Measures of Central Tendency and Dispersion with SPSS

Central Tendency and Variability Program Transcript

JENNIFER 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. Measures of Central Tendency and Dispersion with SPSS

I will also show you how to graph your meanns 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. OK, 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 have 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 in this distribution, I added the value 2, I would have two modes for this distribution. It would be bi-modal. 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 going to 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 dialogue box open in SPSS.

Now you need to choose the variable that you want to find the mode for. I’m going to choose a 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 dialogue 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 variable is the mode. So click on charts. And click on Bar Charts. Click on Continue. And now click on OK to run your analysis.

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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 0 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, the value that appears most often for the variable ethnicity is three, 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.

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 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 will 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 three. So again, for this distribution, my median is three. 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 dialogue box.

So I want to find the median for one of the variables in my data set, and I’m going to choose self esteem first 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 OK. 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

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values in this variable is 1.45. SPSS will also give you a frequency distribution for that variable.

All right. Now let’s learn about the last measure central tendency, the mean. The mean is the average of a set of scores in your distribution. First I’m going to 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 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 10 values. 3, 4, 2, 2, 3, 2, 4, 4, 2, 3. I have 10 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 divide it by the total number of values. In this distribution. And in this case, that is 10. So 29 divided by 10 equals 2.9. 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 you need to choose the variable that you want to find the mean for. I’m going to 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 dialogue box on the right. Now I’m going to click on Options. I’m going to uncheck everything, except for mean. So it should just have mean checked. Now click on Continue. And now click on OK. As you can see, the mean for this variable, peer support is 3.3415.

So for my 150 participants, once I take those values, add those up, and divide by 150, the mean level of peer support for this distribution is 3.3415. Or I would round that up to 3.34, 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 dialogue 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.

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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 going to do anything here. 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 OK. Now 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. Measures of Central Tendency and Dispersion with SPSS

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. 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. 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 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 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 going to 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 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 OK. 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. The variance is the average squared deviation from the mean. It measures

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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 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 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. Measures of Central Tendency and Dispersion with SPSS

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 six. And that value is 7.33. So my sum of squares is 7.33.

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 six values in my distribution minus 1. So that’s 5 equals 1.466.

Or I would round that up to 1.47. So for this example, my sample variance for this distribution of six scores is 1.47. Now let me show you how to calculate the variance using SPSS. Now that I have my SPSS data set window 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 going to 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 dialogue box on the right.

Now click on Options here at the far right. You going to click on Variance, and you’re going to uncheck all of the other boxes. You’re going to click on Continue, and now click on OK. And as you can see SPSS, is going to provide a table that gives you the variance for this variable.

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So for stress, we have 148 participants that have a value for stress. And the variance for the variable stress is 0.522. Now let me show you the last measure of variability, the standard deviation. The standard deviation is just the square root of the variance.

So 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 for the example I gave previously, your standard deviation equals the square root of your sum of squares.

And in this case, your sum of squares for that last example was 7.33 divided by 5. And that equals 1.21. So your standard deviation is 1.21. To calculate the standard deviation in SPSS, click on Analyze, Descriptive Statistics, Descriptives, and it gives you the dialogue 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 OK.

As you can see, SPSS is going to give you a table that tells you the number of participants for the variable of stress, which is 148. And remember, the variance is 0.522. And the standard deviation is now 0.72250, which is the square root of the variance. So the square root of the value 0.522 equals 0.72250, or your standard deviation. Measures of Central Tendency and Dispersion with SPSS

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.

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Frequency Distributions Program Transcript

JENNIFER MORROW: Welcome to frequency distributions. My name is Dr. Jennifer Ann Morrow. Today in this demonstration, I will show you how to conduct frequencies in SPSS. I will also show you how to read a frequency distribution table, and understand the types of percentages displayed in a frequency distribution table.

I will also show you how to create charts and graphs in SPSS. Specifically how to create bar charts, histograms, pie charts, and line graphs. OK, let’s get started. Now that I have my SPSS data set open, the first thing that I’m going to show you how to do is to create frequency distribution tables in SPSS. First, click on Analyze.

Click on Descriptive Statistics, and click on Frequencies. As you can see, the frequency dialog box is now open in SPSS. On your left is the list of variables that you have in your data set. And on the right is the blank analysis box where you will put the variables that you want to use for your analysis.

Now let’s display some frequency tables for a couple of our variables. I’m going to select gender, and then once I click on gender, I’m going to click on the right arrow box. And then I’m also going to select school grade, and then click on the right arrow. Now both of my variables appear in my analysis box on the right.

Remember, you can always click on Paste to put syntax in a syntax window, and run the analysis that way. Or you can just click OK, and have SPSS immediately conduct your analysis. So let’s click on OK. Now an output window appears with the results of your analysis. Let’s just scroll up to the top of your output file. Measures of Central Tendency and Dispersion with SPSS

As you can see in your output window, SPSS will first give you a table that gives the total n for both of the variables. In this data set we have 1,637 participants that have a valid value for gender, and 9 participants that have missing data for this particular variable.

