PL9fwy3NUQKwaIc7KWc8buW344941GStQL/4oMFiRBOjhY_postprocess.srt

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So let's again go back to chapter number one. Last

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time we discussed chapter one, production and data

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collection. And I think we described why learning

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statistics distinguish between some of these

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topics. And also we explained in details types of

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statistics and we mentioned that statistics mainly

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has two types either descriptive statistics which

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means collecting summarizing and obtaining data

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and other type of statistics is called inferential

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statistics or statistical inference and this type

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of statistics we can draw drawing conclusions and

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making decision concerning a population based only

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on a sample. That means we have a sample and

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sample is just a subset of the population or the

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portion of the population and we use the data from

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that sample to make some conclusion about the

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entire population. This type of statistic is

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called inferential statistics. Later, Inshallah,

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we'll talk in details about inferential statistics

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that will start in Chapter 7. Also, we gave some

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definitions for variables, data, and we

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distinguished between population and sample. And

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we know that the population consists of all items

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or individuals about which you want to draw a

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conclusion. But in some cases, it's very hard to

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talk about the population or the entire

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population, so we can select a sample. A sample is

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just a portion or subset of the entire population.

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So we know now the definition of population and

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sample. The other two types, parameter and

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statistics. Parameter is a numerical measure that

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describes characteristics of a population, while

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on the other hand, a sample, a statistic is just

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numerical measures that describe characteristic of

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a sample. So parameter is computed from the

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population while statistic is computed from the

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sample. I think we stopped at this point. Why

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collect data? I mean what are the reasons for

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One of these reasons, for example, a marketing

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research analyst needs to assess the effectiveness

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of a new television advertisement. For example,

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suppose you are a manager and you want to increase

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your salaries or your sales. Now, sales may be

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affected by advertising. So I mean, if you spend

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more on advertising, it means your sales becomes

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larger and larger. So you want to know if this

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variable, I mean if advertisement is an effective

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variable that maybe increase your sales. So that's

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one of the reasons why we use data. The other one,

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for example, pharmaceutical manufacturers needs to

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determine whether a new drug is more effective

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than those currently used. For example, for a

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headache, we use drug A. Now, a new drug is

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produced and you want to see if this new drug is

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more effective than drug A that I mean if headache

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suppose for example is removed after three days by

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using drug A now the question is does B is more

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effective it means it reduces your headache in

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fewer than three days I mean maybe in two days

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That means a drug B is more effective than a drug

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A. So we want to know the difference between these

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two drugs. I mean, we have two samples. Some

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people used drug A and the other used drug B. And

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we want to see if there is a significant

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difference between the times that is used to

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reduce the headache. So that's one of the reasons

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why we use statistics. Sometimes an operation

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manager wants to monitor manufacturing process to

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find out whether the quality of a product being

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manufactured is conforming to a company's

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standards. Do you know what the meaning of

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company's standards?

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The regulations of the firm itself. Another

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example, suppose here in the school last year, we

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teach statistics by using method A. traditional

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method. This year we developed a new method for

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teaching and our goal is to see if the new method

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is better than method A which was used in last

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year. So we want to see if there is a big

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difference between scores or the average scores

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last year and this year. The same you can do for

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your weight. Suppose there are 20 students in this

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class and their weights are high. And our goal is

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to reduce their weights. Suppose they

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have a regime or diet for three months or

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exercise, whatever it is, then after three months,

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we have new weights for these persons. And we want

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to see if the diet is effective. I mean, if the

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average weight was greater than or smaller than

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before diet. Is it clear? So there are many, many

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reasons behind using statistics and collecting

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data. Now, what are the sources of data? Since

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statistics mainly, first step, we have to collect

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data. Now, what are the sources of the data?

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Generally speaking, there are two sources. One is

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called The primary sources and the others

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secondary sources. What do you think is the

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difference between these two? I mean, what's the

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difference between primary and secondary sources?

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The primary source is the collector of the data.

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He is the analyzer. He analyzes it. And then the

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secondary, who collects the data, isn't there.

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That's correct. So the primary sources means the

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researcher by himself. He should collect the data,

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then he can use the data to do his analysis.

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That's for the primary. Now, the primary could be

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data from political survey. You can distribute

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questionnaire, for example, data collected from an

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experiment. I mean maybe control or experimental

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groups. We have two groups, maybe healthy people

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and patient people. So that's experimental group.

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Or observed data. That's the primary sources.

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Secondary sources, the person performing data

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analysis is not the data collector. So he obtained

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the data from other sources. For example, it could

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be analyzing census data or for example, examining

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data from print journals or data published on the

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internet. So maybe he goes to the Ministry of

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Education and he can get some data. So the data is

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already there and he just used the data to do some

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analysis. So that's the difference between a

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primary and secondary sources. So primary, the

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researcher himself, should collect the data by

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using one of the tools, either survey,

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questionnaire, experiment, and so on. But

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secondary, you can use the data that is published

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in the internet, for example, in the books, in

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governments and NGOs and so on. So these are the

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two sources of data. Sources of data fall into

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four categories. Number one, data distributed by

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an organization or an individual. So that's

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secondary source. A design experiment is primary

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because you have to design the experiment, a

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survey. It's also primary. An observational study

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is also a primary source. So you have to

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distinguish between a primary and secondary

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sources. Any question? Comments? Next.

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We'll talk a little bit about types of variables.

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In general, there are two types of variables. One

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is called categorical variables or qualitative

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variables, and the other one is called numerical

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or quantitative variables. Now, for example, if I

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ask you, what's

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your favorite color? You may say white, black,

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red, and so on. What's your marital status? Maybe

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married or unmarried, and so on. Gender, male,

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either male or female, and so on. So this type of

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variable is called qualitative variables. So

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qualitative variables have values that can only be

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placed into categories, such as, for example, yes

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or no. For example, do you like orange?

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The answer is either yes or no. Do you like

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candidate A, for example, whatever his party is?

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Do you like it? Either yes or no, and so on. As I

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mentioned before, gender, marital status, race,

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religions, these are examples of qualitative or

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categorical variables. The other type of variable

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which is commonly used is called numerical or

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quantitative data. Quantitative variables have

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values that represent quantities. For example, if

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I ask you, what's your age? My age is 20 years old

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or 18 years old. What's your weight? Income.

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Height? Temperature? Income. So it's a number.

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Weight, maybe my weight is 70 kilograms. So

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weight, age, height, salary, income, number of

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students, number of phone calls you received on

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your cell phone during one hour, number of

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accidents happened in street and so on. So that's

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the difference between numerical variables and

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qualitative variables.

