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Inshallah we'll start numerical descriptive majors

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for the population. Last time we talked about the

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same majors. I mean the same descriptive measures

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for a sample. And we have already talked about the

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mean, variance, and standard deviation. These are

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called statistics because they are computed from

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the sample. Here we'll see how can we do the same

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measures but for a population, I mean for the

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entire dataset. So descriptive statistics

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described previously in the last two lectures was

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for a sample. Here we'll just see how can we

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compute these measures for the entire population.

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In this case, the statistics we talked about

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before are called And if you remember the first

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lecture, we said there is a difference between

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statistics and parameters. A statistic is a value

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that computed from a sample, but parameter is a

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value computed from population. So the important

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population parameters are population mean,

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variance, and standard deviation. Let's start with

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the first one, the mean, or the population mean.

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As the sample mean is defined by the sum of the

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values divided by the sample size. But here, we

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have to divide by the population size. So that's

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the difference between sample mean and population

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mean. For the sample mean, we use x bar. Here we

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use Greek letter, mu. This is pronounced as mu. So

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mu is the sum of the x values divided by the

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population size, not the sample size. So it's

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quite similar to the sample mean. So mu is the

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population mean, n is the population size, and xi

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is the it value of the variable x. Similarly, for

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the other parameter, which is the variance, the

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variance There is a little difference between the

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sample and population variance. Here, we subtract

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the population mean instead of the sample mean. So

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sum of xi minus mu squared, then divide by this

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population size, capital N, instead of N minus 1.

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So that's the difference between sample and

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population variance. So again, in the sample

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variance, we subtracted x bar. Here, we subtract

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the mean of the population, mu, then divide by

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capital N instead of N minus 1. So the

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computations for the sample and the population

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mean or variance are quite similar. Finally, the

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population standard deviation. is the same as the

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sample population variance and here just take the

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square root of the population variance and again

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as we did as we explained before the standard

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deviation has the same units as the original unit

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so nothing is new we just extend the sample

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statistic to the population parameter and again

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The mean is denoted by mu, it's a Greek letter.

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The population variance is denoted by sigma

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squared. And finally, the population standard

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deviation is denoted by sigma. So that's the

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numerical descriptive measures either for a sample

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or a population. So just summary for these

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measures. The measures are mean variance, standard

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deviation. Population parameters are mu for the

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mean, sigma squared for variance, and sigma for

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standard deviation. On the other hand, for the

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sample statistics, we have x bar for sample mean,

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s squared for the sample variance, and s is the

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sample standard deviation. That's sample

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statistics against population parameters. Any

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question?

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Let's move to new topic, which is empirical role.

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Now, empirical role is just we

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have to approximate the variation of data in case

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of They'll shift. I mean suppose the data is

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symmetric around the mean. I mean by symmetric

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around the mean, the mean is the vertical line

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that splits the data into two halves. One to the

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right and the other to the left. I mean, the mean,

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the area to the right of the mean equals 50%,

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which is the same as the area to the left of the

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mean. Now suppose or consider the data is bell

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-shaped. Bell-shaped, normal, or symmetric? So

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it's not skewed either to the right or to the

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left. So here we assume, okay, the data is bell

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-shaped. In this scenario, in this case, there is

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a rule called 68, 95, 99.7 rule. Number one,

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approximately 68% of the data in a bill shipped

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lies within one standard deviation of the

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population. So this is the first rule, 68% of the

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data or of the observations Lie within a mu minus

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sigma and a mu plus sigma. That's the meaning of

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the data in bell shape distribution is within one

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standard deviation of mean or mu plus or minus

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sigma. So again, you can say that if the data is

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normally distributed or if the data is bell

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shaped, that is 68% of the data lies within one

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standard deviation of the mean, either below or

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above it. So 68% of the data. So this is the first

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rule.

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68% of the data lies between mu minus sigma and mu

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plus sigma.

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The other rule is approximately 95% of the data in

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a bell-shaped distribution lies within two

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standard deviations of the mean. That means this

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area covers between minus two sigma and plus mu

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plus two sigma. So 95% of the data lies between

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minus mu two sigma And finally,

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approximately 99.7% of the data, it means almost

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the data. Because we are saying 99.7 means most of

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the data falls or lies within three standard

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deviations of the mean. So 99.7% of the data lies

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between mu minus the pre-sigma and the mu plus of

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pre-sigma.

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68, 95, 99.7 are fixed numbers. Later in chapter

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6, we will explain in details other coefficients.

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Maybe suppose we are interested not in one of

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these. Suppose we are interested in 90% or 80% or

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85%. This rule just for 689599.7. This rule is

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called 689599

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.7 rule. That is, again, 68% of the data lies

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within one standard deviation of the mean. 95% of

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the data lies within two standard deviations of

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the mean. And finally, most of the data falls

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within three standard deviations of the mean.

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Let's see how can we use this empirical rule for a

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specific example. Imagine that the variable math

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set scores is bell shaped. So here we assume that

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The math status score has symmetric shape or bell

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shape. In this case, we can use the previous rule.

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Otherwise, we cannot. So assume the math status

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score is bell-shaped with a mean of 500. I mean,

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the population mean is 500 and standard deviation

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of 90. And let's see how can we apply the

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empirical rule. So again, meta score has a mean of

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500 and standard deviation sigma is 90. Then we

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can say that 60% of all test takers scored between

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68%.

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So mu is 500. minus sigma is 90. And mu plus

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sigma, 500 plus 90. So you can say that 68% or 230

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of all test takers scored between 410 and 590. So

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68% of all test takers who took that exam scored

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between 14 and 590. That if we assume previously

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the data is well shaped, otherwise we cannot say

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that. For the other rule, 95% of all test takers

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scored between mu is 500 minus 2 times sigma, 500

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plus 2 times sigma. So that means 500 minus 180 is

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320. 500 plus 180 is 680. So you can say that

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approximately 95% of all test takers scored

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between 320 and 680. Finally, you can say that

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all of the test takers, approximately all, because

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when we are saying 99.7 it means just 0.3 is the

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rest, so you can say approximately all test takers

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scored between mu minus three sigma which is 90

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and mu It lost 3 seconds. So 500 minus 3 times 9

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is 270. So that's 230. 500 plus 270 is 770. So we

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can say that 99.7% of all the stackers scored

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between 230 and 770. I will give another example

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just to make sure that you understand the meaning

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of this rule.

