DDBA8437: Central Tendency and Variability Video Podcast Transcript

Transcription

1 DDBA8437: Central Tendency and Variability Video Podcast Transcript JENNIFER ANN MORROW: Today's demonstration will review measures of central tendency and variability. My name is Dr. Jennifer Ann Morrow. First I will review the different measures of central tendency. I will show you how to calculate the mode, the median, and the mean using both formulas and SPSS. I will also show you how to graph your means in SPSS. Second I will demonstrate to you how to calculate three different measures of variability: the range, the variance, and the standard deviation. I will show you how to calculate all of these using formulas as well as using SPSS. Mode JENNIFER ANN MORROW: Okay, let's get started. The first measure of central tendency that I'm going to review for you is the mode. The mode is the score or category that has the greatest frequency in your distribution. Now let me give you an example of how to find the mode in your distribution. If I had the following values for my variable: 3, 6, 4, 3, 5, 1, 2, 3, and 2, what would my mode be? It would be the value 3, because the value 3 appears the most often. It appears three times in this distribution. So my mode... equals 3. What would happen if I had two values that appeared the most often? If for this distribution I added the value 2, I would have two modes for this distribution. It would be bimodal. I would have a mode of 3, and I would also have a mode of 2, because both values appear three times in my distribution. Now let me show you how to calculate the mode in SPSS. First you need to open up a data set in SPSS. Click on File. Click on Open. Click on Data. Now find the data set that you're going to use. Once you have found your data set, click on Open. Make sure your data view window is visible on your screen. Now I'm gonna show you how to find the mode for your variable in SPSS. First click on Analyze. Click on Descriptive Statistics. Click on Frequencies. And now you have the frequency dialog box open in SPSS. Now you need to choose the variable that you want to find the mode for. I'm gonna choose the variable Ethnic, which is the ethnicity of my participants. Click on the variable Ethnic. Click on the right arrow key to move that variable to the dialog box on the right. Now click Statistics. Click on Mode. Click on Continue. You could also ask SPSS to produce a bar chart so you can see which value in your

2 variable is the mode, so click on Charts, and click on Bar Charts. Click on Continue, and now click on Okay to run your analysis. Now you see in your output window SPSS is going to give you a table first telling you the number of participants in this particular data set, which is 150. There are zero participants that are missing a value for this variable ethnicity, and your mode or most frequent score for the variable ethnicity is a 3. If you scroll down further, you'll see the frequency distribution chart, and as you can see, that yes, that value that appears most often for the variable ethnicity is 3, or Caucasian. There are 72 participants out of 150 that are Caucasian in my data set. If you scroll down a bit further, you'll see the bar chart for ethnicity. To find the mode in your bar chart, look at the bar that has the highest value, and here it is Caucasian. It has the highest bar. We know that that is the mode for this particular variable. Now let's learn about other measures of central tendency. Median JENNIFER ANN MORROW: The next measure of central tendency that I'm going to review is the median. The median is the score that divides a distribution exactly in half. I'm going to first show you how to calculate the median in your distribution by hand. The first thing you need to do to find the median in your distribution is to order the values in your variable from lowest to highest, so let me give you an example. 1, 2, 2, 2, 3, 3, 3, 4, 5, and 6. To find the median for this distribution of variables, you need to find the middle value. If you have an even number of values, you take the two middle values and divide that by 2 in order to find the median. In this case, my two middle values are 3 and 3, so 3 plus 3 divided by 2 equals 3, and that is the median for this distribution. But what happens if you have an odd number of values in your distribution? What if I had the values such as this? 1, 2, 2, 2, 3, 3, 3, 4, and 5. What would be my median now? To find the median for your distribution that has an odd number of values, you just choose the middle value. In this case, my middle value is 3. So again, for this distribution, my median is 3. Now let's learn how to find the median in SPSS. Now that I have my SPSS data set open, to calculate the median, you have to go to Analyze, so click on Analyze. Click on Descriptive Statistics. Click on Frequencies. And you'll need to choose the variable that you want to find the median for. First let's click Reset to get rid of this variable ethnicity from the dialog box. So I want to find the median for one of the variables in my data set, and I'm gonna choose Self-Esteem First