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Anyone of you just give me one example of

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qualitative and quantitative variables. Another

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examples. Just give me one example for qualitative

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data.

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Qualitative or quantitative.

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Political party, either party A or party B. So

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suppose there are two parties, so I like party A,

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she likes party B and so on. So party in this case

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is qualitative variable, another one.

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So types of courses, maybe business, economics,

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administration, and so on. So types of courses.

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Another example for quantitative variable or

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numerical variables.

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So production is a numerical variable. Another

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example for quantitative.

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Is that produced by this company? Number of cell

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phones, maybe 20, 17, and so on. Any question?

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Next. So generally speaking, types of data, data

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has two types, categorical and numerical data. As

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we mentioned, marital status, political party, eye

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color, and so on. These are examples of

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categorical variables. On the other hand, a

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numerical variable can be split or divided into

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two parts. One is called discrete and the other is

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continuous, and we have to distinguish between

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these two. For example, Number of students in this

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class, you can say there are 60 or 50 students in

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this class. You cannot say there are 50.5

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students. So number of students is discrete

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because it takes only integers. While for

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continuous type of numerical variables, you can

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say that my weight is 80.5 kilograms. so it makes

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sense that your weight is not exactly 80 kilograms

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it might be 80.6 or 80.5 and so on so discrete

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takes only integers while continuous takes any

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value I mean any real number so that's the

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difference between discrete and continuous number

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of phone or number of calls you have received this

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morning, maybe one, zero, nine, and so on,

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discrete. Number of patients in the hospital,

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discrete, and so on. But when we are talking about

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income, maybe my income is 1,000.5 shekel. It

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could be. It's continuous because my income can be

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any number between, for example, 1,000 and 10,000.

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It takes any value in this interval from 1,000 to

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10,000. So it types of continuous rather than our

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continuous variable. So that's the two types of

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data, categorical and numerical. And numerical

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also has two types, either discrete or continuous.

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Later in Chapter 6, we'll talk more details about

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one of the most distribution statistics, for

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continuous, one which is called normal

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distribution. That will be later, inshallah. As we

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mentioned last time, at the end of each chapter,

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there is a section or sections, sometimes there

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are two sections, talks about computer programs.

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How can we use computer programs in order to

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analyze the data? And as we mentioned last time,

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you should take a course on that. It's called

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Computer and Data Analysis or SPSS course. So we

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are going to skip the computer programs used for

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any chapters in this book. And that's the end of

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chapter number three. Any questions?

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Let's move. quickly on chapter three.

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Chapter three maybe is the easiest chapter in this

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book. It's straightforward. We have some formulas

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to compute some statistical measures. And we

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should know how can we calculate these measures

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and what are the meaning of your results. So

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chapter three talks about numerical descriptive

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measures. In this chapter, you will learn, number

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one, describe the probabilities of central

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tendency, variation, and shape in numerical data.

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In this lecture, we'll talk in more details about

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some of the center tendency measures. Later, we'll

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talk about the variation, or spread, or

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dispersion, and the shape in numerical data. So

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that's part number one. We have to know something

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about center tendency, variation, and the shape of

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the data we have. to calculate descriptive summary

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measures for a population. So we have to calculate

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these measures for the sample. And if we have the

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entire population, we can compute these measures

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also for that population. Then I will introduce in

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more details about something called Paxiplot. How

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can we construct and interpret a Paxiplot? That's,

257
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inshallah, next time on Tuesday. Finally, we'll

258
00:18:11,040 --> 00:18:13,020
see how can we calculate the covariance and

259
00:18:13,020 --> 00:18:15,700
coefficient of variation and coefficient, I'm

260
00:18:15,700 --> 00:18:18,480
sorry, coefficient of correlation. This topic

261
00:18:18,480 --> 00:18:24,280
we'll introduce in more details in chapter 11

262
00:18:24,280 --> 00:18:31,240
later on. So just I will give some brief

263
00:18:31,240 --> 00:18:35,630
notation about coefficient of correlation, how can

264
00:18:35,630 --> 00:18:39,030
we compute the correlation coefficient? What's the

265
00:18:39,030 --> 00:18:41,870
meaning of your result? And later in chapter 11,

266
00:18:42,030 --> 00:18:44,510
we'll talk in more details about correlation and

267
00:18:44,510 --> 00:18:48,930
regression. So these are the objectives of this

268
00:18:48,930 --> 00:18:52,870
chapter. There are some basic definitions before

269
00:18:52,870 --> 00:18:57,410
we start. One is called central tendency. What do

270
00:18:57,410 --> 00:19:00,750
you mean by central tendency? Central tendency is

271
00:19:00,750 --> 00:19:04,990
the extent to which all data value group around a

272
00:19:04,990 --> 00:19:08,890
typical or numerical or central value. So we are

273
00:19:08,890 --> 00:19:12,510
looking for a point that in the center, I mean,

274
00:19:12,810 --> 00:19:18,870
the data points are gathered or collected around a

275
00:19:18,870 --> 00:19:21,670
middle point, and that middle point is called the

276
00:19:21,670 --> 00:19:24,450
central tendency. And the question is, how can we

277
00:19:24,450 --> 00:19:27,780
measure that value? We'll talk in details about

278
00:19:27,780 --> 00:19:32,620
mean, median, and mode in a few minutes. So the

279
00:19:32,620 --> 00:19:35,660
central tendency, in this case, the data values

280
00:19:35,660 --> 00:19:40,080
grouped around a typical or central value. Is it

281
00:19:40,080 --> 00:19:44,380
clear? So we have data set, large data set. Then

282
00:19:44,380 --> 00:19:47,860
these points are gathered or grouped around a

283
00:19:47,860 --> 00:19:51,440
middle point, and this point is called central

284
00:19:51,440 --> 00:19:56,120
tendency, and it can be measured by using mean,

285
00:19:56,420 --> 00:19:59,960
which is the most common one, median and the moon.