For school grade, we have 1,641 participants that have a valid value for school grade, and 5 participants that are missing data for school grade. Now let’s scroll down to the first frequency table. As you can see with our variable gender, we have our variable coded as follows. 0 is for females, and 1 is for males.

In this data set we have 928 females and 709 males, for a total of 1,637 participants that have a valid value for the variable gender. One thing I want to point out to you is the difference between this percent column and the valid percent column.

For the percent column, this is the value that is the total number of participants in one category divided by the total number of participants in your data set. And then multiply

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that value by 100 in order to get the percent. In the case for females, we take 928, divide that by 1,646, and multiply that by 100 in order to get a percent of 56.4.

So in this data set, 56.4% of our participants are females. The valid percent is different in that it takes out all of the missing data before calculating the percentage. So in this case, to get the valid percent for females, you take the value of 928, divide that by 1,637, and multiply that by 100 in order to get the valid percent of 56.7%.

The only time that the values in the percent and the valid percent column will be equal is if you have no missing data for that variable. Now let’s move on to our other frequency distribution table for school grade. Let’s scroll down through our output.

For a variable school grade, we can see we have students from the seventh grade to the 12th grade, as well as one participant that was ungraded. In this data, set we have 1,641 participants that have a response for school grade, and only 5 participants that do not have a response for school grade. Measures of Central Tendency and Dispersion with SPSS

Now I want to show you in this table the cumulative percent column. The cumulative percent column adds the valid percent values for each successive category. For example, if you add the valid percent for seventh graders, 0.2, and the percent for eighth graders, 9.1, you will get a cumulative percent of 9.4.

As you can see, there’s some rounding issues with this. But if you click on the box for the valid percent, you will see the exact number for the valid percent, and not just the value that was rounded up or down. Let me show you what I mean. Double click on your chart, and double click on the valid percent of the seventh graders.

As you can see, the exact value is 0.2437, et cetera. Now double click on the value for eighth graders. And that valid percent is 9.1407, et cetera. If you add those two values together, you will get the cumulative percent here listed as 9.4. If you double click on the box, the exact value, 9.3845, et cetera.

Again, SPSS rounds up to make things easier to read in your table. The last thing I want to mention before moving on is to make sure that you pay attention to the missing data in your variables. It is very important to know how much missing data is in each of your variables.

The more missing data that you have, the fewer participants you will have to conduct your analyses. In the case of school grade, we only have five participants that don’t have a value for school grade, or much less than 1% of our total participants in our data set. Always be aware of the number of missing data points you have for each variable.

Let’s recap. First we learned how to display frequency distributions in SPSS. We talked about the differences between percent and valid percent. We also discussed what a cumulative percent is. Lastly, I mentioned the importance of knowing how much missing data you have in your variables. Now let’s move on to creating charts and graphs.

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Now that I’m back in my SPSS data set, let me show you how to create charts in SPSS. We can create charts in SPSS in two different ways. First, we can ask for them in our frequency distribution dialog window. Or we can go under the graph function in SPSS. First I’m going to show you how to get charts using your frequency dialog box. Measures of Central Tendency and Dispersion with SPSS

Click on Analyze, Descriptive Statistics, and Frequencies. And as you can see, our frequency dialog box will open up, showing our last analysis. Remember, we just did a frequency distribution for gender and school grade. So now let me show you how to create a chart for the variable gender.

So first I’m going to click on school grade, and click on the left arrow button to move that back over to the list on the left, because I only want to look at the variable gender. Now I’m going to show you how to create various charts for the variable gender. Click on Charts. And as you can see, there are a few options here under Charts.

For the variable gender, since it is a categorical variable, or a discrete variable, only bar charts or pie charts are appropriate. It is not appropriate to do a histogram for a variable that is not continuous. So first let me show you how to create a bar chart for gender. Click on bar chart.

You’ll notice here at the bottom where it says chart values, that you have the option of displaying either the frequencies or the percentages for your bar chart. I would like to display the frequencies, so I’ll keep that checked, and then click on Continue. Then I will click on OK. So SPSS will create a chart for gender.

As you can see, SPSS has created a bar chart for the variable gender. And you can modify this chart however you like, again by double clicking the chart in order to get the Chart Editor. So if I double click my chart, I now get the Chart Editor.

And you can do many different things here. You can create new titles. You can change the color of your bars in your chart. Let me close out the Chart Editor and move on to show you how to do another chart. Next, I want to show you how to create a pie chart in SPSS. Click on Analyze, Descriptive Statistics, Frequencies.

We already have the variable gender here in our analysis box. Click on Charts, and now click on pie chart. Again, I’m going to stick with frequencies for this analysis. I’ll click on Continue. And then now, let’s click on OK. SPSS, as you can see, has created a pie chart for my variable, gender of participant.

