we have considered mean as a value for calculating the standard deviation, and standard deviation 09:00.750 --> 09:04.170 is nothing but a square root of readings. 09:05.220 --> 09:08.460 So if you look at the formula, it is just the same. 09:10.470 --> 09:19.710 It is mean minus that for a minus that particular number or you can see the number minus the mean. 09:19.710 --> 09:23.220 You can see it anyway because it is being squared. 09:23.640 --> 09:24.990 So it doesn't make a difference. 09:25.770 --> 09:29.760 And further, your summing all the values up and dividing by n. 09:31.260 --> 09:39.690 Similarly, you're doing the same thing but taking an underscore and the root of square root of the 09:39.930 --> 09:40.830 medians here 09:44.490 --> 09:46.200 if made too complicated for now. 09:46.440 --> 09:49.150 But as we go forward, you'll understand. 09:49.440 --> 09:52.910 So let us try to understand why we have chosen this formula. 09:52.920 --> 09:54.900 How did this formula get formulated? 09:55.590 --> 09:56.730 So let's understand that. 09:59.700 --> 10:01.680 So this is just an example. 10:01.860 --> 10:04.770 So let's say we are considering these numbers. 10:05.640 --> 10:11.760 These are the set of numbers which we have, and we are trying to find out how this particular data 10:11.760 --> 10:12.480 is spread out. 10:14.190 --> 10:19.590 So when we are looking at the spread of this particular data, I can clearly see that the numbers are 10:19.590 --> 10:21.240 ranging from 1 to 16. 10:21.900 --> 10:28.250 There are some numbers which are one and two, which is the lower side of the numbers. 10:28.260 --> 10:34.080 Then I have numbers like seven, 12, 15, 16. 10:36.540 --> 10:38.610 Now these are few numbers. 10:38.730 --> 10:43.700 So you can see that there is a slight difference between the median. 10:43.710 --> 10:50.100 So if I calculate the mean of this particular list of numbers, the mean comes out to be 8.44. 10:50.790 --> 10:54.870 So you can see these are a few numbers which are actually very near to the mean. 10:55.200 --> 11:01.490 But apart from that, all of the numbers are distributed in the dataset. 11:01.500 --> 11:05.550 So there are very low numbers also and very high numbers are also present. 11:06.000 --> 11:13.530 But because these numbers are spread out, that is why the mean is coming out as 8.4. 11:14.790 --> 11:20.100 Now, what should we use to find out how the did they spread? 11:21.300 --> 11:23.610 So let us look this up. 11:23.610 --> 11:30.360 In Excel, women do the calculations in Xs so that you will get an infusion of what we are doing and 11:30.360 --> 11:31.380 why we are doing it. 11:34.140 --> 11:36.540 Now, these were the numbers which we were looking at. 11:36.960 --> 11:39.770 So let us first of all, find out the meaning of these numbers. 11:39.840 --> 11:41.280 The mean will come out when 11:51.360 --> 11:53.400 I'm taking an average of these numbers. 11:54.780 --> 11:56.520 So this is the mean of these numbers. 11:57.240 --> 12:00.630 Now we need to find out the state of these numbers. 12:00.630 --> 12:07.260 So I need to compare these numbers with a particular number to actually find out how the data is being 12:07.380 --> 12:07.920 changed. 12:08.760 --> 12:16.520 I need to have a number with which I can compare these so that I can get an idea of what is this thing? 12:17.010 --> 12:26.100 So what I'm doing is I'll be comparing these for the first number, the last number, and the mean of 12:26.100 --> 12:26.580 the numbers. 12:27.000 --> 12:29.790 And please note that these numbers are sorted in meters. 12:29.790 --> 12:31.280 So the first number and last. 12:31.610 --> 12:36.860 Are actually having a logical meaning that proposed in the last them what is actually the meaning and 12:36.860 --> 12:43.640 max of the list of numbers which we have to first find out that deviation from the first number. 12:43.640 --> 12:50.870 So the deviation from the first number will be this minus this. 12:52.750 --> 12:56.140 Now when we are finding the deviation we want to keep. 12:59.920 --> 13:01.480 This number is constant 13:04.840 --> 13:08.140 and I keep finding out the values. 13:08.470 --> 13:20.020 So here you can see this is basically one minus two, one minus four, one minus seven, one minus seven, 13:20.410 --> 13:25.720 one -12, one -12, one -15, one -16. 