PL9fwy3NUQKwaIc7KWc8buW344941GStQL/0UQx5fYO0DE.srt

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Inshallah we'll start numerical descriptive measures

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for the population. Last time we talked about the 

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same measures. 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 ith 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 a new topic, which is empirical rule.

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Now, empirical rule 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 bell-shaped

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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 a bell-shaped 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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test scores is bell shaped. So here we assume that 

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The math test 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 test 

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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 a 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 the 68% rule. It means that 68%

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of all the students scored

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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 68 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 6 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 6 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.

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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 the 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 the empirical rule

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68959917. And the other one is called the 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 an unknown distribution. For

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this reason, we have to use chip-chip portions. 

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That's all for the empirical rule and the chip-chip rule. 

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The next topic is quartile measures. So far, we

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have discussed central tendency measures, and we

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have talked about mean, median, and more. Then we

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moved to location of variability or spread or

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dispersion. And we talked about range, variance,

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and standardization.

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And we said that outliers affect the mean much

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more than the median. And also, outliers affect

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the range. Here, we'll talk about other measures

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of the data, which is called quartile measures.

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Here, actually, we'll talk about two measures.

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The first one is called the first quartile, and the other

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one is the third quartile. So we have two measures,

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the first and the third quartile. Quartiles split the ranked

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data into four equal segments. I mean, these 

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measures split the data you have into four equal

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parts.

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Q1 has 25% of the data fall below it. I mean 25% 

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of the values lie below Q1. So it means 75% of the

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values are above it. So 25 below and 75 above. But you

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have to be careful that the data is arranged from 

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smallest to largest. So in this case, Q1 is a 

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value that has 25% below it. So Q2 is called the

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median. The median, the value in the middle when 

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we arrange the data from smallest to largest. So 

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that means 50% of the data below and also 50% of 

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the data above. The other measure is called

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the theoretical qualifying. In this case, we have 25%

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of the data above Q3 and 75% of the data below Q3. 

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So quartiles split the ranked 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 the 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 located 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 ranked 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 a 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
a 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 an 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 is 

346
00:29:59,260 --> 00:30:02,360
the value in the middle. How many values do 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 four below and four 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. It's 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 the center. Also, Q3 is not

373
00:32:40,130 --> 00:32:43,170
the 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 the 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 we can compute the 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
a number of data that is seven.

393
00:34:22,070 --> 00:34:25,070
The 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 a whole

396
00:34:33,890 --> 00:34:37,780
number, I mean an integer. Then, in 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 on to the inter-quartile range or

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

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

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 are 2 and 3. So 2 and 3. 

404
00:35:27,400 --> 

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 17. Now, be 

561
00:48:33,990 --> 00:48:38,130
careful here, 17 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
17. 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.