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Dialogue: 0,0:00:05.00,0:00:08.45,Default,,0000,0000,0000,,In this video, we'll be learning about Z scores and standardization.
Dialogue: 0,0:00:08.46,0:00:10.26,Default,,0000,0000,0000,,By learning about both of these topics,
Dialogue: 0,0:00:10.27,0:00:11.82,Default,,0000,0000,0000,,you will learn how to calculate exact
Dialogue: 0,0:00:11.83,0:00:14.26,Default,,0000,0000,0000,,proportions using the standard normal distribution.
Dialogue: 0,0:00:15.29,0:00:17.35,Default,,0000,0000,0000,,What is the standard normal distribution?
Dialogue: 0,0:00:18.09,0:00:21.67,Default,,0000,0000,0000,,The standard normal distribution is a special type of normal distribution that
Dialogue: 0,0:00:21.68,0:00:24.27,Default,,0000,0000,0000,,has a mean of 0 and a standard deviation of 1.
Dialogue: 0,0:00:24.66,0:00:25.63,Default,,0000,0000,0000,,Because of this,
Dialogue: 0,0:00:25.64,0:00:27.93,Default,,0000,0000,0000,,the standard normal distribution is always centered at
Dialogue: 0,0:00:27.94,0:00:30.31,Default,,0000,0000,0000,,0 and has intervals that increase by 1.
Dialogue: 0,0:00:31.11,0:00:34.60,Default,,0000,0000,0000,,Each number on the horizontal axis corresponds to a Z-score.
Dialogue: 0,0:00:34.87,0:00:39.34,Default,,0000,0000,0000,,A Z-score tells us how many standard deviations an observation is from the mean Mu.
Dialogue: 0,0:00:39.80,0:00:40.87,Default,,0000,0000,0000,,For example,
Dialogue: 0,0:00:40.88,0:00:42.91,Default,,0000,0000,0000,,a Z-score of negative 2 tells me that I
Dialogue: 0,0:00:42.92,0:00:45.26,Default,,0000,0000,0000,,am two standard deviations to the left of the mean
Dialogue: 0,0:00:45.27,0:00:50.30,Default,,0000,0000,0000,,and a Z-score of 1.5 tells me that I am one-and-a-half standard deviations to the right of the mean.
Dialogue: 0,0:00:51.04,0:00:52.08,Default,,0000,0000,0000,,Most importantly,
Dialogue: 0,0:00:52.09,0:00:54.14,Default,,0000,0000,0000,,a Z-score allows us to calculate how much
Dialogue: 0,0:00:54.15,0:00:56.74,Default,,0000,0000,0000,,area that specific Z score is associated with.
Dialogue: 0,0:00:56.77,0:01:00.50,Default,,0000,0000,0000,,And we can find out that exact area using something called a Z-score table,
Dialogue: 0,0:01:00.51,0:01:02.74,Default,,0000,0000,0000,,also known as the standard normal table.
Dialogue: 0,0:01:03.20,0:01:05.34,Default,,0000,0000,0000,,This table tells us the total amount of area
Dialogue: 0,0:01:05.35,0:01:07.58,Default,,0000,0000,0000,,contained to the left side of any value of Z.
Dialogue: 0,0:01:07.81,0:01:09.10,Default,,0000,0000,0000,,For this table,
Dialogue: 0,0:01:09.11,0:01:12.16,Default,,0000,0000,0000,,the top row and the first column correspond to Z values
Dialogue: 0,0:01:12.17,0:01:14.81,Default,,0000,0000,0000,,and all the numbers in the middle correspond to areas.
Dialogue: 0,0:01:15.36,0:01:17.70,Default,,0000,0000,0000,,For example, according to the table,
Dialogue: 0,0:01:17.71,0:01:22.99,Default,,0000,0000,0000,,a Z-score of negative 1.95 has an area of 0.0256 to the left of it.
Dialogue: 0,0:01:23.34,0:01:25.11,Default,,0000,0000,0000,,To say this in a more formal manner,
Dialogue: 0,0:01:25.12,0:01:31.33,Default,,0000,0000,0000,,we can say that the proportion of Z less than negative 1.95 is equal to 0.0256.
Dialogue: 0,0:01:32.10,0:01:34.91,Default,,0000,0000,0000,,We can also use the standard normal table to determine
Dialogue: 0,0:01:34.92,0:01:36.98,Default,,0000,0000,0000,,the area to the right of any Z value.
