 Quattals are also measures of variation because they also tell us how spread your data is and where is located as well. So with Quattals, it splits your data into four quotas, four groups of data or four segments of 25% each. Different books use different methods of finding the Quattals as well. So you need to be very careful when you use the books because then also when I explain the Quattals here because I'm assuming the method that I'm sharing with you might be simple for me but you might prefer to use the percentile. You can also use the percentile method to calculate the Quattals. As long as you know that for the first Quattal, it is 25%. For the third Quattal, it's 75% and the median is 50% or it's 50%. So you just need to know those things. So you know that we divide the data into the four segments. So with Quattal 1, Quattal 2, Quattal 3, where first Quattal accounts for 25% of the data with more than 75 are larger than that value and the Quattal 2, which is the same as your median, accounts for 50% are below 50% are up. The third Quattal accounts for 25% are larger values and 75% of the values are smaller than that Quattal. How we find the Quattals, we use the position. Like we did with the median, we use the position but it's different. With the Quattals, to find the first Quattal or to find the Quattal, we need to define or find the position. Then use the position to go find the value. So Quattal 1, we use n plus 1 divided by 4 because there are four Quattals. So n plus 1 divided by 4 to find the position of Quattal 1. And remember also with the Quattals, you need to order your data or sort your data from lowest to highest. Where your n is your sample size, how many there are. So you sort your data, you calculate your first Quattal, which you find the position of the first Quattal. The value on that position, it is your Quattal value, your Quattal value. If they ask you to find the position, you use n plus 1 divided by 4. If they ask you to find the Quattal value, you use the position to go find the value. To find the second Quattal is the same as the median. I'm not going to repeat that because we did the median. You know how to find the position of the median and then finding the median. Finding the third Quattal, we use 3 times n plus 1 divided by 4 because we're looking at the third Quattal. When you calculate or when you find the position, there are rules because when we're dividing by 4, therefore it means we're not going to get a whole number or we might get a whole number or we might not get a whole number, we might get a Quattal or a fraction. So there are rules that you need to take into consideration when you when you calculate in the position. The first one is when you calculate the position and the result is a whole number, then it's easy. The value of that position, it is your Quattal. So if I had 1, 2, 3, 4, 5 and I was calculating Quattal 1 and the position was 2 after I did Quattal 1 is equals to n plus 1 divided by 4. So there are 5 plus 1 divided by 4, which is 6 divided by 4 and it's not going to give me a whole number. I used the wrong, okay, let's not use the calculation because then it's not going to give me the whole number. So let's assume that we calculated the Quattal 1 position by using n plus 1 divided by 4 and we find that the position is on the second position, then that I just say 1, 2, this is my Quattal 1 value and that is the whole number. Then the answer is a fractional value. So like I calculated Quattal 1 and I use n plus 1 divided by 4 and I find the answer to be 2.5. So like we did it with the other one, 3, 4, 5. So if it's 2.5, it's located between the two values. So I must take the average of the two values. So it means it's 2 plus 3 divided by 2, which will give me 2.5 as my Quattal value. So if I change this to 3, so it will have been different because you're going to get confused and say, but it is 2.5, why did we have to calculate 2.5? Remember if my values are not that one. So if these are my values and it's 2.5, so I must just say where is, then I must say 3 plus 3 divided by 2, which also still another example that does not make any difference. So you're still going to get the answer is 3. But if this value here is not 3, let's say it's 4, I'm just going to increase my value so that it makes doesn't end up being. So let's say this is 5 and this is 8 and this is 7, let's say it's like that. Then we say it falls, our position is on the, it's 2.5. So 1, 2.5 is located between those two values. Then we say 3 plus 5 divided by 2, which gives me 8 divided by 2, which means my position, my middle position is 4. And that's how you will use the fractional half. You're not going to get a whole number or a fractional half, but you will get a non-fractional value. What I mean, if it says quarter one, you calculated it and you find the answer is 2.25. So I have 1, 3, 5, 7, and 9. The answer here it says my position is at 2.25. So since my position is at 2.25, so I can go 1, 2.25 can be somewhere closer to there. So since my position is closer to 2, then I can just estimate this to be 2. Then my quarter value will be 3. We just round down. If the answer was 2.75, if it was 2.75, then 2.75 would have been 1, 2.75 would have been somewhere here, which then tells me it is closer to 3. Therefore, we can just round up and estimate that our position is at position 3. Then we count 1, 2, 3, then our quarter one value is 5. The position is still 2.75, it's not 3, it's 2.75. But we estimate that our quarter value, we will find it at position 3. So we just go to position 3 and find the value. That's how you calculate the quarter house. In the next 15 minutes, let's calculate the quarter house. Quarter house, from the data set that we have, we have 11, 12, 13 up until 22. There are nine observations. We can first find the quarter one position by calculating the quarter one position. N is 9 plus 1 divided by 4. The quarter one position is at 2.5, and then we go count. 