13:25.960 --> 13:27.820 These are the values which you're getting here. 13:30.250 --> 13:32.830 Now, I will again take the average of these numbers. 13:33.580 --> 13:35.860 Let me just copy the formulas here. 13:36.460 --> 13:38.430 And here I give the average of these numbers. 13:38.440 --> 13:41.110 I copy this formula here as well, 13:44.080 --> 13:47.170 so that once we have the values, we get the results accordingly. 13:48.130 --> 13:52.300 Now, this is the deviation from the first number. 13:53.170 --> 13:56.830 Now, similarly, I will calculate the deviation from the last number. 13:58.030 --> 14:06.730 So deviation will be what will be this number minus each in every number in the list. 14:07.720 --> 14:10.360 So for that, I will have fixed this number again. 14:10.360 --> 14:11.500 So I put a dollar here. 14:13.360 --> 14:19.390 Now, here, I'm also giving you a slide inside about how we can use formulas in Excel. 14:19.390 --> 14:23.140 So you can take that as a additional point here. 14:23.440 --> 14:28.060 You will get to know how you can perform these in using an Excel. 14:28.060 --> 14:35.500 Also, we will be learning that how we can do this in by hand, how we can do this from using Python. 14:35.920 --> 14:37.150 So in the demo. 14:37.900 --> 14:47.410 So here we have again 16 minus one, 16 minus two, 16 minus four and so on and 16 -16 and last. 14:47.920 --> 14:50.320 So these are the values which we have here and here. 14:50.320 --> 14:56.460 You can see that the deviation from the force number is -7.44. 14:56.470 --> 14:58.290 Here we have 7.55. 14:58.600 --> 15:05.590 If you take more of these numbers or you ignore the same and you can see that these values are somewhat 15:06.250 --> 15:09.220 similar, these values are near to each other as of now. 15:10.060 --> 15:15.730 So it doesn't really make a difference if we are calculating this based on the first number, last number 15:15.730 --> 15:16.180 and so on. 15:16.420 --> 15:18.910 So let's do it on the basis of the mean. 15:18.910 --> 15:31.300 So here I will find of the mean gap as constant minus again all the numbers. 15:31.870 --> 15:35.620 So let me just keep doing it. 15:37.210 --> 15:47.360 So here you can see if I subtract these numbers from the mean, then these numbers again have not changed 15:47.380 --> 15:47.830 much. 15:48.400 --> 15:54.610 So here is with respect to the mean, the deviation from the mean is coming out to be zero. 15:55.000 --> 15:59.590 Now, again, if I take more of these numbers, then it might change a little bit. 16:02.560 --> 16:05.380 Now let us do the same for the mean. 16:05.380 --> 16:10.630 Now, if you see we have followed the values from mean now, then we are calculating the deviation from 16:10.630 --> 16:10.990 mean. 16:11.410 --> 16:19.330 It is coming out to be zero, which is quite understandable because we are finding out the deviation 16:19.330 --> 16:20.740 from the sample values. 16:20.740 --> 16:27.820 So it will be balanced out when we are calculating in a positive direction and in the negative direction 16:27.820 --> 16:28.150 is one. 16:28.930 --> 16:36.040 So what we can do is instead of picking these values, we can take the square of these values so that 16:36.610 --> 16:40.270 these the impact of the sign is not considered. 16:40.600 --> 16:48.270 So we take a square of these values, so it will be free to put this into this set. 16:49.490 --> 16:54.010 So these are the values, and we will do the same for all the numbers. 16:55.000 --> 17:03.070 Now, because we have taken the square of these deviation from the means, what happens is the value 17:03.070 --> 17:04.040 is increased. 17:04.040 --> 17:04.600 Don't skip. 17:04.960 --> 17:13.090 Now we want to bring it back to the normal scale so that we can compare the deviation that the values 17:13.090 --> 17:14.260 which we have here. 17:14.980 --> 17:15.910 So let's do that. 17:15.970 --> 17:17.560 So we'll do a square root 17:20.380 --> 17:21.820 of this particular number. 17:23.290 --> 17:27.650 So it comes out to be 5.2305. 17:28.360 --> 17:37.570 Now, if you will calculate the square root, basically we will conclude the variance and standard deviation 17:37.870 --> 17:39.450 directly for these number. 