3 Semester. Click on that variable: Self-Esteem First Semester. Click on the right arrow key to move that to the dialog box on the right. Now you need to click on Statistics. Click on Median. Click on Continue. And now you can click on Okay. We'll just have to scroll up a little bit. And as you can see, SPSS is going to first give you a table that tells you that you have 149 participants that have a value for Self-Esteem First Semester. You have one participant that does not have a value for Self-Esteem First Semester and that that median for your values in this variable is SPSS will also give you a frequency distribution for that variable. All right, now let's learn about the last measure of central tendency, the mean. Mean JENNIFER ANN MORROW: The mean is the average of a set of scores in your distribution. First I'm gonna show you how to calculate the mean by hand. First you need to know the formula for calculating a mean. To calculate a mean... it is the sum of the values in your distribution divided by the number of participants or values in your distribution. It is sigma, which is this here, which just means "the sum of," X, which is the scores or values in your distribution. The sum of X divided by n, which is the number of values in your distribution. That is the formula for the mean. Now let me give you an example. I have ten values: 3, 4, 2, 2, 3, 2, 4, 4, 2, 3. I have ten values in my distribution. If I add all of these up-- so the sum of the values-- I get 29. So the sum of all my values equals 29. So in order to find the mean... I take the value 29, which is the sum of all the values in my distribution, and divided it by the total number of values in this distribution, and in this case, that is 10. So 29 divided by 10 equals So the mean for this distribution of values is 2.9. Now let's learn how to calculate the mean using SPSS. Now that I have my data set open, to find the mean for my variable, I can click on Analyze, so click on Analyze. Click on Descriptive Statistics. Click on Descriptives. And now I need to choose the variable that you want to find the mean for. I'm gonna choose, let's see, Peer Support, so scroll down until you can see Peer Support. Click on Peer Support. Click on the right arrow key to move that variable to the dialog box on the right. Now I'm going to click on Options. I'm going uncheck everything except for Mean, so it should just have Mean checked. Now click on Continue, and now click on Okay. As you can see, the mean for this variable, peer support, is , so for my 150 participants, once I take those values, add those up, and divide by

4 150, the mean level of peer support for this distribution is , or I would round that up to Just two decimal points. Now let me show you how to graph means in SPSS. One way to graph means is to go to Graphs. Click on Graphs. Click on Line. Click on Simple. Click on Define. Now we need to find the variable in our dialog box on the left. Click on Peer Support. It's about in the middle. Click on Other Statistic. Click on the right arrow key to move that variable Peer Support into the dialog box on the right. And as you can see, SPSS is going to represent the lines as mean of peer support. Now we need to choose a variable that we want to represent the means for. Let's choose the variable school or class, year in school. Click on that variable. Click on the right arrow key to put that in the category access box. Now let's give this graph a title. Click on Titles here at the far right. And let's give our graph a title: Year in School and Peer Support. Now click on Continue. Let's click on Options, and we're not gonna do anything here. If this--this was-- display groups defined by missing values-- if that was checked, I would uncheck that. Click on Continue, and now, let's click on Okay And as you can see, SPSS has produced for you a graph that shows the mean level of peer support for my four different groups of participants: my first-year students, my sophomores, my juniors, and my seniors. So as you can see, it's an easy way to represent the means for your variable to your audience using this type of graph. Recap JENNIFER ANN MORROW: Now let's recap what we just went over. So far, we learned about how to calculate the mode, the median, and the mean. We learned how to calculate all of these both by hand as well as using SPSS, and we also learned how to graph both the mode and the mean. Now let's move on to measures of variability. Range JENNIFER ANN MORROW: The first measure of variability that I want to go over with you is the range. The range is the difference between the largest score and the smallest score in your distribution of scores for your variable. Let me show you how to calculate the range for a variable first by hand. The first thing that you have to do is order your values for your variable from lowest to highest. Let me give you an example: 2, 3, 3, 3, 5, 6, and 8. So I've ordered my

