WEBVTT 00:00:05.000 --> 00:00:08.449 In this video, we'll be learning about Z scores and standardization. 00:00:08.460 --> 00:00:10.260 By learning about both of these topics, 00:00:10.270 --> 00:00:11.819 you will learn how to calculate exact 00:00:11.829 --> 00:00:14.260 proportions using the standard normal distribution. 00:00:15.289 --> 00:00:17.350 What is the standard normal distribution? 00:00:18.090 --> 00:00:21.670 The standard normal distribution is a special type of normal distribution that 00:00:21.680 --> 00:00:24.270 has a mean of 0 and a standard deviation of 1. 00:00:24.659 --> 00:00:25.629 Because of this, 00:00:25.639 --> 00:00:27.930 the standard normal distribution is always centered at 00:00:27.940 --> 00:00:30.309 0 and has intervals that increase by 1. 00:00:31.110 --> 00:00:34.599 Each number on the horizontal axis corresponds to a Z-score. 00:00:34.869 --> 00:00:39.340 A Z-score tells us how many standard deviations an observation is from the mean Mu. 00:00:39.799 --> 00:00:40.869 For example, 00:00:40.880 --> 00:00:42.909 a Z-score of negative 2 tells me that I 00:00:42.919 --> 00:00:45.259 am two standard deviations to the left of the mean 00:00:45.270 --> 00:00:50.299 and a Z-score of 1.5 tells me that I am one-and-a-half standard deviations to the right of the mean. 00:00:51.040 --> 00:00:52.080 Most importantly, 00:00:52.090 --> 00:00:54.139 a Z-score allows us to calculate how much 00:00:54.150 --> 00:00:56.740 area that specific Z score is associated with. 00:00:56.770 --> 00:01:00.500 And we can find out that exact area using something called a Z-score table, 00:01:00.509 --> 00:01:02.740 also known as the standard normal table. 00:01:03.200 --> 00:01:05.339 This table tells us the total amount of area 00:01:05.349 --> 00:01:07.580 contained to the left side of any value of Z. 00:01:07.809 --> 00:01:09.099 For this table, 00:01:09.110 --> 00:01:12.160 the top row and the first column correspond to Z values 00:01:12.169 --> 00:01:14.809 and all the numbers in the middle correspond to areas. 00:01:15.360 --> 00:01:17.699 For example, according to the table, 00:01:17.709 --> 00:01:22.989 a Z-score of negative 1.95 has an area of 0.0256 to the left of it. 00:01:23.339 --> 00:01:25.110 To say this in a more formal manner, 00:01:25.120 --> 00:01:31.330 we can say that the proportion of Z less than negative 1.95 is equal to 0.0256. 00:01:32.099 --> 00:01:34.910 We can also use the standard normal table to determine 00:01:34.919 --> 00:01:36.980 the area to the right of any Z value. 00:01:37.199 --> 00:01:41.319 All we have to do is take one minus the area that corresponds to the Z value. 00:01:41.569 --> 00:01:42.449 For example, 00:01:42.610 --> 00:01:46.239 to determine the area to the right of a Z-score of 0.57, 00:01:46.400 --> 00:01:48.110 all we have to do is find the area that 00:01:48.120 --> 00:01:51.040 corresponds to the Z value and then subtract it from 1. 00:01:51.919 --> 00:01:58.279 According to the table, the Z-score of 0.57 has an area of 0.7157 to the left of it. 00:01:58.819 --> 00:02:05.349 So, 1 minus 0.7157 gives us an area of 0.2843, and that is our answer. 00:02:05.919 --> 00:02:07.389 The reason why we can do this is 00:02:07.400 --> 00:02:09.929 because we have to remember that the normal distribution 00:02:09.940 --> 00:02:14.509 is a density curve and it always has a total area equal to 1 or 100%. 00:02:15.669 --> 00:02:18.649 You can also use the Z-score table to do a reverse look-up, 00:02:18.660 --> 00:02:20.460 which means you can use the table to see 00:02:20.470 --> 00:02:23.130 what Z score is associated with a specific area. 00:02:23.660 --> 00:02:29.229 So, if I wanted to know what value of Z corresponds to an area of 0.8461 to the left of it, 00:02:29.240 --> 00:02:35.130 all we have to do is find 0.8461 on the table and see what value of Z it corresponds to. 00:02:35.449 --> 00:02:38.729 We see that it corresponds to a Z value of 1.02. 00:02:39.940 --> 00:02:42.729 The special thing about the standard normal distribution is 00:02:42.740 --> 00:02:45.820 that any type of normal distribution can be transformed into 00:02:46.000 --> 00:02:46.149 it. 00:02:46.160 --> 00:02:46.850 In other words, 00:02:46.860 --> 00:02:49.600 any normal distribution with any value of Mu 00:02:49.610 --> 00:02:52.070 and Sigma can be transformed into the standard 00:02:52.080 --> 00:02:56.220 normal distribution where you have a Mu of 0 and a standard deviation of 1. 00:02:56.660 --> 00:02:59.399 This conversion process is called standardization. 00:03:00.009 --> 00:03:04.050 The benefit of standardization is that it allows us to use the Z-score table to 00:03:04.059 --> 00:03:07.149 calculate exact areas for any given normally distributed 00:03:07.160 --> 00:03:09.910 population with any value of Mu or Sigma. 00:03:10.779 --> 00:03:13.309 Standardization involves using this formula. 00:03:13.669 --> 00:03:17.350 This formula says that the Z-score is equal to an observation X 00:03:17.360 --> 00:03:22.270 minus the population mean Mu divided by the population of standard deviation Sigma. 