17:39.460 --> 17:43.300 It will come out to be somewhat similar to this, but it is doing that as well. 17:43.420 --> 17:46.690 So let me find the values for you. 17:50.290 --> 17:54.910 So this is variance and I'm taking variance for these numbers. 17:56.260 --> 17:58.030 Similarly, I'm calculating the. 17:58.390 --> 17:59.710 Standard deviation 18:03.100 --> 18:04.660 for these numbers again. 18:07.180 --> 18:10.300 So it is coming out with the exact same number. 18:11.020 --> 18:17.650 Now, I have done scene calculation for two different data sets as well. 18:18.220 --> 18:24.520 So let's have a look at those data sets, everything answering the same how we do the calculation is 18:24.520 --> 18:25.630 also remaining the same. 18:25.960 --> 18:27.070 So let us look at that. 18:28.240 --> 18:34.570 So here you can see that we have we will consider the values which are having similar scaling. 18:34.960 --> 18:38.950 So here you can see the values are distributed evenly. 18:38.980 --> 18:41.760 We can see that the values are not changing. 18:42.060 --> 18:47.200 My instead of having one, two, four, seven, 12, 15, 16, which is a nice spread out. 18:47.800 --> 18:50.620 There is no outlier present in the data as well. 18:50.920 --> 18:57.260 So we are getting the average to be 8.4 for the we do from the first number as -7.4. 18:57.410 --> 18:59.710 Then the last number is 7.5. 19:00.280 --> 19:02.110 This is actually zero. 19:02.260 --> 19:07.270 And when we are taking the square root of the square deviation of the mean, it is coming out to be 19:07.270 --> 19:08.620 5.230. 19:09.250 --> 19:15.070 Now, I have changed these numbers a bit and here I have all these numbers. 19:15.280 --> 19:23.050 Apart from that, I have included this 59, which is an outlier towards the upper limit of the values. 19:23.050 --> 19:26.470 So basically I have increased the maximum value a lot. 19:27.550 --> 19:31.870 So what happens now is the time to give you is the average. 19:31.930 --> 19:36.850 The mean value turns out to be 13.22 instead of 8.44. 19:37.450 --> 19:40.780 Because we have changed just one single value. 19:41.050 --> 19:43.450 It has modified the mean so much. 19:44.710 --> 19:49.390 Instead, if you would have loved that median, the median value would not have changed. 19:50.320 --> 19:51.850 The mean value has changed. 19:52.990 --> 19:59.080 Now, similarly, when we look at the deviation from the force number, it comes out to be -12. 19:59.350 --> 20:04.780 The deviation from the last number comes out to be 45 point something, while the deviation from the 20:04.780 --> 20:10.380 mean comes out, the hero and squared deviation from the mean comes out to be 16.79. 20:10.930 --> 20:15.130 This shows that there is a change in the values. 20:16.300 --> 20:23.590 Now, if you look further here, the deviation from the force number and the last number remains somewhat 20:23.590 --> 20:24.100 seen. 20:24.580 --> 20:31.240 But when we introduce an outlier, the deviation from the force number and last number have a huge difference. 20:32.020 --> 20:36.310 Same thing happens here when I introduce an outlier towards the minimum value. 20:36.640 --> 20:40.780 The deviation from the force number and last number have a huge difference. 20:41.740 --> 20:45.910 While the deviation from the mean remains zero everywhere. 20:48.100 --> 20:50.770 In all the values, the original, the mean remains the same. 20:51.490 --> 20:57.520 But because there is an impact from the positive and negative values, that is something which needs 20:57.520 --> 20:59.110 to be taken care of. 20:59.110 --> 21:02.290 Otherwise we will not get a good information out of it. 21:02.980 --> 21:06.390 You can clearly see that the mean is changing a lot here. 21:06.550 --> 21:11.110 It was 8.4 for we introduced an outlier towards the maximum value. 21:11.110 --> 21:12.460 It changed to 13. 21:12.880 --> 21:16.270 We introduced an outlier towards the negative side of it. 21:16.300 --> 21:18.220 It changed to 2.3. 21:18.220 --> 21:28.240 So mean is highly influenced by a single location and so are the deviation from the first number and 21:28.240 --> 21:29.170 the last numbers. 