5 values for my variable from lowest, which is 2, to highest. Now you take your highest value, which is an 8, and you subtract the lowest value, which is a 2. So 8 minus 2 equals 6, and that is the range for your distribution. Now let's learn how to find the range in SPSS. Now that I have my SPSS data set open, to find the range for my variable, all I have to do is click on Analyze, click on Descriptive Statistics, click on Descriptives. Let's click Reset to get rid of this variable here. Now I'm gonna choose a variable that I want to find the range for. How about we look at year in school, class? Click on Year in School. Click on the right arrow key to move that variable to your dialog box on the right. Click on Options here at the far right. Now, we want to make sure we click on Range. Let's uncheck Standard Deviation and Mean. We don't need that. But let's keep Minimum and Maximum. The minimum is your smallest score in your distribution for this variable, and the maximum is the largest score in your distribution for this variable. Now let's click on Continue. And now let's click on Okay. As you can see, for your variable, class, or year in school, to find the range, SPSS gives you a table that first tells you you have 150 participants that have a value for this variable. The minimum score is 1. The maximum score is 4, so to find the range is 4 minus 1 or 3, so the range for this variable is 3. Now let's learn about another measure of variability, the variance. Variance JENNIFER ANN MORROW: The variance is the average squared deviation from the mean. It measures how spread out or how far apart your scores are from the mean of your distribution. The first thing I'm going to show you is how to calculate the variance by hand. First I need to give you the formula for variance. The formula for a sample variance is... lowercase s squared equals the sum of squares, which is capital S, capital S, divided by n minus 1, which is the number of scores in your distribution minus 1. This equals the variance. In order to calculate the variance, we need to find out what our sum of squares is. The formula for sum of squares is as follows. Sum of squares equals sigma X squared, which is the sum of all the values in your distribution squared... minus--in parentheses-- the sum of X, which is all of the values in your distribution added together. Then you square that value divided by n, which is the number of scores in your distribution. So let me give you an example. I have the values 2, 3, 5, 3, 4, and 5. First I want to find the sum of all the values. If I add all of these together, I get the value of 22, and that is

6 Sigma X, the sum of all the values. If I square that, I get the value 484, so Sigma X squared equals 484. Now let's find the squares of all the individual values. The square of 2 is 4. The square of 3 is 9. The square of 5 is 25. The square of 3 is 9. The square of 4 is 16. And lastly, the square of 5 is 25. If I was to add all of these together, I would get 88. And so 88 is the sum of all the values squared. Now I have the information to find the sum of squares. So my sum of squares... equals 88 minus 484 divided by n, which in this case is 6, and that value is 7.33, so my sum of squares is Now I have enough information to calculate my sample variance. So going back to the formula here at the top, to find my sample variance, let's plug in the values. So my variance equals my sum of squares, which in this case is 7.33 divided by n minus 1. So I have 6 values in my distribution minus 1, so that's 5, equals , or I would round that up to So for this example, my sample variance for this distribution of 6 scores is Now let me show you how to calculate the variance using SPSS. Now that I have my SPSS data set open, to calculate the variance for a variable, just click on Analyze, Descriptive Statistics. Click on Descriptives. Let's click Reset to get rid of that variable, and now choose the variable that you want SPSS to find the variance for. I'm gonna choose the variable Stress, so let me scroll down here to find my variable Stress. Click on Stress. Click on the right arrow key to move it to the dialog box on the right. Now click on Options here at the far right. You're gonna click on Variance, and you're gonna uncheck all of the other boxes. You're gonna click on Continue, and now click on Okay. And as you can see, SPSS is gonna provide at table that gives you the variance for this variable. So for Stress, we have 148 participants that have a value for Stress, and the variance for the variable Stress is Now let me show you the last measure of variability, the standard deviation. Standard Deviation JENNIFER ANN MORROW: The standard deviation is just the square root of the variance, so now let me show you how to calculate the standard deviation first by hand using the formula. To calculate the standard deviation, you use this formula: lowercase s, which stands for Standard Deviation, equals the square root of the sum of squares divided by n minus 1, or the square root of your variance. So using the example we had before, to find the standard deviation, you just take the square root of the variance. So of the example I gave previously, your standard deviation equals the square root of your

7 sum of squares, and in this case, your sum of squares for that last example was 7.33 divided by 5, and that equals So your standard deviation is All right, let's recap. Today we learned how to calculate the range, the variance, and the standard deviation. Oh, excuse me. First let's go back, and let me show you how to calculate the standard deviation using SPSS. I apologize. Let's go back to SPSS and get that. To calculate the standard deviation in SPSS, click on Analyze, Descriptive Statistics, Descriptives, and it gives you the dialog box for your descriptives in SPSS. Let's find the standard deviation for the variable Stress, so keep that variable in your dialog box here on the right. Click on Options, and click on Standard Deviation. Let's keep the box for Variance checked so you can see how SPSS calculates the standard deviation and how that relates to the variance. Click on Continue, and click on Okay. As you can see, SPSS is going to give you a table that tells you the number of participants for the variable Stress, which is 148. And remember, the variance is 0.522, and the standard deviation is now , which is the square root of the variance. So the square root of the value equals or your standard deviation. Recap JENNIFER ANN MORROW: Now let's recap what we just learned. I just showed you how to calculate the range, the variance, and the standard deviation both by using formulas and SPSS. We have now come to the end of this demonstration. Remember, you can practice what we learned today on your own with the data set for these demonstrations or your own data set. Practice calculating the various measures of central tendency and variability. Thank you, and have a great day.