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For business, a statistic goes.

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For business, a statistic example. Suppose the

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scores are bell-shaped. So we are assuming the

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data is bell-shaped. with mean of 75 and standard

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deviation of 5.

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Also, let's assume that 100 students took

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the exam. So we have 100 students. Last year took

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the exam of business statistics. The mean was 75.

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And standard deviation was 5. And let's see how it

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can tell about 6 to 8% rule. It means that 6 to 8%

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of all the students score

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between mu minus sigma. Mu is 75. minus sigma and

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the mu plus sigma.

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So that means 68 students, because we have 100, so

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you can say 68 students scored between 70 and 80.

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So 60 students out of 100 scored between 70 and

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80. About 95 students out of 100 scored between 75

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minus 2 times 5. 75 plus 2 times 5. So that gives

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65.

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The minimum and the maximum is 85. So you can say

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that around 95 students scored between 65 and 85.

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Finally, maybe you can see all students. Because

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when you're saying 99.7, it means almost all the

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students scored between 75 minus 3 times Y. and 75

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plus three times one. So that's six days in two

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nights. Now let's look carefully at these three

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intervals. The first one is seven to eight, the

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other one 65 to 85, then six to 90. When we are

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more confident,

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When we are more confident here for 99.7%, the

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interval becomes wider. So this is the widest

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interval. Because here, the length of the interval

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is around 10. The other one is 20. Here is 30. So

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the last interval has the highest width. So as the

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confidence coefficient increases, the length of

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the interval becomes larger and larger because it

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starts with 10, 20, and we end with 30. So that's

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another example of empirical load. And again, here

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we assume the data is bell shape. Let's move. to

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another one when the data is not in shape. I mean,

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if we have data and that data is not symmetric. So

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that rule is no longer valid. So we have to use

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another rule. It's called shape-example rule.

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Any questions before we move to the next topic?

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At shape and shape rule, it says that regardless

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of how the data are distributed, I mean, if the

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data is not symmetric or

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not bell-shaped, then we can say that at least

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Instead of saying 68, 95, or 99.7, just say around

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1 minus 1 over k squared. Multiply this by 100.

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All of the values will fall within k. So k is

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number of standard deviations. I mean number of

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signals. So if the data is not bell shaped, then

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you can say that approximately at least 1 minus 1

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over k squared times 100% of the values will fall

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within k standard deviations of the mean. In this

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case, we assume that k is greater than 1. I mean,

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you cannot apply this rule if k equals 1. Because

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if k is 1. Then 1 minus 1 is 0. That makes no

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sense. For this reason, k is above 1 or greater

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than 1. So this rule is valid only for k greater

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than 1. So you can see that at least 1 minus 1

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over k squared of the data or of the values will

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fall within k standard equations. So now, for

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example, suppose k equals 2.

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When k equals 2, we said that 95% of the data

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falls within two standard ratios. That if the data

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is bell shaped. Now what's about if the data is

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not bell shaped? We have to use shape shape rule.

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So 1 minus 1 over k is 2. So 2, 2, 2 squared. So 1

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minus 1 fourth. That gives. three quarters, I

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mean, 75%. So instead of saying 95% of the data

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lies within one or two standard deviations of the

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mean, if the data is bell-shaped, if the data is

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not bell-shaped, you have to say that 75% of the

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data falls within two standard deviations. For

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bell shape, you are 95% confident there. But here,

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you're just 75% confident. Suppose k is 3. Now for

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k equal 3, we said 99.7% of the data falls within

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three standard deviations. Now here, if the data

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is not bell shape, 1 minus 1 over k squared. 1

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minus 1

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over 3 squared is one-ninth. One-ninth is 0.11. 1

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minus 0.11 means 89% of the data, instead of

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saying 99.7. So 89% of the data will fall within

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three standard deviations of the population mean.

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regardless of how the data are distributed around

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them. So here, we have two scenarios. One, if the

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data is symmetric, which is called empirical rule

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68959917. And the other one is called shape-by

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-shape rule, and that regardless of the shape of

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the data.

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Excuse me? Yes. In this case, you don't know the

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distribution of the data. And the reality is

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sometimes the data has unknown distribution. For

252
00:21:58,650 --> 00:22:02,590
this reason, we have to use chip-chip portions.

253
00:22:05,410 --> 00:22:09,830
That's all for empirical rule and chip-chip rule.

254
00:22:11,230 --> 00:22:18,150
The next topic is quartile measures. So far, we

255
00:22:18,150 --> 00:22:24,330
have discussed central tendency measures, and we

256
00:22:24,330 --> 00:22:28,450
have talked about mean, median, and more. Then we

257
00:22:28,450 --> 00:22:32,830
moved to location of variability or spread or

258
00:22:32,830 --> 00:22:37,810
dispersion. And we talked about range, variance,

259
00:22:37,950 --> 00:22:38,890
and standardization.

260
00:22:41,570 --> 00:22:48,230
And we said that outliers affect the mean much

261
00:22:48,230 --> 00:22:51,470
more than the median. And also, outliers affect

262
00:22:51,470 --> 00:22:55,730
the range. Here, we'll talk about other measures

263
00:22:55,730 --> 00:22:59,570
of the data, which is called quartile measures.

264
00:23:01,190 --> 00:23:03,450
Here, actually, we'll talk about two measures.

265
00:23:04,270 --> 00:23:10,130
First one is called first quartile, And the other

266
00:23:10,130 --> 00:23:14,150
one is third quartile. So we have two measures,

267
00:23:15,470 --> 00:23:26,030
first and third quartile. Quartiles split the rank

268
00:23:26,030 --> 00:23:32,930
data into four equal segments. I mean, these

269
00:23:32,930 --> 00:23:37,190
measures split the data you have into four equal

270
00:23:37,190 --> 00:23:37,730
parts.