1, 2.5 is located between 12 and 13. And then since it's located between the two values, we take an average of the two values. Finding, we already did quarter one, so let's look at quarter two. Finding quarter two position, remember the quarter two position, we use N plus 1 divided by 2 because it's the same as the median. So our N is 9 plus 1 divided by 2, it's on position 5. 1, 2, 3, 4, 5, our value is 16. So our quarter two, which is the same as our median, is 16. Finding quarter three, we use the formula 3 times N plus 1 divided by 4. 3 times N, our N is 9 plus 1 divided by 4, 9 plus 1 is 10, 10 times 3, 30 divided by 4, we get 7.5. 1, 2, 3, 4, 5, 6, 7.5, it's between 18 and 21. Therefore, we take the average, which is 18 plus 21 divided by 2, which gives us the value of quarter three as 19.5. Quarter one and quarter three are not measures of central location, as we don't discuss them in measures of central location. And quarter two, it's part of the measures of central location because it is the same as your median. When we have the quarter, we can calculate the interquartal range. When we calculate the interquartal range, we do not use the position, but we use the values. So interquartal range, we use the value of quarter three minus the value of quarter one. Remember, we take the smallest quarter and the highest quarter. The quarter three is the highest one and quarter one is the smallest one. So it will be high minus low. So quarter three minus quarter one value, and it will tell us the spread of your data. Same as with the range, it's a measure of variability because also with the quarter or interquartal is not affected by outliers or extreme values. And also it includes also not only the interquartal, but also quarter one and quarter three. And we call those resistant measures because they don't get affected by the outliers. So how do we calculate the quarter or interquartal? Remember, we did calculate quarter one and quarter two values. We found that quarter one value was 12.5 and quarter three value was 19.5. Interquartal range, 19.5 minus 12.5 gives us seven and that is our quarter. I'm not going to ask you to do this exercise. We can do it on Saturday. I want to wrap up the section. You can come back and watch the video before Saturday so that then you are away. So this question will be in on Saturday. So when we have the interquartal range. Remember, we can create what we call the five number summary. And our five number summary, describe the center, the spread, and the shape of your data. And we will use a box plot after because with our five number summary, we have the smallest value. We have quarter one value. We have quarter two value and we have quarter three value. And also we have the highest value. And all of them, they create a five number summary. And when we plot the box plot, all these values are clearly identified on the box plot. And they will show you the spread of the data. So let's get this example that we have here. If we have this data set, we are able to create the minimum value, which is our smallest value. And we have our maximum value where we have our whiskers. So the whiskers of a box plot gives us the minimum value and the maximum value. Sometimes box plot don't have the whisker. But generally we do create the whiskers on the box plot. The box plot has a box. That's where the name comes from. This box is defined by the quartiles. The start of the quartile, which is quartile one, starts the box from the lowest of, from minimum to the start of the box. The middle of the box gives us the quartile two value. And the end of the box gives us the quartile three, which also includes the 25% of the data from maximum. To calculate interquartile range, we take your quartile three value and your quartile one value. Sorry, my data also on this one is stretched out. So it's 70 cross points with this. So I didn't check when I did this. So this is 45 and this is 80 and that is 12. So quartile three is 57. Quartile one is 57 minus 30 gives us the interquartile range. 