286
00:20:01,020 --> 00:20:04,480
Next is the variation, which is the amount of

287
00:20:04,480 --> 00:20:09,420
dispersion. Variation is the amount of dispersion

288
00:20:09,420 --> 00:20:13,900
or scattering of values. And we'll use, for

289
00:20:13,900 --> 00:20:18,400
example, range, variance or standard deviation in

290
00:20:18,400 --> 00:20:22,960
order to compute the variation. Finally, We have

291
00:20:22,960 --> 00:20:26,300
data, and my question is, what's the shape of the

292
00:20:26,300 --> 00:20:29,920
data? So the shape is the pattern of distribution

293
00:20:29,920 --> 00:20:35,220
of values from the lowest value to the highest. So

294
00:20:35,220 --> 00:20:39,400
that's the three definitions we need to know

295
00:20:39,400 --> 00:20:44,580
before we start. So we'll start with the easiest

296
00:20:44,580 --> 00:20:48,680
one, measures of central tendency. As I mentioned,

297
00:20:49,160 --> 00:20:55,110
there are three measures. median and moon. And our

298
00:20:55,110 --> 00:20:58,270
goal or we have two goals actually. We have to

299
00:20:58,270 --> 00:21:02,290
know how to compute these measures. Number two,

300
00:21:03,270 --> 00:21:06,390
which one is better? The mean or the median or the

301
00:21:06,390 --> 00:21:06,550
moon?

302
00:21:11,310 --> 00:21:14,770
So the mean sometimes called the arithmetic mean.

303
00:21:15,680 --> 00:21:20,020
Or in general, just say the mean. So often we use

304
00:21:20,020 --> 00:21:26,860
the mean. And the mean is just sum

305
00:21:26,860 --> 00:21:33,220
of the values divided by the sample size. So it's

306
00:21:33,220 --> 00:21:36,800
straightforward. We have, for example, three data

307
00:21:36,800 --> 00:21:42,180
points. And your goal is to find the average or

308
00:21:42,180 --> 00:21:45,890
the mean of these points. They mean it's just some

309
00:21:45,890 --> 00:21:50,230
of these values divided by the sample size. So for

310
00:21:50,230 --> 00:21:54,570
example, if we have a data X1, X2, X3 up to Xn. So

311
00:21:54,570 --> 00:21:59,650
the average is denoted by X bar. This one is

312
00:21:59,650 --> 00:22:04,530
pronounced as X bar and X bar is just sum of Xi.

313
00:22:05,010 --> 00:22:08,250
It is summation, you know this symbol, summation

314
00:22:08,250 --> 00:22:11,350
of sigma, summation of Xi and I goes from one to

315
00:22:11,350 --> 00:22:14,490
N. divided by N which is the total number of

316
00:22:14,490 --> 00:22:19,710
observations or the sample size. So it means X1

317
00:22:19,710 --> 00:22:23,290
plus X2 all the way up to XN divided by N gives

318
00:22:23,290 --> 00:22:28,530
the mean or the arithmetic mean. So X bar is the

319
00:22:28,530 --> 00:22:32,690
average which is the sum of values divided by the

320
00:22:32,690 --> 00:22:36,270
number of observations. So that's the first

321
00:22:36,270 --> 00:22:38,830
definition. For example,

322
00:22:42,180 --> 00:22:46,920
So again, the mean is the most common measure of

323
00:22:46,920 --> 00:22:51,780
center tendency. Number two, the definition of the

324
00:22:51,780 --> 00:22:55,440
mean. Sum of values divided by the number of

325
00:22:55,440 --> 00:23:02,960
values. That means the mean takes all the values,

326
00:23:04,140 --> 00:23:09,740
then divided by N. it makes sense that the mean is

327
00:23:09,740 --> 00:23:13,380
affected by extreme values or outliers. I mean, if

328
00:23:13,380 --> 00:23:17,840
the data has outliers or extreme values, I mean by

329
00:23:17,840 --> 00:23:21,400
extreme values, large or very, very large values

330
00:23:21,400 --> 00:23:24,980
and small, small values. Large values or small

331
00:23:24,980 --> 00:23:31,100
values are extreme values. Since the mean takes

332
00:23:31,100 --> 00:23:33,420
all these values and sums all together, doesn't

333
00:23:33,420 --> 00:23:38,550
divide by n, that means The mean is affected by

334
00:23:38,550 --> 00:23:41,350
outliers or by extreme values. For example,

335
00:23:42,030 --> 00:23:45,110
imagine we have simple data as 1, 2, 3, 4, and 5.

336
00:23:46,110 --> 00:23:49,830
Simple example. Now, what's the mean? The mean is

337
00:23:49,830 --> 00:23:53,570
just add these values, then divide by the total

338
00:23:53,570 --> 00:23:56,910
number of observations. In this case, the sum of

339
00:23:56,910 --> 00:24:01,710
these is 15. N is five because there are five

340
00:24:01,710 --> 00:24:05,920
observations. So X bar is 15 divided by 5, which

341
00:24:05,920 --> 00:24:10,240
is 3. So straightforward. Now imagine instead of

342
00:24:10,240 --> 00:24:16,480
5, this number 5, we have a 10. Now 10, there is a

343
00:24:16,480 --> 00:24:21,400
gap between 4, which is the second largest, and

344
00:24:21,400 --> 00:24:25,600
the maximum, which is 10. Now if we add these

345
00:24:25,600 --> 00:24:30,540
values, 1, 2, 3, 4, and 10, then divide by 5, the

346
00:24:30,540 --> 00:24:36,680
mean will be 4. If you see here, we just added one

347
00:24:36,680 --> 00:24:41,060
value, or I mean, we replaced five by 10, and the

348
00:24:41,060 --> 00:24:44,700
mean changed dramatically from three to four.

349
00:24:45,520 --> 00:24:48,860
There is big change between three and four, around

350
00:24:48,860 --> 00:24:55,560
25% more. So that means outliers or extreme values

351
00:24:55,560 --> 00:25:01,200
affected the mean. So take this information in

352
00:25:01,200 --> 00:25:03,560
your mind because later we'll talk a little bit

353
00:25:03,560 --> 00:25:07,360
about another one. So the mean is affected by

354
00:25:07,360 --> 00:25:13,100
extreme values. Imagine another example. Suppose

355
00:25:13,100 --> 00:25:20,060
we have data from 1 to 9. 1, 2, 3, 4, 6, 7, 8, 9.