271
00:23:42,850 --> 00:23:48,690
Q1 has 25% of the data fall below it. I mean 25%

272
00:23:48,690 --> 00:23:56,410
of the values lie below Q1. So it means 75% of the

273
00:23:56,410 --> 00:24:04,410
values above it. So 25 below and 75 above. But you

274
00:24:04,410 --> 00:24:07,370
have to be careful that the data is arranged from

275
00:24:07,370 --> 00:24:12,430
smallest to largest. So in this case, Q1. is a

276
00:24:12,430 --> 00:24:19,630
value that has 25% below it. So Q2 is called the

277
00:24:19,630 --> 00:24:22,450
median. The median, the value in the middle when

278
00:24:22,450 --> 00:24:26,250
we arrange the data from smallest to largest. So

279
00:24:26,250 --> 00:24:31,190
that means 50% of the data below and also 50% of

280
00:24:31,190 --> 00:24:36,370
the data above. The other measure is called

281
00:24:36,370 --> 00:24:41,730
theoretical qualifying. In this case, we have 25%

282
00:24:41,730 --> 00:24:47,950
of the data above Q3 and 75% of the data below Q3.

283
00:24:49,010 --> 00:24:54,410
So quartiles split the rank data into four equal

284
00:24:54,410 --> 00:25:00,190
segments, Q1 25% to the left, Q2 50% to the left,

285
00:25:00,970 --> 00:25:08,590
Q3 75% to the left, and 25% to the right. Before,

286
00:25:09,190 --> 00:25:13,830
we explained how to compute the median, and let's

287
00:25:13,830 --> 00:25:18,850
see how can we compute first and third quartile.

288
00:25:19,750 --> 00:25:23,650
If you remember, when we computed the median,

289
00:25:24,350 --> 00:25:28,480
first we locate the position of the median. And we

290
00:25:28,480 --> 00:25:33,540
said that the rank of n is odd. Yes, it was n plus

291
00:25:33,540 --> 00:25:37,800
1 divided by 2. This is the location of the

292
00:25:37,800 --> 00:25:41,100
median, not the value. Sometimes the value may be

293
00:25:41,100 --> 00:25:44,900
equal to the location, but most of the time it's

294
00:25:44,900 --> 00:25:48,340
not. It's not the case. Now let's see how can we

295
00:25:48,340 --> 00:25:54,130
locate the fair support. The first quartile after

296
00:25:54,130 --> 00:25:56,690
you arrange the data from smallest to largest, the

297
00:25:56,690 --> 00:26:01,290
location is n plus 1 divided by 2. So that's the

298
00:26:01,290 --> 00:26:06,890
location of the first quartile. The median, as we

299
00:26:06,890 --> 00:26:10,390
mentioned before, is located in the middle. So it

300
00:26:10,390 --> 00:26:15,210
makes sense that if n is odd, the location of the

301
00:26:15,210 --> 00:26:20,490
median is n plus 1 over 2. Now, for the third

302
00:26:20,490 --> 00:26:27,160
quartile position, The location is N plus 1

303
00:26:27,160 --> 00:26:31,160
divided by 4 times 3. So 3 times N plus 1 divided

304
00:26:31,160 --> 00:26:39,920
by 4. That's how can we locate Q1, Q2, and Q3. So

305
00:26:39,920 --> 00:26:42,080
one more time, the median, the value in the

306
00:26:42,080 --> 00:26:46,260
middle, and it's located exactly at the position N

307
00:26:46,260 --> 00:26:52,590
plus 1 over 2 for the range data. Q1 is located at

308
00:26:52,590 --> 00:26:56,770
n plus one divided by four. Q3 is located at the

309
00:26:56,770 --> 00:26:59,670
position three times n plus one divided by four.

310
00:27:03,630 --> 00:27:07,490
Now, when calculating the rank position, we can

311
00:27:07,490 --> 00:27:14,690
use one of these rules. First, if the result of

312
00:27:14,690 --> 00:27:18,010
the location, I mean, is a whole number, I mean,

313
00:27:18,250 --> 00:27:24,050
if it is an integer. Then the rank position is the

314
00:27:24,050 --> 00:27:28,590
same number. For example, suppose the rank

315
00:27:28,590 --> 00:27:34,610
position is four. So position number four is your

316
00:27:34,610 --> 00:27:38,450
quartile, either first or third or second

317
00:27:38,450 --> 00:27:42,510
quartile. So if the result is a whole number, then

318
00:27:42,510 --> 00:27:48,350
it is the rank position used. Now, if the result

319
00:27:48,350 --> 00:27:52,250
is a fractional half, I mean if the right position

320
00:27:52,250 --> 00:27:58,830
is 2.5, 3.5, 4.5. In this case, average the two

321
00:27:58,830 --> 00:28:02,050
corresponding data values. For example, if the

322
00:28:02,050 --> 00:28:10,170
right position is 2.5. So the rank position is 2

323
00:28:10,170 --> 00:28:13,210
.5. So take the average of the corresponding

324
00:28:13,210 --> 00:28:18,950
values for the rank 2 and 3. So look at the value.

325
00:28:19,280 --> 00:28:24,740
at rank 2, value at rank 3, then take the average

326
00:28:24,740 --> 00:28:29,300
of the corresponding values. That if the rank

327
00:28:29,300 --> 00:28:31,280
position is fractional.

