Description
Discussion Board: Interpreting & Analyzing Central Tendencies
20 points
About: How many hours per day do people have to relax? With our busy lives, it is often hard to find time for ourselves, to unwind and do something relaxing.
In this discussion board, we will be using real data to examine the question: How much time do people in the United States have to relax?
Warning: We will be applying the concepts of central tendency in this discussion board. If you have not read and taken notes on our chapters 3 and lecture videos make sure to do so before starting this assignment. Click here to look at our video lesson (once finished, click next to access the lecture notes)
Background & Results:
These results come from a national survey called the General Social Survey. They provide us with an insight on American habits and behaviors. People from across the U.S. of all backgrounds, ages, social class, and genders were asked several questions including the following question:
After an average work day, about how many hours do you have to relax or pursue activities that you enjoy? Here are the results!
Table 1: Statistics for Variable: Hours to Relax Per Day
Variable
N
(Sample Size)
Mean Median Mode
Hours to Relax Per Day
1,242
3.92 hours
3.0 hours
2 hours
Directions:
For your original post in paragraph format write a minimum of 200 words and provide answers and statements to the following prompts:
(Make sure to answer the questions in paragraph format)
Examine the results for central tendency and interpret the findings. On average how many hours did people have to relax? Use the mean, median, and the mode to answer this question
Interpret the findings for each the mode, median, and the mean
How many hours a day do you have to relax? Are you above or below the average?
Lastly, what additional variables can influence the variable: hours relax per day?
Unformatted Attachment Preview
In this lesson, we cover measures of central tendency, our learning objectives are to define all of the measures of central tendency, as well as explain the relative strengths and weaknesses. We also want to be able to determine the criteria for selecting a measure of central tendency. On top of that, we want to find or calculate the mean median mode and percentiles of a given distribution. Lastly, we want to be able to tell the shape of a distribution. The first measure of central tendency we’ll talk about is the moat, keep in mind that one of the things that measures of central tendency all have in common is that they basically tell us what is typical or what is common about the set of data that we’re looking at. So the way that we define the mode is the most typical or most common slash popular category that we can find in the data. In this example, we’re looking at the opinions of Americans and how they feel about the importance of the institution of marriage to societies will be. What we can glean from it is that the mode is very important. That was the most popular answer. In fact, sixty eight point six percent of the respondents said that marriage is something really important. And what this tells us about the data is that Americans feel strongly about the importance of the institution of marriage. There are cases when we have a bimodal distribution and this occurs when there are more than one mode’s. So let’s say, for instance, both very important and not important. Both had 412 frequencies. That would mean we have a bimodal distribution and that would tell us that Americans are split about how they feel about marriage. So we learned how to locate the mode by looking at a frequency distribution table. It’s really quite simple as well to find the mode from a raw data set. So that’s what we’ll do now. So we just want to count up all of the instances of males here. There are four males and there are. Five female, so we would say that her mode is. FENA.
Fairly simple. This is the formula for the mean where Y bar is the symbol for the new.
And this is Sigma.
It basically is telling us to. Add something, it stands for some of and it tells you to add whatever it falls that.
OK, so with that said, what the formula is telling us is to add up all of the whys, which is all of our scores and divided by and our sample size. Let’s look at an example.
War equals the sum of Y divided by and that’s our formula. And so what we would do is add up all of these numbers and then divide by an our end is our sample size. So in this case, there are one, two, three, four, five, six, seven, seven respondents.
After adding up these numbers, we end up with 23 hour and seven. So therefore, 23 divided by seven is roughly around three point three per lebar. That’s three point three.
So sometimes we’ll be dealing with the data set and there will appear to be some outliers, basically an outlier is a piece of data that stands out, far out and away from the rest of the data. Let’s look at an example. If you notice the first eight numbers and this data set roughly flowed around 35 years of age.
It’s not until these last two digits that we come across some outliers, these outliers will do something that we refer to as skew the calculation of the need. For instance, if we were to calculate the mean of this data set, we would end up with forty six point five is forty six point five is not representative of most of the numbers in our data set. It’s much higher than most of these numbers here. So one thing that we can do to see how this works is to calculate the average without the the outliers. And you’ll see that the average for just these eight data points here is much lower than the average with the outliers. When meat is brought down, it’s because we have a negative skew. And when the mean is brought up by extreme scores, it’s because we have a positively skewed distribution figure, shows us a perfectly symmetrical distribution in a perfectly symmetrical distribution. The mean median in the mode all coincide in the middle. This is a quality of measures of central tendency that when we have a perfectly symmetrical distribution, all of the measures of central tendency lie right in the middle. A negatively skewed distribution happens when you have some really low outliers.