Dialogue: 0,0:01:37.20,0:01:41.32,Default,,0000,0000,0000,,All we have to do is take one minus the area that corresponds to the Z value.
Dialogue: 0,0:01:41.57,0:01:42.45,Default,,0000,0000,0000,,For example,
Dialogue: 0,0:01:42.61,0:01:46.24,Default,,0000,0000,0000,,to determine the area to the right of a Z-score of 0.57,
Dialogue: 0,0:01:46.40,0:01:48.11,Default,,0000,0000,0000,,all we have to do is find the area that
Dialogue: 0,0:01:48.12,0:01:51.04,Default,,0000,0000,0000,,corresponds to the Z value and then subtract it from 1.
Dialogue: 0,0:01:51.92,0:01:58.28,Default,,0000,0000,0000,,According to the table, the Z-score of 0.57 has an area of 0.7157 to the left of it.
Dialogue: 0,0:01:58.82,0:02:05.35,Default,,0000,0000,0000,,So, 1 minus 0.7157 gives us an area of 0.2843, and that is our answer.
Dialogue: 0,0:02:05.92,0:02:07.39,Default,,0000,0000,0000,,The reason why we can do this is
Dialogue: 0,0:02:07.40,0:02:09.93,Default,,0000,0000,0000,,because we have to remember that the normal distribution
Dialogue: 0,0:02:09.94,0:02:14.51,Default,,0000,0000,0000,,is a density curve and it always has a total area equal to 1 or 100%.
Dialogue: 0,0:02:15.67,0:02:18.65,Default,,0000,0000,0000,,You can also use the Z-score table to do a reverse look-up,
Dialogue: 0,0:02:18.66,0:02:20.46,Default,,0000,0000,0000,,which means you can use the table to see
Dialogue: 0,0:02:20.47,0:02:23.13,Default,,0000,0000,0000,,what Z score is associated with a specific area.
Dialogue: 0,0:02:23.66,0:02:29.23,Default,,0000,0000,0000,,So, if I wanted to know what value of Z corresponds to an area of 0.8461 to the left of it,
Dialogue: 0,0:02:29.24,0:02:35.13,Default,,0000,0000,0000,,all we have to do is find 0.8461 on the table and see what value of Z it corresponds to.
Dialogue: 0,0:02:35.45,0:02:38.73,Default,,0000,0000,0000,,We see that it corresponds to a Z value of 1.02.
Dialogue: 0,0:02:39.94,0:02:42.73,Default,,0000,0000,0000,,The special thing about the standard normal distribution is
Dialogue: 0,0:02:42.74,0:02:45.82,Default,,0000,0000,0000,,that any type of normal distribution can be transformed into
Dialogue: 0,0:02:46.00,0:02:46.15,Default,,0000,0000,0000,,it.
Dialogue: 0,0:02:46.16,0:02:46.85,Default,,0000,0000,0000,,In other words,
Dialogue: 0,0:02:46.86,0:02:49.60,Default,,0000,0000,0000,,any normal distribution with any value of Mu
Dialogue: 0,0:02:49.61,0:02:52.07,Default,,0000,0000,0000,,and Sigma can be transformed into the standard
Dialogue: 0,0:02:52.08,0:02:56.22,Default,,0000,0000,0000,,normal distribution where you have a Mu of 0 and a standard deviation of 1.
Dialogue: 0,0:02:56.66,0:02:59.40,Default,,0000,0000,0000,,This conversion process is called standardization.
Dialogue: 0,0:03:00.01,0:03:04.05,Default,,0000,0000,0000,,The benefit of standardization is that it allows us to use the Z-score table to
Dialogue: 0,0:03:04.06,0:03:07.15,Default,,0000,0000,0000,,calculate exact areas for any given normally distributed
Dialogue: 0,0:03:07.16,0:03:09.91,Default,,0000,0000,0000,,population with any value of Mu or Sigma.
Dialogue: 0,0:03:10.78,0:03:13.31,Default,,0000,0000,0000,,Standardization involves using this formula.
Dialogue: 0,0:03:13.67,0:03:17.35,Default,,0000,0000,0000,,This formula says that the Z-score is equal to an observation X
Dialogue: 0,0:03:17.36,0:03:22.27,Default,,0000,0000,0000,,minus the population mean Mu divided by the population of standard deviation Sigma.