328
00:28:34,380 --> 00:28:37,900
So if the result is whole number, just take it as

329
00:28:37,900 --> 00:28:41,160
it is. If it is a fractional half, take the

330
00:28:41,160 --> 00:28:44,460
corresponding data values and take the average of

331
00:28:44,460 --> 00:28:49,110
these two values. Now, if the result is not a

332
00:28:49,110 --> 00:28:53,930
whole number or a fraction of it. For example,

333
00:28:54,070 --> 00:29:01,910
suppose the location is 2.1. So the position is 2,

334
00:29:02,390 --> 00:29:06,550
just round, up to the nearest integer. So that's

335
00:29:06,550 --> 00:29:11,350
2. What's about if the position rank is 2.6? Just

336
00:29:11,350 --> 00:29:16,060
rank up to 3. So that's 3. So that's the rule you

337
00:29:16,060 --> 00:29:21,280
have to follow if the result is a number, a whole

338
00:29:21,280 --> 00:29:27,200
number, I mean integer, fraction of half, or not

339
00:29:27,200 --> 00:29:31,500
real number, I mean, not whole number, or fraction

340
00:29:31,500 --> 00:29:35,540
of half. Look at this specific example. Suppose we

341
00:29:35,540 --> 00:29:40,180
have this data. This is ordered array, 11, 12, up

342
00:29:40,180 --> 00:29:45,680
to 22. And let's see how can we compute These

343
00:29:45,680 --> 00:29:46,240
measures.

344
00:29:50,080 --> 00:29:51,700
Look carefully here.

345
00:29:55,400 --> 00:29:59,260
First, let's compute the median. The median and

346
00:29:59,260 --> 00:30:02,360
the value in the middle. How many values we have?

347
00:30:02,800 --> 00:30:08,920
There are nine values. So the middle is number

348
00:30:08,920 --> 00:30:15,390
five. One, two, three, four, five. So 16. This

349
00:30:15,390 --> 00:30:23,010
value is the median. Now look at the values below

350
00:30:23,010 --> 00:30:29,650
the median. There are 4 and 4 below and above the

351
00:30:29,650 --> 00:30:34,970
median. Now let's see how can we compute Q1. The

352
00:30:34,970 --> 00:30:38,250
position of Q1, as we mentioned, is N plus 1

353
00:30:38,250 --> 00:30:42,630
divided by 4. So N is 9 plus 1 divided by 4 is 2

354
00:30:42,630 --> 00:30:50,330
.5. 2.5 position, it means you have to take the

355
00:30:50,330 --> 00:30:54,490
average of the two corresponding values, 2 and 3.

356
00:30:55,130 --> 00:31:01,010
So 2 and 3, so 12 plus 13 divided by 2. That gives

357
00:31:01,010 --> 00:31:08,390
12.5. So this is Q1.

358
00:31:08,530 --> 00:31:18,210
So Q1 is 12.5. Now what's about Q3? The Q3, the

359
00:31:18,210 --> 00:31:27,810
rank position, Q1 was 2.5. So Q3 should be three

360
00:31:27,810 --> 00:31:32,410
times that value, because it's three times A plus

361
00:31:32,410 --> 00:31:36,090
1 over 4. That means the rank position is 7.5.

362
00:31:36,590 --> 00:31:39,410
That means you have to take the average of the 7

363
00:31:39,410 --> 00:31:44,890
and 8 position. 7 and 8 is 18.

364
00:31:45,880 --> 00:31:56,640
which is 19.5. So that's Q3, 19.5.

365
00:32:00,360 --> 00:32:09,160
So this is Q3. This value is Q1. And this value

366
00:32:09,160 --> 00:32:15,910
is? Now, Q2 is the center. is located in the

367
00:32:15,910 --> 00:32:18,570
center because, as we mentioned, four below and

368
00:32:18,570 --> 00:32:22,950
four above. Now what's about Q1? Q1 is not in the

369
00:32:22,950 --> 00:32:28,150
center of the entire data. Because Q1, 12.5, so

370
00:32:28,150 --> 00:32:31,830
two points below and the others maybe how many

371
00:32:31,830 --> 00:32:34,750
above two, four, six, seven observations above it.

372
00:32:35,390 --> 00:32:40,130
So that means Q1 is not center. Also Q3 is not

373
00:32:40,130 --> 00:32:43,170
center because two observations above it and seven

374
00:32:43,170 --> 00:32:48,780
below it. So that means Q1 and Q3 are measures of

375
00:32:48,780 --> 00:32:52,480
non-central location, while the median is a

376
00:32:52,480 --> 00:32:56,080
measure of central location. But if you just look

377
00:32:56,080 --> 00:33:03,720
at the data below the median, just focus on the

378
00:33:03,720 --> 00:33:09,100
data below the median, 12.5 lies exactly in the

379
00:33:09,100 --> 00:33:13,130
middle of the data. So 12.5 is the center of the

380
00:33:13,130 --> 00:33:18,090
data. I mean, Q1 is the center of the data below

381
00:33:18,090 --> 00:33:22,810
the overall median. The overall median was 16. So

382
00:33:22,810 --> 00:33:27,490
the data before 16, the median for this data is 12

383
00:33:27,490 --> 00:33:31,770
.5, which is the first part. Similarly, if you

384
00:33:31,770 --> 00:33:36,870
look at the data above Q2,

385
00:33:37,770 --> 00:33:42,190
now 19.5. is located in the middle of the line. So

386
00:33:42,190 --> 00:33:46,470
Q3 is a measure of center for the data above the

387
00:33:46,470 --> 00:33:48,390
line. Make sense?

388
00:33:51,370 --> 00:33:56,430
So that's how can we compute first, second, and

389
00:33:56,430 --> 00:34:03,510
third part. Any questions? Yes, but it's a whole

390
00:34:03,510 --> 00:34:09,370
number. Whole number, it means any integer. For

391
00:34:09,370 --> 00:34:14,450
example, yeah, exactly, yes. Suppose we have

392
00:34:14,450 --> 00:34:18,090
number of data is seven.

393
00:34:22,070 --> 00:34:25,070
Number of observations we have is seven. So the

394
00:34:25,070 --> 00:34:29,730
rank position n plus one divided by two, seven

395
00:34:29,730 --> 00:34:33,890
plus one over two is four. Four means the whole

396
00:34:33,890 --> 00:34:37,780
number, I mean an integer. then this case just use

397
00:34:37,780 --> 00:34:45,280
it as it is. Now let's see the benefit or the

398
00:34:45,280 --> 00:34:48,680
feature of using Q1 and Q3.