356
00:25:21,040 --> 00:25:26,690
Now the mean of these values, some divide by n. If

357
00:25:26,690 --> 00:25:31,970
you sum 1 through 9, summation is 45. Divide by 9,

358
00:25:32,510 --> 00:25:36,230
which is 5. So the sum of these values divided by

359
00:25:36,230 --> 00:25:41,590
N gives the average, so the average is 5. Now

360
00:25:41,590 --> 00:25:46,670
suppose we add 100 to the end of this data. So the

361
00:25:46,670 --> 00:25:53,670
sum will be 145 divided by 10, that's 14.5. Now

362
00:25:53,670 --> 00:25:58,850
the mean was 5. Then after we added 100, it

363
00:25:58,850 --> 00:26:05,470
becomes 14.5. Imagine the mean was 5, it changed

364
00:26:05,470 --> 00:26:11,650
to 14.5. It means around three times. So that

365
00:26:11,650 --> 00:26:17,510
means outliers affect the mean much more than the

366
00:26:17,510 --> 00:26:19,890
other one. We'll talk a little later about it,

367
00:26:19,990 --> 00:26:23,950
which is the median. So keep in mind outliers

368
00:26:25,290 --> 00:26:34,790
affected the mean in this case. Any question? Is

369
00:26:34,790 --> 00:26:41,590
it clear? Yes. So, one more time. The mean is

370
00:26:41,590 --> 00:26:46,990
affected by extreme values. So that's for the

371
00:26:46,990 --> 00:26:50,910
mean. The other measure of center tendency is

372
00:26:50,910 --> 00:26:57,600
called the median. Now, what's the median? What's

373
00:26:57,600 --> 00:27:00,760
the definition of the median from your previous

374
00:27:00,760 --> 00:27:05,880
studies? What's the median? I mean, what's the

375
00:27:05,880 --> 00:27:09,360
definition of the median? Now the middle value,

376
00:27:09,760 --> 00:27:12,980
that's correct, but after we arrange the data from

377
00:27:12,980 --> 00:27:17,040
smallest to largest or largest to smallest, so we

378
00:27:17,040 --> 00:27:20,160
should arrange the data, then we can figure out

379
00:27:20,160 --> 00:27:24,280
the median. So the median is the middle point, but

380
00:27:24,280 --> 00:27:27,060
after we arrange the data from smallest to largest

381
00:27:27,060 --> 00:27:30,030
or vice versa. So that's the definition of the

382
00:27:30,030 --> 00:27:33,930
median. So in an ordered array, so we have to have

383
00:27:33,930 --> 00:27:39,230
order array, the median is the middle number. The

384
00:27:39,230 --> 00:27:42,810
middle number means 50 percent of the data below

385
00:27:42,810 --> 00:27:50,370
and 50 percent above the median because it's

386
00:27:50,370 --> 00:27:52,190
called the median, the value in the middle after

387
00:27:52,190 --> 00:27:55,990
you arrange the data from smallest to largest.

388
00:28:00,130 --> 00:28:02,770
Suppose I again go back to the previous example

389
00:28:02,770 --> 00:28:09,690
when we have data 1, 2, 3, 4, and 5. Now for this

390
00:28:09,690 --> 00:28:14,210
specific example as we did before, now the data is

391
00:28:14,210 --> 00:28:18,670
already ordered. The value in the middle is 3

392
00:28:18,670 --> 00:28:22,330
because there are two pillows.

393
00:28:24,860 --> 00:28:27,300
And also there are the same number of observations

394
00:28:27,300 --> 00:28:33,140
above it. So 3 is the median. Now again imagine we

395
00:28:33,140 --> 00:28:37,320
replace 5, which is the maximum value, by another

396
00:28:37,320 --> 00:28:42,140
one which is extreme one, for example 10. In this

397
00:28:42,140 --> 00:28:47,600
case, the median is still 3. Because the median is

398
00:28:47,600 --> 00:28:49,380
just the value of the middle after you arrange the

399
00:28:49,380 --> 00:28:53,900
data. So it doesn't matter what is the highest or

400
00:28:53,900 --> 00:28:58,860
the maximum value is, the median in this case is

401
00:28:58,860 --> 00:29:03,700
three. It doesn't change. That means the median is

402
00:29:03,700 --> 00:29:08,020
not affected by extreme values. Or to be more

403
00:29:08,020 --> 00:29:12,910
precise, we can say that The median is affected by

404
00:29:12,910 --> 00:29:18,990
outlier, but not the same as the mean. So affect

405
00:29:18,990 --> 00:29:23,610
the mean much more than the median. I mean, you

406
00:29:23,610 --> 00:29:26,550
cannot say for this example, yes, the median is

407
00:29:26,550 --> 00:29:29,310
not affected because the median was three, it

408
00:29:29,310 --> 00:29:33,590
becomes three. But in another examples, there is

409
00:29:33,590 --> 00:29:36,750
small difference between all.

410
00:29:40,770 --> 00:29:44,850
Extreme values affected the mean much more than

411
00:29:44,850 --> 00:29:51,450
the median. If the dataset has extreme values, we

412
00:29:51,450 --> 00:29:54,510
have to use, what do you think, the mean or the

413
00:29:54,510 --> 00:29:58,090
median? The median. So in case or in the presence

414
00:29:58,090 --> 00:30:01,910
of extreme values or outliers, we have to use the

415
00:30:01,910 --> 00:30:07,010
median, not the mean. But in general, we use If

416
00:30:07,010 --> 00:30:10,770
the data is free of outliers, I mean if the data

417
00:30:10,770 --> 00:30:16,410
has not extreme values, then you can use the mean.

418
00:30:16,510 --> 00:30:19,230
The mean is much better than the median in this

419
00:30:19,230 --> 00:30:22,490
case. But if the data has extreme values or

420
00:30:22,490 --> 00:30:27,190
outliers, we should use the median instead of the

421
00:30:27,190 --> 00:30:31,310
mean. Any question? So these are the most common

422
00:30:31,310 --> 00:30:36,710
center tendency measures in statistics, the mean

423
00:30:36,710 --> 00:30:42,390
and the median. And keep in mind, your data should

424
00:30:42,390 --> 00:30:46,170
be numeric. I mean, you cannot use the mean or the

425
00:30:46,170 --> 00:30:50,730
median for qualitative or categorical data, for

426
00:30:50,730 --> 00:30:54,310
example, gender, males or females. You cannot say

427
00:30:54,310 --> 00:30:59,490
the mean of gender or sex is 1.5. It doesn't make

428
00:30:59,490 --> 00:31:05,490
sense. It should be numerical data to use the mean

429
00:31:05,490 --> 00:31:07,590
or the median. So the mean and the median is used

430
00:31:07,590 --> 00:31:11,210
only for numerical data. And we have to

431
00:31:11,210 --> 00:31:14,170
distinguish between mean and median. Mean is used

432
00:31:14,170 --> 00:31:16,870
for data that has not outliers or extreme values,

433
00:31:17,370 --> 00:31:21,450
while the median is used for data that has

434
00:31:21,450 --> 00:31:25,230
outliers or extreme values. Sometimes better to

435
00:31:25,230 --> 00:31:27,990
report both. I mean, sometimes better to report

436
00:31:27,990 --> 00:31:33,450
mean and the median. So you just say the sales for

437
00:31:33,450 --> 00:31:40,560
this company is, for example, $500,000. And the

438
00:31:40,560 --> 00:31:43,900
median, for example, is 550,000. You can see that.