45 is our median, which is our quartile two. And 12 is our minimum. That is our quartile one and 57 is our quartile three and 70 is our maximum. And this is what we call a box plot or a box, box risk a plot. And it also gives us the five number summaries. We can use these measures to describe the shape of your data. So if we look at the quartiles, the median, and the maximum values and the minimum values, we can also tell how the data is spread. Shape of the data. Left scoot, the mean minus the smallest value. So if we take the mean, so if I draw the box, I'm just going to draw it here, I'm just going to draw there. So remember, this is the smallest value. This is quartile one. This is quartile two. This is quartile three. And this is the maximum, the highest. So left scoot says the median minus the smallest value. So you take the median value and minus the smallest value. If it is more than, if it's bigger, then your largest value, which will be the highest value minus the median. So you take the highest value minus the median. If it's the median of this, the difference between the median and the smallest value, if it's bigger than the largest value or the difference of the largest value in the median, the data is skewed. The same, quartile one, the difference between quartile one and the smallest value. If it's bigger than the difference between quartile three and the highest value, this left scoot, you can just go and understand the whole graph, no, the whole table. For symmetric, the median, which is quartile, remember, quartile two is the median. So the median, which is quartile two, minus the smallest value. If it's the same as your largest value minus the median, then your data is symmetric. Quartile one minus the smallest value. If it's the same as quartile or largest value minus quartile three, then your data is symmetric. Same will apply to your data being a right skewed. By looking at the difference between the median and the smallest value, if it's less than, then your data is left skewed. And with that, I'm not going to ask you to do any calculations. I'm going to wrap up. You have learned how to describe the properties of central tendency, which are the mean, the median, the mode, and how to do the calculation of finding them. You've learned how to describe the properties of the variation, which is the range, the highest minus the smallest, the variance, which is the sum of your values minus your mean squared divided by n minus one for the sample variance, and divide by n for the population variance. You also learned the standard deviation, and we did some examples of how to calculate the standard deviation, which is the square root of your, your standard deviation is the square root of your variance, and it tells you how far apart your data is from the mean, or how variable are your data from the mean. We also looked at the coefficient of variation, which gives you the relative variation, the relative variation, which is your standard deviation divided by the sample mean multiplied by 100, and we always represent it as a percentage. And we did an example and an exercise on that. We also looked at the quartiles, which gives you the distribution of the data in terms of the number summary. And remember to find in the quartile, we first need to find the position, and remember the rules. If it is a whole number at that position, that's where you find the quarter. At a fractional half, you take the average of the two values that the position is located. The non-fractional half, which is 0.25, you round down your value to the nearest value. If it's 0.75, you round up to the closest upper value of that position. We also looked at the five number summary, which are made up of smallest value, quartile 1, quartile 2, quartile 3, and highest value. And we know that quartile 2 is the same as the median. We also learned how to calculate the interquartile range, which is the highest quartile minus the smallest quartile, which is quartile 3 value, not the position, quartile 3 value minus quartile 1 value. And I also highlighted that some books, prescribed books, they do not use the quartiles, but they use the percentage, which is the same, because 1 divided by 4 gives you 25 percentile. So your 25 percentile will give you the position of your first quartile. And if you want to find the 75 percentile, you will get quartile number 3. If you want 50 percentile, you will get the value of quartile 3 or the median. So depending on which prescribed book you use, you need to understand how to get the position and the values of the quartiles. And we constructed a box plot and we learned the distribution of the measures of variation or the measures using the quartiles. With that, it concludes today's session. Any question? Ma'am, I think I left to ask on WhatsApp after catching up. It seems like the visuals were delayed. My network was acting up as well. Okay, you can also re-watch the video and then ask any question you have. The video should be up today or tonight or early in the morning. Let me just not promise, but tomorrow the video should be up on my Unisa and also on Teams for those who have data. You can watch them on Teams or you can watch them on my Unisa. Will you also have time? Maybe I'm not sure exactly when I'll be able to post the answers for those questions from Saturday. Will you at least have time to check maybe? I'm always on WhatsApp. My WhatsApp is 24-7. Okay. Yes. Okay, so if there is no content-related questions, I'm going to stop the recording and then I will hang around for any question you have. So for now, let me stop the recording.