Dialogue: 0,0:03:23.41,0:03:26.75,Default,,0000,0000,0000,,So, suppose that we gathered data from last year's final chemistry exam
Dialogue: 0,0:03:26.76,0:03:29.85,Default,,0000,0000,0000,,and found that it followed a normal distribution with a mean of 60 and
Dialogue: 0,0:03:30.01,0:03:31.52,Default,,0000,0000,0000,,a standard deviation of 10.
Dialogue: 0,0:03:31.98,0:03:34.21,Default,,0000,0000,0000,,If we were to draw this normal distribution,
Dialogue: 0,0:03:34.22,0:03:37.66,Default,,0000,0000,0000,,we would have 60 located at the center of the distribution because it
Dialogue: 0,0:03:37.67,0:03:41.03,Default,,0000,0000,0000,,is the value of the mean and each interval would increase by 10,
Dialogue: 0,0:03:41.04,0:03:43.40,Default,,0000,0000,0000,,since that is the value of the standard deviation.
Dialogue: 0,0:03:43.95,0:03:46.91,Default,,0000,0000,0000,,To convert this distribution to the standard normal distribution,
Dialogue: 0,0:03:46.92,0:03:48.27,Default,,0000,0000,0000,,we will use the formula.
Dialogue: 0,0:03:48.50,0:03:50.63,Default,,0000,0000,0000,,The value of Mu is equal to 60 and
Dialogue: 0,0:03:50.82,0:03:52.77,Default,,0000,0000,0000,,the value of Sigma is equal to 10.
Dialogue: 0,0:03:53.25,0:03:56.73,Default,,0000,0000,0000,,We can then take each value of X and plug it into the equation.
Dialogue: 0,0:03:56.95,0:03:59.71,Default,,0000,0000,0000,,If I plug in 60, I will get a value of 0.
Dialogue: 0,0:03:59.94,0:04:02.95,Default,,0000,0000,0000,,If I plug in 50, I will get a value of negative 1.
Dialogue: 0,0:04:03.15,0:04:06.26,Default,,0000,0000,0000,,If I plug in 40, I will get a value of negative 2.
Dialogue: 0,0:04:06.60,0:04:08.36,Default,,0000,0000,0000,,If we do this for each value,
Dialogue: 0,0:04:08.37,0:04:12.40,Default,,0000,0000,0000,,you can see that we end up with the same values as a standard normal distribution.
Dialogue: 0,0:04:12.70,0:04:14.43,Default,,0000,0000,0000,,When doing this conversion process,
Dialogue: 0,0:04:14.44,0:04:17.81,Default,,0000,0000,0000,,the mean of the normal distribution will always be converted to 0
Dialogue: 0,0:04:17.82,0:04:21.27,Default,,0000,0000,0000,,and the standard deviation will always correspond to a value of 1.
Dialogue: 0,0:04:21.86,0:04:24.57,Default,,0000,0000,0000,,It's important to remember that this will happen with any normal
Dialogue: 0,0:04:24.58,0:04:27.62,Default,,0000,0000,0000,,distribution no matter what value the Mu and Sigma are.
Dialogue: 0,0:04:28.29,0:04:32.37,Default,,0000,0000,0000,,Now, if I asked you what proportion of students score less than 49 on the exam,
Dialogue: 0,0:04:32.38,0:04:34.47,Default,,0000,0000,0000,,it is this area that we are interested in.
Dialogue: 0,0:04:34.79,0:04:35.59,Default,,0000,0000,0000,,However,
Dialogue: 0,0:04:35.64,0:04:38.03,Default,,0000,0000,0000,,the proportion of X less than 49 is
Dialogue: 0,0:04:38.04,0:04:40.63,Default,,0000,0000,0000,,unknown until we use the standardization formula.
Dialogue: 0,0:04:40.93,0:04:45.77,Default,,0000,0000,0000,,After plugging in 49 into this formula, we end up with a value of negative 1.1.
Dialogue: 0,0:04:46.03,0:04:50.48,Default,,0000,0000,0000,,As a result, we will be looking for the proportion of Z less than negative 1.1.
Dialogue: 0,0:04:50.79,0:04:51.49,Default,,0000,0000,0000,,And finally,
Dialogue: 0,0:04:51.50,0:04:55.83,Default,,0000,0000,0000,,we can use the Z score table to determine how much area is associated with the Z score.