399
00:34:55,180 --> 00:35:01,300
So let's move at the inter-equilateral range or

400
00:35:01,300 --> 00:35:01,760
IQ1.

401
00:35:08,020 --> 00:35:14,580
2.5 is the position. So the rank data of the rank

402
00:35:14,580 --> 00:35:19,180
data. So take the average of the two corresponding

403
00:35:19,180 --> 00:35:25,700
values of this one, which is 2 and 3. So 2 and 3.

404
00:35:27,400 --> 00:35:31,940
The average of these two values is 12.5. One more

405
00:35:31,940 --> 00:35:40,920
time, 2.5 is not the value. It is the rank

406
00:35:40,920 --> 00:35:47,880
position of the first quartile. So in this case, 2

407
00:35:47,880 --> 00:35:57,740
.5 takes position 2 and 3. The average of these

408
00:35:57,740 --> 00:36:02,580
two rank positions the corresponding one, which

409
00:36:02,580 --> 00:36:10,080
are 12 and 13. So 12 for position number 2, 13 for

410
00:36:10,080 --> 00:36:13,580
the other one. So the average is just divided by

411
00:36:13,580 --> 00:36:16,660
2. That will give 12.5.

412
00:36:28,760 --> 00:36:34,900
Next, again, the inter-quartile range, which is

413
00:36:34,900 --> 00:36:44,160
denoted by IQR. Now IQR is the distance between Q3

414
00:36:44,160 --> 00:36:48,000
and Q1. I mean the difference between Q3 and Q1 is

415
00:36:48,000 --> 00:36:53,460
called the inter-quartile range. And this one

416
00:36:53,460 --> 00:36:56,680
measures the spread in the middle 50% of the data.

417
00:36:57,680 --> 00:36:59,060
Because if you imagine that,

418
00:37:02,250 --> 00:37:10,250
This is Q1 and Q3. IQR is the distance between

419
00:37:10,250 --> 00:37:14,130
these two values. Now imagine that we have just

420
00:37:14,130 --> 00:37:19,570
this data, which represents 50%.

421
00:37:21,540 --> 00:37:25,440
And IQR, the definition is a Q3. So we have just

422
00:37:25,440 --> 00:37:31,480
this data, for example. And IQ3 is Q3 minus Q1. It

423
00:37:31,480 --> 00:37:37,080
means IQ3 is the maximum minus the minimum of the

424
00:37:37,080 --> 00:37:41,540
50% of the middle data. So it means this is your

425
00:37:41,540 --> 00:37:46,980
range, new range. After you've secluded 25% to the

426
00:37:46,980 --> 00:37:52,450
left of Q1, And also you ignored totally 25% of

427
00:37:52,450 --> 00:37:57,070
the data above Q3. So that means you're focused on

428
00:37:57,070 --> 00:38:00,630
50% of the data. And just take the average of

429
00:38:00,630 --> 00:38:04,070
these two points, I'm sorry, the distance of these

430
00:38:04,070 --> 00:38:07,670
two points Q3 minus Q1. So you will get the range.

431
00:38:07,990 --> 00:38:11,170
But not exactly the range. It's called, sometimes

432
00:38:11,170 --> 00:38:16,390
it's called mid-spread range. Because mid-spread,

433
00:38:16,510 --> 00:38:19,910
because we are talking about middle of the data,

434
00:38:19,990 --> 00:38:22,430
50% of the data, which is located in the middle.

435
00:38:23,110 --> 00:38:28,550
So do you think in this case, outliers actually,

436
00:38:29,090 --> 00:38:32,930
they are extreme values, the data below Q1 and

437
00:38:32,930 --> 00:38:38,150
data above Q3. That means inter-quartile range, Q3

438
00:38:38,150 --> 00:38:42,410
minus Q1, is not affected by outliers. Because you

439
00:38:42,410 --> 00:38:49,150
ignored the small values And the high values. So

440
00:38:49,150 --> 00:38:53,890
IQR is not affected by outliers. So in case of

441
00:38:53,890 --> 00:38:58,930
outliers, it's better to use IQR. Because the

442
00:38:58,930 --> 00:39:01,610
range is maximum minus minimum. And as we

443
00:39:01,610 --> 00:39:05,030
mentioned before, the range is affected by

444
00:39:05,030 --> 00:39:11,650
outliers. So IQR is again called the mid-spread

445
00:39:11,650 --> 00:39:17,940
because it covers the middle 50% of the data. IQR

446
00:39:17,940 --> 00:39:20,120
again is a measure of variability that is not

447
00:39:20,120 --> 00:39:23,900
influenced or affected by outliers or extreme

448
00:39:23,900 --> 00:39:26,680
values. So in the presence of outliers, it's

449
00:39:26,680 --> 00:39:34,160
better to use IQR instead of using the range. So

450
00:39:34,160 --> 00:39:39,140
again, median and the range are not affected by

451
00:39:39,140 --> 00:39:43,180
outliers. So in case of the presence of outliers,

452
00:39:43,340 --> 00:39:46,380
we have to use these measures, one as measure of

453
00:39:46,380 --> 00:39:49,780
central and the other as measure of spread. So

454
00:39:49,780 --> 00:39:54,420
measures like Q1, Q3, and IQR that are not

455
00:39:54,420 --> 00:39:57,400
influenced by outliers are called resistant

456
00:39:57,400 --> 00:40:01,980
measures. Resistance means in case of outliers,

457
00:40:02,380 --> 00:40:06,120
they remain in the same position or approximately

458
00:40:06,120 --> 00:40:09,870
in the same position. Because outliers don't

459
00:40:09,870 --> 00:40:13,870
affect these measures. I mean, don't affect Q1,

460
00:40:14,830 --> 00:40:20,130
Q3, and consequently IQR, because IQR is just the

461
00:40:20,130 --> 00:40:24,990
distance between Q3 and Q1. So to determine the