Meaning you have a set of scores and then you have some scores that are extremely low, far and below what’s typical of the distribution. If you notice what happens is the mean is lower than the median when you have a negatively skewed distribution as seen in figure B. The that’s because these low outliers are bringing down the me and where I’m sorry, the mean right, the mean is brought down by these really low outliers. And so we call this a negative distribution and the scope of it takes on this shape with a tail and over to the left.
Figures she portrays a positively skewed distribution in that we have high outliers that are bringing our means up, so it’s bringing our me up and it to misrepresent the data. So when the mean is
brought up by these high outliers, you have what’s referred to as a positively skewed distribution. And notice that in this example, the need is actually higher than the median. So what this is telling us is that the army is being kind of skewed upwards because of these high outliers, just like in a negatively skewed distribution, the mean is also also skewed and this time the other direction skewed downwards. So what this tells us is that we need to use the median as our measure of central tendency whenever we have a skewed distribution.
This is depicted in this chart here. So this chart can be used to consider what measure of central tendency to use and when we will use it and when we will not use it more. Most importantly, you want to take a look at and consider what level of measurement you’re dealing with if you’re dealing with interval ratio variables. You’re going to be able to to calculate the mean and the mean is going to be the best way to describe this information as long as you have a symmetrical, symmetrical distribution. How do you know if you have a symmetrical distribution? Well, if your median isn’t too far away from your mean, you have a fairly symmetrical distribution. So let’s talk about the median. Now, the median can be performed on an interval ratio variable or an ordinary variable. It cannot be performed on a nominal variable rate. The median cannot be performed on a nominal variable. And the mean to go back to talk about this cannot be performed on an ordinal variable.
The most compatible measure of central tendency is the mode because you can perform it on nominal or or interval ratio.
And the other thing to consider is what you’re trying to convey with this information. Are you trying to portray the most popular answer in the survey, then you want to go with the mood. Are you trying to portray a unbiassed, a middle middle of the road answer? Then you probably want to go with the media. If you’re trying to portray how all of the scores average out, then you will go with me. So that’s it for now. There are going to be another video that goes through problem sets to show you exactly how we would calculate each one of these and various situations. – So let’s apply these concepts to actual data. Before we begin, let’s review levels of measurement. So there’s nominal data, which is made up of categories that cannot be ranked. So for example, if people are asked, what do you use for transportation? And people could say the bus, cars, bikes, et cetera. These are categories that cannot be ranked. Ordinal data is made up of categories that can be ranked. And typically, they’re measured from low to high or least to most. So for example, you could look at what degree people have, and you could ask people, what’s your highest educational degree? And they can say high school, bachelor’s degree, master’s, PhD. And notice that you could have from low to high, and you can rank them. Least to most is also a way that these categories can be ranked and count as ordinal. So for example, you could ask people, how much do you agree that education should be funded more? And people could say, I disagree, I agree somewhat, or I strongly agree. And so notice that I’m looking at the least amount of agreement to the most. So this is ordinal as well. Lastly, we have interval ratio. And interval ratio has to come in numerical units. So if we ask people, how old were you when you had your first child? How many hours of TV do you watch a day? How often have you– how many times have you been to the dentist in the last two years? So people will give you numbers. I’ve been to the dentist two times. I have three children. I’ve watched four hours of TV today. So when you have individual numerical units, that’s interval ratio. The reason I bring this up is because depending on the data, we’re going to have to limit the type of measures that we use. With nominal data, we can only use mode. Ordinal data, we can use the mode and the median. And with interval ratio data, we can use all three, mode, median, and the mean. So just keep that in mind. With interval ratio data, the one that comes in numerical units, we can use all three. We can do the mode, the median, and the mean. And it’s typically advised that you do all three. Ordinal data, we have only the mode and the median. And then with the nominal data, we can only do the mode. So let’s jump into this first problem. First thing we want to do is figure out what type of data we have here. It says find all possible measures of central tendency, so we may not have to use all of them. So this data, looking at it, it is definitely not interval ratio because there’s no numerical units. And let’s think if it’s ordinal. Well, if it was ordinal, we’d have to be able to rank them from a low to a high or least to most. And there’s nothing about the type of streaming service that you prefer that allows us to say it goes from low to high. None of these streaming services have more or less streaming service than the rest. They’re just different