Dialogue: 0,0:04:56.13,0:05:01.11,Default,,0000,0000,0000,,According to the table, there is an area of 0.1357 to the left of this Z value.
Dialogue: 0,0:05:01.16,0:05:06.63,Default,,0000,0000,0000,,This means that the proportion of Z less than negative 1.1 is 0.1357.
Dialogue: 0,0:05:06.96,0:05:09.47,Default,,0000,0000,0000,,This value is in fact the same proportion of
Dialogue: 0,0:05:09.48,0:05:12.43,Default,,0000,0000,0000,,individuals that scored less than 49 on the exam.
Dialogue: 0,0:05:12.60,0:05:14.65,Default,,0000,0000,0000,,As a result, this is the answer.
Dialogue: 0,0:05:15.16,0:05:16.65,Default,,0000,0000,0000,,Let's do one more example,
Dialogue: 0,0:05:16.98,0:05:20.00,Default,,0000,0000,0000,,when measuring the heights of all students at a local university,
Dialogue: 0,0:05:20.13,0:05:22.57,Default,,0000,0000,0000,,it was found that it was normally distributed with a mean
Dialogue: 0,0:05:22.58,0:05:26.31,Default,,0000,0000,0000,,height of 5.5 feet and a standard deviation of 0.5 feet.
Dialogue: 0,0:05:26.56,0:05:30.91,Default,,0000,0000,0000,,What proportion of students are between 5.81 feet and 6.3 feet tall?
Dialogue: 0,0:05:31.33,0:05:32.74,Default,,0000,0000,0000,,Before we solve this question,
Dialogue: 0,0:05:32.75,0:05:35.68,Default,,0000,0000,0000,,it's always a good habit to first write down important information.
Dialogue: 0,0:05:36.04,0:05:39.68,Default,,0000,0000,0000,,So, we have a Mu of 5.5 feet and a Sigma of 0.5 feet.
Dialogue: 0,0:05:39.94,0:05:41.79,Default,,0000,0000,0000,,We are also looking for the proportion of
Dialogue: 0,0:05:41.80,0:05:45.43,Default,,0000,0000,0000,,individuals between 5.81 feet and 6.3 feet tall.
Dialogue: 0,0:05:45.57,0:05:47.83,Default,,0000,0000,0000,,This corresponds to this highlighted area.
Dialogue: 0,0:05:48.17,0:05:51.46,Default,,0000,0000,0000,,To determine this area, we need to standardize the distribution,
Dialogue: 0,0:05:51.56,0:05:53.87,Default,,0000,0000,0000,,so we will use the standardization formula.
Dialogue: 0,0:05:55.22,0:05:59.63,Default,,0000,0000,0000,,Plugging in 5.81 to this formula gives us a Z-score of 0.62.
Dialogue: 0,0:05:59.98,0:06:04.11,Default,,0000,0000,0000,,And plugging in 6.3 into the formula gives us a Z-score of 1.6.
Dialogue: 0,0:06:05.60,0:06:07.72,Default,,0000,0000,0000,,According to the standard normal table,
Dialogue: 0,0:06:07.73,0:06:12.69,Default,,0000,0000,0000,,the Z-score of 0.62 corresponds to an area of 0.7324,
Dialogue: 0,0:06:13.06,0:06:17.77,Default,,0000,0000,0000,,and the Z-score of 1.6 corresponds to an area of 0.9452.
Dialogue: 0,0:06:18.55,0:06:22.58,Default,,0000,0000,0000,,To find the proportion of values between 0.62 and 1.6,
Dialogue: 0,0:06:22.59,0:06:25.30,Default,,0000,0000,0000,,we must subtract the smaller area from the bigger area.
Dialogue: 0,0:06:25.55,0:06:31.40,Default,,0000,0000,0000,,So, 0.9452 minus 0.7324 gives us 0.2128.
Dialogue: 0,0:06:31.63,0:06:38.42,Default,,0000,0000,0000,,As a result, the proportion of students between 5.81 feet and 6.3 feet tall is 0.2128.
Dialogue: 0,0:06:39.36,0:06:40.82,Default,,0000,0000,0000,,If you found this video helpful,
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Dialogue: 0,0:06:44.43,0:06:47.46,Default,,0000,0000,0000,,You can also visit our website at simplelearningpro.com
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Dialogue: 0,0:06:50.36,0:06:51.34,Default,,0000,0000,0000,,Thanks for watching.