462
00:40:24,990 --> 00:40:29,430
value of IQR, you have first to compute Q1, Q3,

463
00:40:29,750 --> 00:40:35,780
then take the difference between these two. So,

464
00:40:36,120 --> 00:40:41,120
for example, suppose we have a data, and that data

465
00:40:41,120 --> 00:40:51,400
has Q1 equals 30, and Q3 is 55. Suppose for a data

466
00:40:51,400 --> 00:41:00,140
set, that data set has Q1 30, Q3 is 57. The IQR,

467
00:41:00,800 --> 00:41:07,240
or Inter Equal Hyper Range, 57 minus 30 is 27. Now

468
00:41:07,240 --> 00:41:12,460
what's the range? The range is maximum for the

469
00:41:12,460 --> 00:41:17,380
largest value, which is 17 minus 12. That gives

470
00:41:17,380 --> 00:41:21,420
58. Now look at the difference between the two

471
00:41:21,420 --> 00:41:26,900
ranges. The inter-quartile range is 27. The range

472
00:41:26,900 --> 00:41:29,800
is 58. There is a big difference between these two

473
00:41:29,800 --> 00:41:35,750
values because range depends only on smallest and

474
00:41:35,750 --> 00:41:40,190
largest. And these values could be outliers. For

475
00:41:40,190 --> 00:41:44,410
this reason, the range value is higher or greater

476
00:41:44,410 --> 00:41:48,410
than the required range, which is just the

477
00:41:48,410 --> 00:41:54,050
distance of the 50% of the middle data. For this

478
00:41:54,050 --> 00:41:59,470
reason, it's better to use the range in case of

479
00:41:59,470 --> 00:42:03,940
outliers. Make sense? Any question?

480
00:42:08,680 --> 00:42:19,320
Five-number summary are smallest

481
00:42:19,320 --> 00:42:27,380
value, largest value, also first quartile, third

482
00:42:27,380 --> 00:42:32,250
quartile, and the median. These five numbers are

483
00:42:32,250 --> 00:42:35,870
called five-number summary, because by using these

484
00:42:35,870 --> 00:42:41,590
statistics, smallest, first, median, third

485
00:42:41,590 --> 00:42:46,010
quarter, and largest, you can describe the center

486
00:42:46,010 --> 00:42:52,590
spread and the shape of the distribution. So by

487
00:42:52,590 --> 00:42:56,450
using five-number summary, you can tell something

488
00:42:56,450 --> 00:43:00,090
about it. The center of the data, I mean the value

489
00:43:00,090 --> 00:43:02,070
in the middle, because the median is the value in

490
00:43:02,070 --> 00:43:06,550
the middle. Spread, because we can talk about the

491
00:43:06,550 --> 00:43:11,070
IQR, which is the range, and also the shape of the

492
00:43:11,070 --> 00:43:15,450
data. And let's see, let's move to this slide,

493
00:43:16,670 --> 00:43:18,530
slide number 50.

494
00:43:21,530 --> 00:43:25,090
Let's see how can we construct something called

495
00:43:25,090 --> 00:43:31,850
box plot. Box plot. Box plot can be constructed by

496
00:43:31,850 --> 00:43:34,990
using the five number summary. We have smallest

497
00:43:34,990 --> 00:43:37,550
value. On the other hand, we have the largest

498
00:43:37,550 --> 00:43:43,430
value. Also, we have Q1, the first quartile, the

499
00:43:43,430 --> 00:43:47,510
median, and Q3. For symmetric distribution, I mean

500
00:43:47,510 --> 00:43:52,490
if the data is bell-shaped. In this case, the

501
00:43:52,490 --> 00:43:56,570
vertical line in the box which represents the

502
00:43:56,570 --> 00:43:59,730
median should be located in the middle of this

503
00:43:59,730 --> 00:44:05,510
box, also in the middle of the entire data. Look

504
00:44:05,510 --> 00:44:11,350
carefully at this vertical line. This line splits

505
00:44:11,350 --> 00:44:16,070
the data into two halves, 25% to the left and 25%

506
00:44:16,070 --> 00:44:19,960
to the right. And also this vertical line splits

507
00:44:19,960 --> 00:44:24,720
the data into two halves, from the smallest to

508
00:44:24,720 --> 00:44:29,760
largest, because there are 50% of the observations

509
00:44:29,760 --> 00:44:34,560
lie below, and 50% lies above. So that means by

510
00:44:34,560 --> 00:44:37,840
using box plot, you can tell something about the

511
00:44:37,840 --> 00:44:42,520
shape of the distribution. So again, if the data

512
00:44:42,520 --> 00:44:48,270
are symmetric around the median, And the central

513
00:44:48,270 --> 00:44:53,910
line, this box, and central line are centered

514
00:44:53,910 --> 00:44:57,550
between the endpoints. I mean, this vertical line

515
00:44:57,550 --> 00:45:00,720
is centered between these two endpoints. between

516
00:45:00,720 --> 00:45:04,180
Q1 and Q3. And the whole box plot is centered

517
00:45:04,180 --> 00:45:07,100
between the smallest and the largest value. And

518
00:45:07,100 --> 00:45:10,840
also the distance between the median and the

519
00:45:10,840 --> 00:45:14,320
smallest is roughly equal to the distance between

520
00:45:14,320 --> 00:45:19,760
the median and the largest. So you can tell

521
00:45:19,760 --> 00:45:22,660
something about the shape of the distribution by

522
00:45:22,660 --> 00:45:26,780
using the box plot.