types. So what we’re dealing with here is nominal. Categories that cannot be ranked. So all we can do is find the mode for nominal data. So all we’re going to do is try to see which one is the most popular category, and looking at this, it appears to be Netflix. Netflix appears six times, and that’s higher than any other subscription service. So our answer is mode equals Netflix and that’s it. We don’t include the median, we don’t include the mean because we cannot calculate those. Why can’t we calculate them? Because it’s nominal data. Looking at the second one, again, let’s figure out what level we’re working at so we don’t try to calculate something that’s not there. So we always calculate the mode, even in nominal data. So I wrote mode, but let’s figure out what level of measurement this is. So you might be tempted to say it’s interval ratio because we see numerical units a couple times, but it’s actually ordinal. And the reason it’s ordinal is because in order for it to be interval ratio, it has to be individual numerical units. In other words, they can’t be clumped together or grouped together like this. Some of the other categories don’t have any numerical units either. So that also gives it away, this is going to be ordinal. So it’s categories that can be ranked. So we can go from low to high here. How often do you go to the movies? Never is the lowest amount, less than once a month is the next amount, one to two times a month is the third amount, and three or more times a month is the highest amount that you could pick. So we can tell the amounts just not the exact amounts. All right. So now that we know it’s ordinal, we know that we can calculate the mode and the median. Starting with the mode, we just have to find the most popular category. So let’s look at this for a second and try to figure that out. OK. So less than once a month shows up 1, 2, 3 times. One or two times a month shows up 1, 2, 3, 4. Never shows up 1, 2, 3, 4 as well. And then three or more times a month only shows up once. So in this case, one and two times a month and never show up in an equal amount of time. So we’re going to have a bimodal distribution– one and two times a month and never. So we actually include both because they have a tie. Another thing to remember, too, is we always include the actual category and not the number of categories. So the mode is the actual categories and not 4. Even though we might think, oh, 4 is the answer because it appeared four times, that’s not correct. The correct answer is always the actual category that we’re looking at. These categories are known as people’s scores. And so we want to report scores and not frequencies. So when it comes to the median, the first thing we’re going to do is list the scores from low to high. And we have to include all of them. So we’re going to include– let’s see, we have 2, 4, 6, 8, 10, 11 scores here. We’re going to include them from low to high. So we have– we’re going to start with the nevers. And they appear– I thought they appeared four times. Yes. So here’s my– OK. So I went ahead and moved up the scores so that we could see them all. So never is the lowest category, so I include all four of these. And then now that I’ve included them, I’m just going to put them in bold so we know that they’re gone. And then we include the less-than-once-a-month category. And so that’s going to appear a total of three times. All right, so these are gone. Now it’s the one to two times, so I’m going to just write 1 to 2. And those appear four times. And so these are gone. And we’re only going to have one left, which is the three or more times a month. So I just put 3-plus. I’ve included all of my data from low to high. Now my job– my second step is to find the middle score. So there’s different ways to do this. In a simple data set like this, I just simply try to locate whichever score’s in the middle. For example, I’ll make sure to cross out the equal amounts on the left as I do on the right. So I knocked out 3 on the right and I knocked out 3 on the left. I’m going to knock out two more and see if I can get down to the middle one. So notice that I have two left in the middle. Usually what you have to do is add them up and find the average. So in this case, though, it’s made up of categories. So our median is going to be a category, and in this case, the categories are the same. The category– the two categories in the middle are less than once a month, so that’s going to be my answer. There are some cases where there are two different categories. So for example, let’s say that the two categories in the middle was less than once a month and one or two. In that case, we would have two medians, and that’s the only situation where we would actually have two medians. Otherwise, if it’s the same category like this, we just list that category. OK. So on the next video, I’m going to continue this worksheet, and then also show how to calculate the mean on a frequency table.- OK. So we’re going to continue with the last problem on here, which you can clearly tell is interval ratio data. The reason that I can tell is because it comes in numerical units. And these are individual numerical units. People were asked, how many movies did you watch at home in the past month? And people could say, I watched five, and I watched two. They were told to provide an actual numerical unit. So let’s get started. We know that we can calculate the mode and we can calculate the median and the mean. Let’s start with the mode. This is the score that appears the most often. So let’s take a look, and it appears that 1 is going to be our most popular answer. It appears more often than any other score, so our mode is one movie. Next up is the median. And we know that what we have to do for this one is, two, list the numbers from low to high. So we go 0 three times, 1 four times, 2 appears once, 3 appears twice, 4 appears once, 5 appears twice. Let me make sure I have all of them. 3, 7, 8, 10, 13. 