439
00:31:45,680 --> 00:31:46,400
Is it clear?

440
00:31:51,440 --> 00:31:55,560
If you have a small data, it's straightforward and

441
00:31:55,560 --> 00:31:59,180
it's very easy to locate the median. But in case

442
00:31:59,180 --> 00:32:02,120
of large dataset, how can we locate the median?

443
00:32:02,340 --> 00:32:06,640
It's not easy. Just look at the data and you can

444
00:32:06,640 --> 00:32:11,200
say this is the median. It's not easy task. So we

445
00:32:11,200 --> 00:32:15,820
need a rule that locate the median. The location

446
00:32:15,820 --> 00:32:18,020
of the median when the values are in numerical

447
00:32:18,020 --> 00:32:23,580
order from smallest to largest is N plus one

448
00:32:23,580 --> 00:32:26,140
divided by two. That's the position of the median.

449
00:32:26,640 --> 00:32:28,860
If we go back a little bit to the previous

450
00:32:28,860 --> 00:32:34,980
example, here N was five. So the location was

451
00:32:34,980 --> 00:32:40,000
number three, because n plus one divided by two,

452
00:32:40,120 --> 00:32:43,120
five plus one divided by two is three. So location

453
00:32:43,120 --> 00:32:47,340
number three is the median. Location number one is

454
00:32:47,340 --> 00:32:50,840
one, in this case, then two, then three. So

455
00:32:50,840 --> 00:32:53,740
location number three is three. But maybe this

456
00:32:53,740 --> 00:32:57,280
number is not three, and other value maybe 3.1 or

457
00:32:57,280 --> 00:33:02,440
3.2. But the location is number three. Is it

458
00:33:02,440 --> 00:33:08,470
clear? So that's the location. If it is odd, you

459
00:33:08,470 --> 00:33:13,270
mean by odd number, five, seven and so on. So if

460
00:33:13,270 --> 00:33:17,090
the number of values is odd, the median is the

461
00:33:17,090 --> 00:33:21,210
middle number. Now let's imagine if we have even

462
00:33:21,210 --> 00:33:24,570
number of observations. For example, we have one,

463
00:33:24,610 --> 00:33:28,270
two, three, four, five and six. So imagine numbers

464
00:33:28,270 --> 00:33:32,390
from one up to six. What's the median? Now three

465
00:33:32,390 --> 00:33:35,610
is not the median because there are two

466
00:33:35,610 --> 00:33:43,390
observations below three. And three above it. And

467
00:33:43,390 --> 00:33:46,210
four is not the median because three observations

468
00:33:46,210 --> 00:33:53,290
below, two above. So three and four is the middle

469
00:33:53,290 --> 00:33:56,870
value. So just take the average of two middle

470
00:33:56,870 --> 00:34:01,570
points, And that will be the median. So if n is

471
00:34:01,570 --> 00:34:07,990
even, you have to locate two middle points. For

472
00:34:07,990 --> 00:34:10,310
example, n over 2, in this case, we have six

473
00:34:10,310 --> 00:34:13,910
observations. So divide by 2, not n plus 1 divided

474
00:34:13,910 --> 00:34:17,970
by 2, just n over 2. So n over 2 is 3. So place

475
00:34:17,970 --> 00:34:22,930
number 3, and the next one, place number 4, these

476
00:34:22,930 --> 00:34:25,930
are the two middle points. Take the average of

477
00:34:25,930 --> 00:34:32,300
these values, then that's your median. So if N is

478
00:34:32,300 --> 00:34:37,080
even, you have to be careful. You have to find two

479
00:34:37,080 --> 00:34:40,860
middle points and just take the average of these

480
00:34:40,860 --> 00:34:45,100
two. So if N is even, the median is the average of

481
00:34:45,100 --> 00:34:49,200
the two middle numbers. Keep in mind, when we are

482
00:34:49,200 --> 00:34:54,600
saying N plus 2, N plus 2 is just the position of

483
00:34:54,600 --> 00:34:58,670
the median, not the value, location. Not the

484
00:34:58,670 --> 00:35:07,770
value. Is it clear? Any question? So location is

485
00:35:07,770 --> 00:35:10,150
not the value. Location is just the place or the

486
00:35:10,150 --> 00:35:13,450
position of the medium. If N is odd, the position

487
00:35:13,450 --> 00:35:17,710
is N plus one divided by two. If N is even, the

488
00:35:17,710 --> 00:35:20,870
positions of the two middle points are N over two

489
00:35:20,870 --> 00:35:23,090
and the next term or the next point.

490
00:35:28,390 --> 00:35:32,510
Last measure of center tendency is called the

491
00:35:32,510 --> 00:35:32,750
mood.

492
00:35:35,890 --> 00:35:39,010
The definition of the mood, the mood is the most

493
00:35:39,010 --> 00:35:44,250
frequent value. So sometimes the mood exists,

494
00:35:45,230 --> 00:35:48,570
sometimes the mood does not exist. Or sometimes

495
00:35:48,570 --> 00:35:53,730
there is only one mood, in other cases maybe there

496
00:35:53,730 --> 00:35:58,730
are several moods. So a value that occurs most

497
00:35:58,730 --> 00:36:03,010
often is called the mood. The mood is not affected

498
00:36:03,010 --> 00:36:07,610
by extreme values. It can be used for either

499
00:36:07,610 --> 00:36:11,190
numerical or categorical data. And that's the

500
00:36:11,190 --> 00:36:13,910
difference between mean and median and the mood.

501
00:36:14,590 --> 00:36:16,930
Mean and median is used just for numerical data.

502
00:36:17,430 --> 00:36:21,270
Here, the mood can be used for both, categorical

503
00:36:21,270 --> 00:36:25,610
and numerical data. Sometimes, as I mentioned,

504
00:36:25,930 --> 00:36:29,570
there may be no mood or the mood does not exist.

505
00:36:30,130 --> 00:36:34,190
In other cases, there may be several events. So

506
00:36:34,190 --> 00:36:36,870
the mood is the value that has the most frequent.