00:03:23.410 --> 00:03:26.750 So, suppose that we gathered data from last year's final chemistry exam 00:03:26.759 --> 00:03:29.850 and found that it followed a normal distribution with a mean of 60 and 00:03:30.009 --> 00:03:31.520 a standard deviation of 10. 00:03:31.979 --> 00:03:34.210 If we were to draw this normal distribution, 00:03:34.220 --> 00:03:37.660 we would have 60 located at the center of the distribution because it 00:03:37.669 --> 00:03:41.029 is the value of the mean and each interval would increase by 10, 00:03:41.039 --> 00:03:43.399 since that is the value of the standard deviation. 00:03:43.949 --> 00:03:46.910 To convert this distribution to the standard normal distribution, 00:03:46.919 --> 00:03:48.270 we will use the formula. 00:03:48.500 --> 00:03:50.630 The value of Mu is equal to 60 and 00:03:50.820 --> 00:03:52.770 the value of Sigma is equal to 10. 00:03:53.250 --> 00:03:56.729 We can then take each value of X and plug it into the equation. 00:03:56.949 --> 00:03:59.710 If I plug in 60, I will get a value of 0. 00:03:59.940 --> 00:04:02.949 If I plug in 50, I will get a value of negative 1. 00:04:03.149 --> 00:04:06.259 If I plug in 40, I will get a value of negative 2. 00:04:06.600 --> 00:04:08.360 If we do this for each value, 00:04:08.369 --> 00:04:12.399 you can see that we end up with the same values as a standard normal distribution. 00:04:12.699 --> 00:04:14.429 When doing this conversion process, 00:04:14.440 --> 00:04:17.809 the mean of the normal distribution will always be converted to 0 00:04:17.820 --> 00:04:21.269 and the standard deviation will always correspond to a value of 1. 00:04:21.858 --> 00:04:24.570 It's important to remember that this will happen with any normal 00:04:24.579 --> 00:04:27.619 distribution no matter what value the Mu and Sigma are. 00:04:28.290 --> 00:04:32.369 Now, if I asked you what proportion of students score less than 49 on the exam, 00:04:32.380 --> 00:04:34.470 it is this area that we are interested in. 00:04:34.790 --> 00:04:35.589 However, 00:04:35.640 --> 00:04:38.029 the proportion of X less than 49 is 00:04:38.040 --> 00:04:40.630 unknown until we use the standardization formula. 00:04:40.929 --> 00:04:45.769 After plugging in 49 into this formula, we end up with a value of negative 1.1. 00:04:46.029 --> 00:04:50.480 As a result, we will be looking for the proportion of Z less than negative 1.1. 00:04:50.790 --> 00:04:51.489 And finally, 00:04:51.500 --> 00:04:55.829 we can use the Z score table to determine how much area is associated with the Z score. 00:04:56.130 --> 00:05:01.109 According to the table, there is an area of 0.1357 to the left of this Z value. 00:05:01.160 --> 00:05:06.630 This means that the proportion of Z less than negative 1.1 is 0.1357. 00:05:06.959 --> 00:05:09.470 This value is in fact the same proportion of 00:05:09.480 --> 00:05:12.429 individuals that scored less than 49 on the exam. 00:05:12.600 --> 00:05:14.649 As a result, this is the answer. 00:05:15.160 --> 00:05:16.649 Let's do one more example, 00:05:16.980 --> 00:05:20.000 when measuring the heights of all students at a local university, 00:05:20.130 --> 00:05:22.570 it was found that it was normally distributed with a mean 00:05:22.579 --> 00:05:26.309 height of 5.5 feet and a standard deviation of 0.5 feet. 00:05:26.559 --> 00:05:30.910 What proportion of students are between 5.81 feet and 6.3 feet tall? 00:05:31.329 --> 00:05:32.739 Before we solve this question, 00:05:32.750 --> 00:05:35.679 it's always a good habit to first write down important information. 00:05:36.040 --> 00:05:39.679 So, we have a Mu of 5.5 feet and a Sigma of 0.5 feet. 00:05:39.940 --> 00:05:41.790 We are also looking for the proportion of 00:05:41.799 --> 00:05:45.429 individuals between 5.81 feet and 6.3 feet tall. 00:05:45.570 --> 00:05:47.829 This corresponds to this highlighted area. 00:05:48.170 --> 00:05:51.459 To determine this area, we need to standardize the distribution, 00:05:51.559 --> 00:05:53.869 so we will use the standardization formula. 00:05:55.220 --> 00:05:59.630 Plugging in 5.81 to this formula gives us a Z-score of 0.62. 00:05:59.980 --> 00:06:04.109 And plugging in 6.3 into the formula gives us a Z-score of 1.6. 00:06:05.600 --> 00:06:07.720 According to the standard normal table, 00:06:07.730 --> 00:06:12.690 the Z-score of 0.62 corresponds to an area of 0.7324, 00:06:13.059 --> 00:06:17.769 and the Z-score of 1.6 corresponds to an area of 0.9452. 00:06:18.549 --> 00:06:22.579 To find the proportion of values between 0.62 and 1.6, 00:06:22.589 --> 00:06:25.299 we must subtract the smaller area from the bigger area. 00:06:25.549 --> 00:06:31.399 So, 0.9452 minus 0.7324 gives us 0.2128. 00:06:31.630 --> 00:06:38.420 As a result, the proportion of students between 5.81 feet and 6.3 feet tall is 0.2128. 00:06:39.359 --> 00:06:40.820 If you found this video helpful, 00:06:40.829 --> 00:06:43.920 consider supporting us on Patreon to help us make more videos. 00:06:44.429 --> 00:06:47.459 You can also visit our website at simplelearningpro.com 00:06:47.470 --> 00:06:50.290 to get access to many study guides and practice questions. 00:06:50.359 --> 00:06:51.339 Thanks for watching.