523
00:45:32,870 --> 00:45:36,110
The graph in the middle. Here median and median

524
00:45:36,110 --> 00:45:40,110
are the same. The box plot, we have here the

525
00:45:40,110 --> 00:45:43,830
median in the middle of the box, also in the

526
00:45:43,830 --> 00:45:47,390
middle of the entire data. So you can say that the

527
00:45:47,390 --> 00:45:50,210
distribution of this data is symmetric or is bell

528
00:45:50,210 --> 00:45:55,750
-shaped. It's normal distribution. On the other

529
00:45:55,750 --> 00:46:00,110
hand, if you look here, you will see that the

530
00:46:00,110 --> 00:46:06,160
median is not in the center of the box. It's near

531
00:46:06,160 --> 00:46:12,580
Q3. So the left tail, I mean, the distance between

532
00:46:12,580 --> 00:46:16,620
the median and the smallest, this tail is longer

533
00:46:16,620 --> 00:46:20,600
than the right tail. In this case, it's called

534
00:46:20,600 --> 00:46:24,850
left skewed or skewed to the left. or negative

535
00:46:24,850 --> 00:46:29,510
skewness. So if the data is not symmetric, it

536
00:46:29,510 --> 00:46:35,630
might be left skewed. I mean, the left tail is

537
00:46:35,630 --> 00:46:40,590
longer than the right tail. On the other hand, if

538
00:46:40,590 --> 00:46:45,950
the median is located near Q1, it means the right

539
00:46:45,950 --> 00:46:49,930
tail is longer than the left tail, and it's called

540
00:46:49,930 --> 00:46:56,470
positive skewed or right skewed. So for symmetric

541
00:46:56,470 --> 00:47:00,310
distribution, the median in the middle, for left

542
00:47:00,310 --> 00:47:04,570
or right skewed, the median either is close to the

543
00:47:04,570 --> 00:47:09,930
Q3 or skewed distribution to the left, or the

544
00:47:09,930 --> 00:47:14,910
median is close to Q1 and the distribution is

545
00:47:14,910 --> 00:47:20,570
right skewed or has positive skewness. That's how

546
00:47:20,570 --> 00:47:25,860
can we tell spread center and the shape by using

547
00:47:25,860 --> 00:47:28,460
the box plot. So center is the value in the

548
00:47:28,460 --> 00:47:32,860
middle, Q2 or the median. Spread is the distance

549
00:47:32,860 --> 00:47:38,340
between Q1 and Q3. So Q3 minus Q1 gives IQR. And

550
00:47:38,340 --> 00:47:41,880
finally, you can tell something about the shape of

551
00:47:41,880 --> 00:47:45,140
the distribution by just looking at the scatter

552
00:47:45,140 --> 00:47:46,440
plot.

553
00:47:49,700 --> 00:47:56,330
Let's look at This example, and suppose we have

554
00:47:56,330 --> 00:48:02,430
small data set. And let's see how can we construct

555
00:48:02,430 --> 00:48:05,750
the MaxPlot. In order to construct MaxPlot, you

556
00:48:05,750 --> 00:48:09,510
have to compute minimum first or smallest value,

557
00:48:09,810 --> 00:48:14,650
largest value. Besides that, you have to compute

558
00:48:14,650 --> 00:48:21,110
first and third part time and also Q2. For this

559
00:48:21,110 --> 00:48:27,570
simple example, Q1 is 2, Q3 is 5, and the median

560
00:48:27,570 --> 00:48:33,990
is 3. Smallest is 0, largest is 1 7. Now, be

561
00:48:33,990 --> 00:48:38,130
careful here, 1 7 seems to be an outlier. But so

562
00:48:38,130 --> 00:48:44,190
far, we don't explain how can we decide if a data

563
00:48:44,190 --> 00:48:47,550
value is considered to be an outlier. But at least

564
00:48:47,550 --> 00:48:53,080
1 7. is a suspected value to be an outlier, seems

565
00:48:53,080 --> 00:48:57,200
to be. Sometimes you are 95% sure that that point

566
00:48:57,200 --> 00:49:00,160
is an outlier, but you cannot tell, because you

567
00:49:00,160 --> 00:49:04,060
have to have a specific rule that can decide if

568
00:49:04,060 --> 00:49:07,400
that point is an outlier or not. But at least it

569
00:49:07,400 --> 00:49:12,060
makes sense that that point is considered maybe an

570
00:49:12,060 --> 00:49:14,700
outlier. But let's see how can we construct that

571
00:49:14,700 --> 00:49:18,190
first. The box plot. Again, as we mentioned, the

572
00:49:18,190 --> 00:49:21,630
minimum value is zero. The maximum is 27. The Q1

573
00:49:21,630 --> 00:49:27,830
is 2. The median is 3. The Q3 is 5. Now, if you

574
00:49:27,830 --> 00:49:32,010
look at the distance between, does this vertical

575
00:49:32,010 --> 00:49:35,790
line lie between the line in the middle or the

576
00:49:35,790 --> 00:49:40,090
center of the box? It's not exactly. But if you

577
00:49:40,090 --> 00:49:45,260
look at this line, vertical line, and the location

578
00:49:45,260 --> 00:49:50,600
of this with respect to the minimum and the

579
00:49:50,600 --> 00:49:56,640
maximum. You will see that the right tail is much

580
00:49:56,640 --> 00:50:01,560
longer than the left tail because it starts from 3

581
00:50:01,560 --> 00:50:06,180
up to 27. And the other one, from zero to three,

582
00:50:06,380 --> 00:50:09,760
is a big distance between three and 27, compared

583
00:50:09,760 --> 00:50:13,140
to the other one, zero to three. So it seems to be

584
00:50:13,140 --> 00:50:16,600
this is quite skewed, so it's not at all

585
00:50:16,600 --> 00:50:23,700
symmetric, because of this value. So maybe by

586
00:50:23,700 --> 00:50:25,580
using MaxPlot, you can tell that point is

587
00:50:25,580 --> 00:50:31,440
suspected to be an outlier. It has a very long

588
00:50:31,440 --> 00:50:32,800
right tail.

589
00:50:35,560 --> 00:50:41,120
So let's see how can we determine if a point is an

590
00:50:41,120 --> 00:50:50,400
outlier or not. Sometimes we can use box plot to

591
00:50:50,400 --> 00:50:53,840
determine if the point is an outlier or not. The

592
00:50:53,840 --> 00:51:00,860
rule is that a value is considered an outlier It

593
00:51:00,860 --> 00:51:04,780
is more than 1.5 times the entire quartile range

594
00:51:04,780 --> 00:51:11,420
below Q1 or above it. Let's explain the meaning of

595
00:51:11,420 --> 00:51:12,260
this sentence.