507
00:36:37,490 --> 00:36:43,650
For example, if you look at this data, one is

508
00:36:43,650 --> 00:36:48,370
repeated once, three is the same one time, five is

509
00:36:48,370 --> 00:36:52,290
repeated twice. seven is one nine is repeated

510
00:36:52,290 --> 00:36:57,330
three times and so on so in this case nine is the

511
00:36:57,330 --> 00:37:00,290
mood because the mood again is the most frequent

512
00:37:00,290 --> 00:37:05,030
value on

513
00:37:05,030 --> 00:37:08,550
the right side there are some values zero one two

514
00:37:08,550 --> 00:37:12,830
three up to six now each one is repeated once so

515
00:37:12,830 --> 00:37:15,350
in this case the mood does not exist I mean there

516
00:37:15,350 --> 00:37:22,310
is no mood So generally speaking, the mood is the

517
00:37:22,310 --> 00:37:26,310
value that you care most often. It can be used for

518
00:37:26,310 --> 00:37:29,790
numerical or categorical data, not affected by

519
00:37:29,790 --> 00:37:32,970
extreme values or outliers. Sometimes there is

520
00:37:32,970 --> 00:37:36,150
only one mood as this example. Sometimes the mood

521
00:37:36,150 --> 00:37:40,390
does not exist. Or sometimes there are several

522
00:37:40,390 --> 00:37:45,190
moods. And so that's the definitions for mean,

523
00:37:46,430 --> 00:37:52,540
median, and the mood. I will give just a numerical

524
00:37:52,540 --> 00:37:56,380
example to know how can we compute these measures.

525
00:37:57,420 --> 00:38:01,540
This data, simple data, just for illustration, we

526
00:38:01,540 --> 00:38:07,580
have house prices. We have five data points, $2

527
00:38:07,580 --> 00:38:10,940
million. This is the price of house A, for

528
00:38:10,940 --> 00:38:15,880
example. House B price is 500,000. The other one

529
00:38:15,880 --> 00:38:19,120
is 300,000. And two houses have the same price as

530
00:38:19,120 --> 00:38:25,850
100,000. Now, just to compute the mean, add these

531
00:38:25,850 --> 00:38:29,350
values or sum these values, which is three

532
00:38:29,350 --> 00:38:34,030
million, divide by number of houses here, there

533
00:38:34,030 --> 00:38:38,550
are five houses, so just three thousand divided by

534
00:38:38,550 --> 00:38:44,170
five, six hundred thousand. The median, the value

535
00:38:44,170 --> 00:38:46,150
in the median, after you arrange the data from

536
00:38:46,150 --> 00:38:51,470
smallest to largest, Or largest smallest. This

537
00:38:51,470 --> 00:38:55,410
data is already arranged from largest smallest or

538
00:38:55,410 --> 00:38:58,150
smallest large. It doesn't matter actually. So the

539
00:38:58,150 --> 00:39:02,930
median is $300,000. Make sense? Because there are

540
00:39:02,930 --> 00:39:09,490
two house prices above and two below. So the

541
00:39:09,490 --> 00:39:13,610
median is $300,000. Now if you look at these two

542
00:39:13,610 --> 00:39:21,350
values, the mean for this data equals 600,000 and

543
00:39:21,350 --> 00:39:26,690
the median is 300,000. The mean is double the

544
00:39:26,690 --> 00:39:31,750
median. Do you think why there is a big difference

545
00:39:31,750 --> 00:39:36,030
in this data between the mean and the median?

546
00:39:36,190 --> 00:39:42,290
Which one? Two million dollars is extreme value,

547
00:39:42,510 --> 00:39:45,940
very large number. I mean, if you compare two

548
00:39:45,940 --> 00:39:48,860
million dollars with the other data sets or other

549
00:39:48,860 --> 00:39:51,320
data values, you will see there is a big

550
00:39:51,320 --> 00:39:53,260
difference between two million and five hundred.

551
00:39:53,620 --> 00:39:56,280
It's four times, plus about three hundred

552
00:39:56,280 --> 00:39:59,780
thousands, around seven times and so on. For this

553
00:39:59,780 --> 00:40:07,880
value, the mean is affected. Exactly. The median

554
00:40:07,880 --> 00:40:11,740
is resistant to outliers. It's affected but little

555
00:40:11,740 --> 00:40:17,100
bit. For this reason, we have to use the median.

556
00:40:17,300 --> 00:40:20,720
So the median makes more sense than using the

557
00:40:20,720 --> 00:40:24,480
mean. The mode is just the most frequent value,

558
00:40:24,660 --> 00:40:28,720
which is 100,000, because this value is repeated

559
00:40:28,720 --> 00:40:33,820
twice. So that's the whole story for central

560
00:40:33,820 --> 00:40:40,720
tendency measures, mean, median, and 1. Now the

561
00:40:40,720 --> 00:40:45,640
question again is which measure to use? The mean

562
00:40:45,640 --> 00:40:49,280
is generally used. The most common center tendency

563
00:40:49,280 --> 00:40:53,420
is the mean. We can use it or we should use it

564
00:40:53,420 --> 00:40:59,920
unless extreme values exist. I mean if the data

565
00:40:59,920 --> 00:41:03,960
set has no outliers or extreme values, we have to

566
00:41:03,960 --> 00:41:06,240
use the mean instead of the median.

567
00:41:09,810 --> 00:41:14,670
The median is often used since the median is not

568
00:41:14,670 --> 00:41:18,330
sensitive to extreme values. I mean, the median is

569
00:41:18,330 --> 00:41:22,030
resistant to outliers. It remains nearly in the

570
00:41:22,030 --> 00:41:26,490
same position if the dataset has outliers. But the

571
00:41:26,490 --> 00:41:29,850
median will be affected either to the right or to

572
00:41:29,850 --> 00:41:34,350
the left tail. So we have to use the median if the

573
00:41:34,350 --> 00:41:40,060
data has extreme values. For example, median home

574
00:41:40,060 --> 00:41:44,100
prices for the previous one may be reported for a

575
00:41:44,100 --> 00:41:48,000
region that is less sensitive to outliers. So the

576
00:41:48,000 --> 00:41:52,880
mean is more sensitive to outliers than the

577
00:41:52,880 --> 00:41:56,520
median. Sometimes, I mean in some situations, it

578
00:41:56,520 --> 00:41:58,760
makes sense to report both the mean and the

579
00:41:58,760 --> 00:42:01,860
median. Just say the mean for this data for home