3, 6, 9– it appears I am missing one. So it’s always good to double-check your work. This is a good example of that. Let’s find out where I went wrong. I’m missing a 7. So I’m going to add that to the end and we’re good to go. All right. So to find the middle score, there’s many different ways that you can do it. But basically, narrowing it down so that you’re down to the middle score. Make sure to delete the same amount on the left as you do on the right. So in this case, I deleted 6 on the left, 6 on the right, and I’m left with exactly two middle scores, 1 and 2. What do I do now? Well, I’m just going to have to find the middle of those two middle scores. So our median is going to be 1.5 movies. And that’s because 1.5 is in the middle of 1 and 2. If you want to do the math version of this, what you do is you add up the scores and you divide by 2. So whatever your two scores are– in this case, 1 and 2, you’re going to add them up and then divide by 2. So you still get the same answer, 1.5. And now let’s look at the mean. For the mean, there is a formula known as the y bar formula. I can’t really draw out all the symbols on here, so you’re going to want to check the book for all the symbols, but basically, it’s y bar equals– OK, actually after pausing and working on this for a bit, I was able to find a way to include symbols using the equation function on Word. So this is the equation Y bar equals sigma Y divided by n. So let’s break these symbols down starting with Y bar. So Y bar represents the mean. And then this symbol, which is the Greek letter, letter sigma, tells us sum up all of the blank. So it’s telling us sum up this following series. Series of what? And in this case, it’s the series of Y. So Y is going to represent individual scores. So Y represents all of these. Could be any of them and all of them. N is going to be the sample size or the total number of scores in our sample. So it’s literally telling us to add up all of the Y-scores and divide by the total sample size. So all we have to do is take our calculator, add up all of our scores. And so once we do that, if we were to put all of this into a calculator– and I’m combining some of the scores, so that’s why the numbers seem a little weird, we should get 33. So if you take all these numbers and add them up one by one, you’ll get 33. After that, it says divide by N. Our N is the total sample size, and in this case, we have a sample size of 3, 6, 9, 12, 14 scores. So we’re going to do 33 divided by 14. And in this case, we get 2.3571, et cetera. I’m going to round that to 2.4. Wherever you want to round it is fine as long as we’re rounding correctly. 2.4 movies. If the number next to the decimal place we want to round to is 5 or higher, we round up. So this got rounded to 2.4. If the number next to the decimal where we want to round is 4 or lower, then we round down. So that’s the rule for rounding. So we have 2.4 for that. So let’s go ahead and look at another way of representing data, and that’s on frequency tables. So I am going to skip to the last one here and show how to find the mode, median, and the mean on this. And the reason I’m skipping to the last one is because it’s interval ratio data and we can actually do all three. So if you know how to do the mode, median, and mean on interval ratio data, you’re able to do it on ordinal and nominal data as well. All right, so first the mode. It’s the most popular category, and for this, I’m going to look at which score appeared the most often. The frequency table is measuring the number of classes people are enrolled in. It could be 0, 1, 2, 3, 4, or 5 classes. These are known as our scores. We’re going to find the one that was the most popular, and that’s going to be 5 classes. So our mode is 5 classes. For the median, what we’re going to do is look at cumulative percentage and spot where we meet or surpass the 50% mark. So in this case, we meet or surpass 50 in this spot here. So our median is going to be 4 classes. For the mean, what we’re going to do is find sigma y by creating a new column. So you’ll notice that there’s a third column here, F times y. Basically, we multiply the scores times F and get F times y. The next thing we want to do is add up all of our F times y. OK, so after adding up all the F times y, I got 127. So what this is saying is that in total, there were 127 classes that people took. We surveyed 35 people, we asked them how many classes are you taking? Some of them said 0, some of them said all sorts of numbers. And in total, they took 127 classes. So we’re going to– all we have to do now is take these 127 classes and divide by the 35 people that we surveyed. So 127 divided by 35. So 127 divided by the 35 people is going to come out to an average of 3.6 classes per person. So 3.6 classes is going to be our mathematical average. If you’re wondering where this number came from, 127, again, it comes from multiplying the number of people by the number of classes that they took. So for example, this one person took 0 classes, so they contributed 0 classes. These 10 people, for example, took four classes each. So how many classes did they contribute in total? If 10 people took four classes, they contributed a total of 40 classes. These 10 people– these 10 people attributed 4, 40 classes. And so that’s why we do this. And then basically what we’re doing is getting the total number of classes that everybody took and dividing it by the number of people. So there’s measures of central tendency.
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