596
00:51:15,260 --> 00:51:20,100
First, let's compute something called lower.

597
00:51:23,740 --> 00:51:28,540
The lower limit is

598
00:51:28,540 --> 00:51:38,680
not the minimum. It's Q1 minus 1.5 IQR. This is

599
00:51:38,680 --> 00:51:39,280
the lower limit.

600
00:51:42,280 --> 00:51:47,560
So it's 1.5 times IQR below Q1. This is the lower

601
00:51:47,560 --> 00:51:50,620
limit. The upper limit,

602
00:51:54,680 --> 00:51:57,460
Q3,

603
00:51:58,790 --> 00:52:06,890
plus 1.5 times IQR. So we computed lower and upper

604
00:52:06,890 --> 00:52:13,350
limit by using these rules. Q1 minus 1.5 IQR. So

605
00:52:13,350 --> 00:52:20,510
it's 1.5 times IQR below Q1 and 1.5 times IQR

606
00:52:20,510 --> 00:52:25,070
above Q1. Now, any value.

607
00:52:31,150 --> 00:52:38,610
Is it smaller than the

608
00:52:38,610 --> 00:52:45,990
lower limit or

609
00:52:45,990 --> 00:52:53,290
greater than the

610
00:52:53,290 --> 00:52:54,150
upper limit?

611
00:52:58,330 --> 00:53:04,600
Any value. smaller than the lower limit and

612
00:53:04,600 --> 00:53:13,260
greater than the upper limit is considered to

613
00:53:13,260 --> 00:53:20,720
be an outlier. This is the rule how can you tell

614
00:53:20,720 --> 00:53:24,780
if the point or data value is outlier or not. Just

615
00:53:24,780 --> 00:53:27,100
compute lower limit and upper limit.

616
00:53:29,780 --> 00:53:35,580
So lower limit, Q1 minus 1.5IQ3. Upper limit, Q3

617
00:53:35,580 --> 00:53:38,620
plus 1.5. This is a constant.

618
00:53:43,200 --> 00:53:47,040
Now let's go back to the previous example, which

619
00:53:47,040 --> 00:53:53,800
was, which Q1 was, what's the value of Q1? Q1 was

620
00:53:53,800 --> 00:53:57,680
2. Q3 is 5.

621
00:54:00,650 --> 00:54:05,230
In order to turn an outlier, you don't need the

622
00:54:05,230 --> 00:54:11,150
value, the median. Now, Q3 is 5, Q1 is 2, so IQR

623
00:54:11,150 --> 00:54:21,050
is 3. That's the value of IQR. Now, lower limit, A

624
00:54:21,050 --> 00:54:31,830
times 2 minus 1.5 times IQR3. So that's minus 2.5.

625
00:54:33,550 --> 00:54:41,170
U3 plus U3 is 3. It's 5, sorry. It's 5 plus 1.5.

626
00:54:41,650 --> 00:54:48,570
That gives 9.5. Now, any point or any data value,

627
00:54:49,450 --> 00:54:55,950
any data value falls below minus 2.5. I mean

628
00:54:55,950 --> 00:55:00,380
smaller than minus 2.5. Or greater than 9.5 is an

629
00:55:00,380 --> 00:55:05,420
outlier. If you look at the data you have, we have

630
00:55:05,420 --> 00:55:09,520
0 up to 9. So none of these is considered to be an

631
00:55:09,520 --> 00:55:16,200
outlier. But what's about 27? 27 is greater than,

632
00:55:16,260 --> 00:55:23,160
much bigger than actually 9.5. So for that data,

633
00:55:24,020 --> 00:55:27,920
27 is an outlier. So this is the way how can we

634
00:55:27,920 --> 00:55:36,120
compute the outlier for the sample. Another

635
00:55:36,120 --> 00:55:39,620
method. The score is another method to determine

636
00:55:39,620 --> 00:55:43,600
if that point is an outlier or not. So, so far we

637
00:55:43,600 --> 00:55:48,300
have two rules. One by using quartiles and the

638
00:55:48,300 --> 00:55:50,540
other, as we mentioned last time, by using the

639
00:55:50,540 --> 00:55:54,200
score. And for these scores, if you remember, any

640
00:55:54,200 --> 00:56:00,030
values below lie Below minus three. And above

641
00:56:00,030 --> 00:56:03,430
three is considered to be irrelevant. That's

642
00:56:03,430 --> 00:56:07,950
another example. That's another way to figure out

643
00:56:07,950 --> 00:56:09,190
if the data is irrelevant.

644
00:56:13,730 --> 00:56:17,110
You can apply the two rules either for the sample

645
00:56:17,110 --> 00:56:20,190
or the population. If you have the entire data,

646
00:56:20,890 --> 00:56:23,950
you can also determine out there for the entire

647
00:56:23,950 --> 00:56:29,110
dataset, even if that data is the population. But

648
00:56:29,110 --> 00:56:34,490
most of the time, we select a sample, which is a

649
00:56:34,490 --> 00:56:37,790
subset or a portion of that population.

650
00:56:40,570 --> 00:56:41,290
Questions?

651
00:56:53,360 --> 00:57:00,000
And locating outliers. So again, outlier is any

652
00:57:00,000 --> 00:57:05,000
value that is above the upper limit or below the

653
00:57:05,000 --> 00:57:08,340
lower limit. And also we can use this score also

654
00:57:08,340 --> 00:57:12,680
to determine if that point is outlier or not. Next

655
00:57:12,680 --> 00:57:16,340
time, Inshallah, we will go over the covariance

656
00:57:16,340 --> 00:57:19,420
and the relationship and I will give some practice

657
00:57:19,420 --> 00:57:22,180
problems for Chapter 3.