580
00:42:01,860 --> 00:42:07,570
prices is 600,000 while the median is 300,000. If

581
00:42:07,570 --> 00:42:10,150
you look at these two figures, you can tell that

582
00:42:10,150 --> 00:42:13,830
there exists outlier or the outlier exists because

583
00:42:13,830 --> 00:42:17,230
there is a big difference between the mean and the

584
00:42:17,230 --> 00:42:24,310
median. So that's all for measures of central

585
00:42:24,310 --> 00:42:28,830
tendency. Again, we explained three measures,

586
00:42:29,450 --> 00:42:33,930
arithmetic mean, median, and mode. And arithmetic

587
00:42:33,930 --> 00:42:38,990
mean again is denoted by X bar is pronounced as X

588
00:42:38,990 --> 00:42:44,410
bar and just summation of X divided by N. So

589
00:42:44,410 --> 00:42:48,070
summation Xi, i goes from 1 up to N divided by the

590
00:42:48,070 --> 00:42:52,170
total number of observations. The median, as we

591
00:42:52,170 --> 00:42:55,690
mentioned, is the value in the middle in ordered

592
00:42:55,690 --> 00:42:59,150
array. After you arrange the data from smallest to

593
00:42:59,150 --> 00:43:01,930
largest or vice versa, then the median is the

594
00:43:01,930 --> 00:43:06,330
value in the middle. The mode is the most frequent

595
00:43:06,330 --> 00:43:09,030
observed value. And we have to know that mean and

596
00:43:09,030 --> 00:43:13,870
median are used only for numerical data, while the

597
00:43:13,870 --> 00:43:17,510
mode can be used for both numerical and

598
00:43:17,510 --> 00:43:24,290
categorical data. That's all about measures of

599
00:43:24,290 --> 00:43:27,210
central tendency. Any question?

600
00:43:33,210 --> 00:43:40,230
Let's move to measures of variation. It's another

601
00:43:40,230 --> 00:43:43,750
type of measures. It's called measures of

602
00:43:43,750 --> 00:43:47,490
variation, sometimes called measures of spread.

603
00:43:50,490 --> 00:43:53,850
Now, variation can be computed by using range,

604
00:43:55,590 --> 00:44:00,850
variance, standard deviation, and coefficient of

605
00:44:00,850 --> 00:44:08,430
variation. So we have four types, range, variance,

606
00:44:09,250 --> 00:44:12,050
standard deviation, and coefficient of variation.

607
00:44:13,710 --> 00:44:16,150
Now, measures of variation give information on the

608
00:44:16,150 --> 00:44:19,410
spread. Now, this is the first difference between

609
00:44:19,410 --> 00:44:24,210
central tendency measures and measures of

610
00:44:24,210 --> 00:44:28,270
variation. That one measures the central value or

611
00:44:28,270 --> 00:44:30,790
the value in the middle. Here, it measures the

612
00:44:30,790 --> 00:44:36,310
spread. Or variability. Or dispersion of the data.

613
00:44:36,450 --> 00:44:40,310
Do you know what is dispersion? Dispersion.

614
00:44:40,630 --> 00:44:45,590
Tabaad. So major variation given formation with

615
00:44:45,590 --> 00:44:48,350
the spread. Spread or variation or dispersion of

616
00:44:48,350 --> 00:44:52,250
the data values. Now if you look at these two bell

617
00:44:52,250 --> 00:44:52,650
shapes.

618
00:44:55,670 --> 00:44:59,170
Both have the same center. The center I mean the

619
00:44:59,170 --> 00:45:01,730
value in the middle. So the value in the middle

620
00:45:01,730 --> 00:45:06,990
here for figure

621
00:45:06,990 --> 00:45:10,150
graph number one is the same as the value for the

622
00:45:10,150 --> 00:45:16,270
other graph. So both graphs have the same center.

623
00:45:17,430 --> 00:45:20,670
But if you look at the spread, you will see that

624
00:45:20,670 --> 00:45:26,230
figure A is less spread than figure B. Now if you

625
00:45:26,230 --> 00:45:29,720
look at this one, the spread here, is much less

626
00:45:29,720 --> 00:45:34,120
than the other one. Even they have the same

627
00:45:34,120 --> 00:45:39,260
center, the same mean, but figure A is more spread

628
00:45:39,260 --> 00:45:45,140
than figure B. It means that the variation in A is

629
00:45:45,140 --> 00:45:49,920
much less than the variation in figure B. So it

630
00:45:49,920 --> 00:45:55,960
means that the mean is not sufficient to describe

631
00:45:55,960 --> 00:45:59,970
your data. Because maybe you have two datasets and

632
00:45:59,970 --> 00:46:03,330
both have the same mean, but the spread or the

633
00:46:03,330 --> 00:46:07,350
variation is completely different. Again, maybe we

634
00:46:07,350 --> 00:46:10,250
have two classes of statistics, class A and class

635
00:46:10,250 --> 00:46:13,230
B. The center or the mean or the average is the

636
00:46:13,230 --> 00:46:16,150
same for each one. For example, maybe the average

637
00:46:16,150 --> 00:46:19,810
of this class is 70. The average of class B is

638
00:46:19,810 --> 00:46:26,640
also 70. But the scores are scattered. or spread

639
00:46:26,640 --> 00:46:32,580
out in class A maybe much more than in class B. So

640
00:46:32,580 --> 00:46:34,280
the mean is not sufficient to describe the data.

641
00:46:34,360 --> 00:46:37,100
You have to say that the mean equals such and such

642
00:46:37,100 --> 00:46:41,000
and the spread. And one of these measures we'll

643
00:46:41,000 --> 00:46:44,500
talk later about range and variance standard

644
00:46:44,500 --> 00:46:49,030
deviation. So I mean, The mean by itself is not

645
00:46:49,030 --> 00:46:51,890
sufficient to describe the data. You have to use

646
00:46:51,890 --> 00:46:55,730
something else to measure the variation or the

647
00:46:55,730 --> 00:46:57,950
spread of the data. Make sense?