21:31.620 --> 21:36.500 While the squared deviation from the mean if you see it is 20 here. 21:37.090 --> 21:40.020 Here it is 16 and here it is five. 21:40.470 --> 21:45.900 So these values are giving us a good idea of how the details are spread. 21:47.250 --> 21:54.630 So that is why we use the standard deviation to see how the the how the values are spread. 21:56.580 --> 22:02.490 The calculation which we have done here is exactly the same, which is to be done for the calculation 22:02.490 --> 22:04.440 of standard deviation and variance. 22:04.830 --> 22:10.320 The value which we have calculated by doing the square root is the standard deviation. 22:10.560 --> 22:16.740 And if we would have kept it as it is, that is, we would have found no deviation from this from the 22:16.740 --> 22:17.100 mean. 22:17.430 --> 22:24.570 That is the mean value minus each and every number. 22:24.870 --> 22:28.680 I'm thinking the sum of that deviation from the mean. 22:29.040 --> 22:33.180 If we were taking this division of that mean after doing the sum. 22:33.510 --> 22:41.820 So we are basically taking the difference, doing squaring, squaring all these mean deviations and 22:41.820 --> 22:49.950 then taking an average of that, basically dividing by the total number of values which we have. 22:50.250 --> 22:51.990 And that is the variance. 22:52.200 --> 22:55.830 And then when we take a square root of it, it is the standard deviation. 22:55.840 --> 22:57.720 That is exactly what needs to be done. 22:57.750 --> 22:58.920 That is the bottom line. 22:59.100 --> 23:05.530 This is the best way to remember this because this actually tells you why we are doing this. 23:07.080 --> 23:09.060 So this is the formula for obedience. 23:09.750 --> 23:16.350 And here you can clearly see that in probability theory that this experience is the average of the squared 23:16.350 --> 23:19.770 differences from the mean informally. 23:19.890 --> 23:25.740 It measures how what a set of numbers are spread out from their average values that you have seen that 23:25.740 --> 23:27.690 this is changing a lot. 23:28.350 --> 23:32.940 We are checking from the based on the basis of the plus number, the last number. 23:32.940 --> 23:37.260 But you mean gives us the most reliable values. 23:37.470 --> 23:42.300 That is why we use this Formula One variance and standard deviation. 23:43.680 --> 23:46.440 Now, if you look, there are two formulas here. 23:46.950 --> 23:52.170 First is for the population and second one is for the sum. 23:52.590 --> 23:58.380 Similarly, we have here one for the population and one for the sample. 23:58.620 --> 24:06.150 The only difference between the formula for its population and some food is that the for the values 24:06.150 --> 24:11.640 are divided by n minus one instead of in when we are calculating for the sample variance. 24:13.950 --> 24:19.440 Now why this is done, you will get to know in the next section where we are taking this business connection. 24:24.570 --> 24:33.840 Now why have we divided the standard deviation by in minus one instead of dividing it by in which is 24:33.840 --> 24:35.310 done in case of population? 24:35.640 --> 24:38.550 So this is done because of vessel selection. 24:38.790 --> 24:45.600 So vessels correction refers to the PN minus one found in several formulas, including the sample readings 24:45.600 --> 24:47.760 and sample standard deviation formula. 24:48.090 --> 24:56.970 This correction is made to correct for the fact that these sample statistics tend to underestimate the 24:56.970 --> 24:59.430 actual pattern we found in the population. 24:59.700 --> 25:07.050 Now, because the standard deviation and the variance which we are calculating for the sample is kind 25:07.050 --> 25:14.220 of considering, not considering the entire picture and might be a little lower than the actual variance 25:14.220 --> 25:17.670 or actual standard deviation of the population. 25:17.880 --> 25:27.810 So what we do is we reduce the in value in the sample so that when we reduce the denominator, the entire 25:27.810 --> 25:32.670 value of the variance and the standard deviation kind of increases a little. 25:32.940 --> 25:33.540 So it. 25:34.780 --> 25:40.900 Balances out and makes it closer to the population, standard deviation and the population readings. 25:42.210 --> 25:45.090 So this is why we use the vessels.