As you can see, SPSS will automatically create a slice of the pie for those participants that have missing data. You can easily modify the chart so it does not display participants, that have missing data. Double click on your chart. Click on Edit. Click on Properties. Click on Categories. Click on where it says missing. Measures of Central Tendency and Dispersion with SPSS

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Click on the red X So it takes out the pie slice for the variable, the category missing. And then click on Apply. We’ll close this box. And as you can see now, you have a pie chart that just displays participants that have a value for female, and a value for male.

Lastly, under the frequencies dialogue box, let me show you how to create a histogram. Click on Analyze, Descriptive Statistics, Frequencies. Let’s click reset to get rid of the last analysis that we conducted, so we can start fresh. Let’s go over to our box on the left, and scroll down, and click on hours per week at job.

Second to last variable here. Click on the right arrow key. And now we’re going to click on Charts, and click on histograms. And I always like to display the normal curve, so click on the box that says with normal curve. Click on Continue, and then click on OK.

And as you can see, SPSS has created a histogram for this variable, number of hours per week at job. A histogram is appropriate for this type of variable because it is a continuous variable. Next, I would like to show you how to create charts using the graphs function in SPSS.

Click on graphs, and you’ll see that there’s a variety of graphs available to you in SPSS. First, I want to show you how to create a bar graph under this function. So scroll down and click on bar, then you see you have three different choices, simple, clustered, and stacked. Click on simple, and then click on define.

Now let’s click on a variable to create a bar chart. I’m going to choose gender, so click on gender, and then move that over here to where it says Category Axis. You can also answer a title for your bar chart. If you go down here going to right and click on Titles, you can answer a title for your chart. I’ll enter Gender of Participant.

Once you’ve entered your title, click on Continue. Now click on OK. As you can see, you’ll get a bar chart very similar to the one we’ve already created under the frequency dialogue box. We can also ask for histograms under the graphs menu in SPSS. Click on Analyze. I mean click on Graphs, excuse me.

Click on histogram. And now we get the histogram dialogue box in SPSS. Click on a variable that is continuous. How about I choose hours per week doing chores? And then click the right arrow key for the variable box, moving that variable hours per week doing chores into the variable window. Measures of Central Tendency and Dispersion with SPSS

Again, I always like to display the normal curve. Click on the titles box at the right so you can enter a title. Hours per week doing chores. Once you’ve entered your title, click Continue, and then click OK. As you can see, SPSS will display a histogram for this continuous variable, hours per week doing chores.

You also have the freedom with this type of chart to double click to get the Chart Editor to modify your chart in many ways. Now let me show you how to ask for a pie chart in

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SPSS. Click on Graphs, click on pie, then click on Define. Now choose a variable that you want to create a pie chart four. Measures of Central Tendency and Dispersion with SPSS

Let me choose the variable whom do they live with. Click on that variable. Click on the right arrow key for define slices by, so the variable whom do they live with appears there. Click here on the right, titles to enter a title for your pie chart. Whom do they live with. Once you’ve entered your title, click on Continue.

And then click on OK. And as you can see for this variable, whom do they live with, there are a variety of categories. For mother and stepfather, father and stepmother, et cetera. And again, you have the freedom in SPSS to double click on the chart to modify any of the properties using your Chart Editor.

Lastly, I want to show you how to create a line graph in SPSS. Click on Graphs, click on line, and you see you have three choices, simple, multiple, and drop line. I’ll choose simple, and then click on Define. Now I have to choose a variable. Let’s click on a variable, hours per week at job.

Over here on the right, click other statistic. Then click the right arrow box next to the variable window, and this will now display the mean for hours per week at someone’s job. This will represent, then, the mean hours per week in your line graph.

I would like SPSS to display a line graph displaying the mean hours per week at a job for both males and females. So I click on my variable over here on the left, gender. And click on the right arrow key for Category Axis.

This will then tell SPSS to give me a line graph that displays the mean hours per week for females, and the mean hours per week for males. Again, I can always go to the box on the right, click on titles to enter a title for this line graph. Gender and hours per week at job. Once you’ve entered your title, click on Continue.

And then click on OK. SPSS will display a line graph that connects the mean for females for hours per week at job, and the mean for males. As you can see here in your graph, the mean for females at hours per week at job is a little bit above 2.5 hours per week. And for males, the mean is a little bit above 3.0 hours per week.

SPSS automatically connects those two means and draws a line representing the mean hours per week for both females and males. Now let’s recap. We went over displaying percentages versus frequencies in your charts and graphs. We next learned how to display bar charts, histograms, pie charts, and line graphs.

We are now at the end of this demonstration. Remember to practice what we learned today. On your own, use SPSS to create frequency tables, and charts and graphs. Practice modifying your charts or graphs using your Chart Editor. Thanks, and have a great day. Measures of Central Tendency and Dispersion with SPSS

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