648
00:47:02,170 --> 00:47:05,670
The first measure of variation, the easiest one,

649
00:47:05,810 --> 00:47:11,230
is called the range. The range is the simplest

650
00:47:11,230 --> 00:47:15,590
measure of variation. The range is just the

651
00:47:15,590 --> 00:47:19,750
difference or the distance between the largest and

652
00:47:19,750 --> 00:47:23,550
the smallest value. For example, suppose the

653
00:47:23,550 --> 00:47:27,070
minimum score for this class is 40 and the maximum

654
00:47:27,070 --> 00:47:33,230
is 90. So the range is 50, 90 minus 40. Now

655
00:47:33,230 --> 00:47:38,850
imagine that the minimum score for this class is

656
00:47:38,850 --> 00:47:47,330
60 and the maximum is 80, so 20. If we replace 80

657
00:47:47,330 --> 00:47:51,450
by 100, I mean the minimum is 60 and the maximum

658
00:47:51,450 --> 00:47:57,030
is 100, it's 40. That means a range is affected by

659
00:47:57,030 --> 00:48:02,170
outliers because it depends only on two values.

660
00:48:03,480 --> 00:48:06,100
maximum and minimum value. So it should be

661
00:48:06,100 --> 00:48:09,320
affected by outliers. So range is sensitive to

662
00:48:09,320 --> 00:48:12,780
outliers. So if the data has the data set has

663
00:48:12,780 --> 00:48:15,660
outliers, then in this case, you have to avoid

664
00:48:15,660 --> 00:48:19,640
using range because range only based on two

665
00:48:19,640 --> 00:48:23,480
values. So it should be affected by outliers. Now

666
00:48:23,480 --> 00:48:25,880
for the for simple example, suppose we have this

667
00:48:25,880 --> 00:48:32,360
data. The minimum value is one. I mean, the

668
00:48:32,360 --> 00:48:34,680
smallest value is one, and the largest or the

669
00:48:34,680 --> 00:48:38,880
maximum is 13. So it makes sense that the range of

670
00:48:38,880 --> 00:48:41,840
the data is the difference between these two

671
00:48:41,840 --> 00:48:48,540
values. So 13 minus one is 12. Now, imagine that

672
00:48:48,540 --> 00:48:58,040
we just replace 13 by 100. So the new range will

673
00:48:58,040 --> 00:49:03,820
be equal to 100 minus 199. So the previous range

674
00:49:03,820 --> 00:49:08,340
was 12. It becomes now 99 after we replace the

675
00:49:08,340 --> 00:49:12,100
maximum by 100. So it means that range is affected

676
00:49:12,100 --> 00:49:18,740
by extreme values. So the mean and range both are

677
00:49:18,740 --> 00:49:23,040
sensitive to outliers. So you have to link between

678
00:49:26,410 --> 00:49:30,210
measures of center tendency and measures of

679
00:49:30,210 --> 00:49:33,130
variation. Mean and range are affected by

680
00:49:33,130 --> 00:49:37,910
outliers. The mean and range are affected by

681
00:49:37,910 --> 00:49:41,450
outliers. This is an example. So it's very easy to

682
00:49:41,450 --> 00:49:49,550
compute the mean. Next, if you look at why the

683
00:49:49,550 --> 00:49:51,190
range can be misleading.

684
00:49:53,830 --> 00:49:56,810
Sometimes you report the range and the range does

685
00:49:56,810 --> 00:50:00,310
not give an appropriate answer or appropriate

686
00:50:00,310 --> 00:50:04,450
result because number

687
00:50:04,450 --> 00:50:06,790
one ignores the way in which the data are

688
00:50:06,790 --> 00:50:10,770
distributed. For example, if you look at this

689
00:50:10,770 --> 00:50:15,430
specific data, we have data seven, eight, nine,

690
00:50:15,590 --> 00:50:18,110
ten, eleven and twelve. So the range is five.

691
00:50:19,270 --> 00:50:21,910
Twelve minus seven is five. Now if you look at the

692
00:50:21,910 --> 00:50:26,360
other data, The smallest value was seven.

693
00:50:29,600 --> 00:50:33,260
And there is a gap between the smallest and the

694
00:50:33,260 --> 00:50:38,220
next smallest value, which is 10. And also we have

695
00:50:38,220 --> 00:50:44,480
12 is repeated three times. Still the range is the

696
00:50:44,480 --> 00:50:48,140
same. Even there is a difference between these two

697
00:50:48,140 --> 00:50:53,640
values, between two sets. we have seven, eight,

698
00:50:53,760 --> 00:50:57,020
nine up to 12. And then the other data, we have

699
00:50:57,020 --> 00:51:02,180
seven, 10, 11, and 12 three times. Still, the

700
00:51:02,180 --> 00:51:06,360
range equals five. So it doesn't make sense to

701
00:51:06,360 --> 00:51:09,620
report the range as a measure of variation.

702
00:51:10,520 --> 00:51:12,640
Because if you look at the distribution for this

703
00:51:12,640 --> 00:51:15,500
data, it's completely different from the other

704
00:51:15,500 --> 00:51:20,860
dataset. Even though it has the same range. So

705
00:51:20,860 --> 00:51:25,220
range is not used in this case. Look at another

706
00:51:25,220 --> 00:51:25,680
example.

707
00:51:28,300 --> 00:51:32,920
We have data. All the data ranges, I mean, starts

708
00:51:32,920 --> 00:51:38,680
from 1 up to 5. So the range is 4. If we just

709
00:51:38,680 --> 00:51:46,200
replace the maximum, which is 5, by 120. So the

710
00:51:46,200 --> 00:51:49,190
range is completely different. the range becomes

711
00:51:49,190 --> 00:51:55,010
119. So that means range

712
00:51:55,010 --> 00:51:59,230
is sensitive to outliers. So we have to avoid

713
00:51:59,230 --> 00:52:06,030
using range in case of outliers or extreme values.

714
00:52:08,930 --> 00:52:14,410
I will stop at the most important one, the

715
00:52:14,410 --> 00:52:18,350
variance, for next time inshallah. Up to this

716
00:52:18,350 --> 00:52:19,310
point, any questions?

717
00:52:22,330 --> 00:52:29,730
Okay, stop at this point if

718
00:52:29,730 --> 00:52:30,510
you have any question.

719
00:52:35,430 --> 00:52:39,430
So later we'll discuss measures of variation and

720
00:52:39,430 --> 00:52:44,810
variance, standard deviation up to the end of this

721
00:52:44,810 --> 00:52:45,090
chapter.

722
00:52:54,630 --> 00:53:00,690
So again, the range is sensitive to outliers. So

723
00:53:00,690 --> 00:53:03,850
we have to avoid using range in this case. And

724
00:53:03,850 --> 00:53:06,270
later we'll talk about the variance, which is the

725
00:53:06,270 --> 00:53:09,750
most common measures of variation for next time,

726
00:53:09,830 --> 00:53:10,130
inshallah.