Week 3 - Assignment: Analyze Numerical Summaries of Data Using Excel and SPSS

CHAPTER 5

Understanding Central Tendencies

There are three measures of central tendency; they are the mmean, median, and the mmode.

Mean

The mean is the average value of a set of data. It can also be thought of as the center of gravity of data; that is, the sum of the distances from the mean for each data value on one side of the mean will equal the sum of the distances from the mean for each data value on the other side. The mean will balance the data; in fact if the differences of all the data values and the mean are added together, then the sum will be zero. The formu-

la for the mean is x

n �

where capital sigma, � is the summation sym-

bol and simply means “add values together,” x represents the individual values, and n is the total number of values.

Example 1 Find the mean of the following data. Also compute a new variable which is the difference of the mean and the x values and show the sum of that variable is zero. 6, 18, 10, 12, 9, 5, 4, 3, 15, 13, 12, 16, 18, 15, 14, 13, 17, 7, 6, 7 (chpt3example1.sav). (This problem can be done using a hand-held calculator before using SPSS.)

Solution

Enter the data into a SPSS data editor or open the data file chpt3 example1.sav. To find the mean use Analyze � Descriptive Statistics � Frequencies, move the x variable into the variable window, click on the tab Statistics, and put a check mark in the mean box; see Figure 5.1.

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56 DATA ANALYSIS WITH SPSS SOFTWARE

Figure 5.1

Click on Continue then OK. The mean is 11; see Figure 5.2.

Statistics

X

N Valid 20

Missing 0

Mean 11.00

Figure 5.2

(To avoid the display frequencies tables, deselect display frequencies tables after Analyze � Descriptive Statistics � Frequencies.)

As a matter of academic understanding of the mean, we’ll also find

the mean by use of the formula 1 .

n

i i

x x

n = �

= To find the sum of the data

values, use Analyze � Descriptive Statistics � Frequencies, click on the tab Statistics, and put a check mark in the sum box. Hit OK. The sum is 220; see Figure 5.3. The total number of data values is 20 (note the statistics output box has 20 next to N, which is the number of data val- ues) so the mean is 220/20 = 11.

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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UNDERSTANDING CENTRAL TENDENCIES 57

Statistics

X

N Valid 20

Missing 0

Sum 220

Figure 5.3

Let’s check to make sure the sum of the differences of all the x values and the mean is zero. Go back to the data editor and click on Transform � Compute, call the new variable diff (difference), and create the for- mula diff = x − 11; see Figure 5.4.

Figure 5.4

Click OK and a new variable will be created with the difference of x and the mean. Now find the sum of the diff column (use the procedure above). Is it zero? This illustrates that the mean is the center of gravity. (We will assume everyone is familiar with the concept of center of gravity because we’re sure everyone played with their utensils at the dinner table at one time in their past.) (Teaching tip: this data set can also be used to illustrate the concepts of median and mode. A 21st data value can be added for further illustration. These concepts are explained below.)

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58 DATA ANALYSIS WITH SPSS SOFTWARE

The symbol for the mean is x (x bar), so the equation for the mean

can be written as , x

x n �

= where the summation symbol means to add

all the x values.

Example 2a Use Chapter 1 Example 2 data set to find the mean income.

Solution

Open the data file chptlexample2.sav, use Analyze � Descriptive Statis- tics � Frequencies, move the income variable to the variables window, click on statistics tab, and check the statistic mean. Click OK. Is the answer 51,000?

As we saw in the previous chapters, whenever we use the explore function we get all the descriptive statistics, including the mean. If you need more practice finding the mean, then access data sets from the Internet or elsewhere to find means. (Use Analyze � Descriptive Statistics � Frequencies, put the variable of interest in the variable window, click on statistics, and check mean, continue, and OK.) Sometimes finding the mean is the only job required of a statistician.

Median

The median is the number (which may or may not be a data value) that divides the data in half. That is, half the data values are less than the median and half the data values are greater than the median. We have two different formulas for finding the median. To find the median we first rank the data—that is, put the data in order—normally from lowest to highest, ascending order, but the data can also be put in descending order, from highest to lowest. The key point is the data must be put in order.

Median for an odd number of data values

If the number of data values is odd, then the median is simply the data value in the middle. For example, if the number of data values is 9 then the median is the 5th data value, the middlemost data value after the data have been ranked. Now you can see why it’s important to rank the data values, otherwise we wouldn’t know what the fifth data value is if

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UNDERSTANDING CENTRAL TENDENCIES 59

the data aren’t in order. In general if the number of data values, n, is

odd, then the median is in the position of the ( 1)

2 n +

data value of the

ranked data values.

Example 2b Find the median income in this set.

Gender Position Income Female President $100,000

Male Vice President $50,000

Male Manager $40,000

Female Worker 1 $35,000

Male Worker 2 $30,000

Solution

The data set Income is already ranked and has 5 data values, thus the median is the 3rd data value. The median is $40,000. Let’s let SPSS find the median for us. Open the data set chpt1example2. sav or the data set you saved from that example, use Analyze � Descriptive Statistics � Frequencies, move the Income variable into the variable window, click on Statistics, check the box Median, click on continue, and hit OK. The output box will have Median 40000.00.

Median for an even number of data values

If the number of data values is even, then the median is the average of the middle two ranked data values. Thus, if N is even, then the median

is the sum of the 2 n

data value plus the ( 1)

2 n +

data value divided by 2.

It is the average of the two middlemost data values. In this case the median may or may not be a data value.

Example 3 Suppose the data set in Example 2 are revised as such:

Gender Position Income Female President $100,000

Male Vice President $50,000

Male Manager $40,000

Female Worker 1 $35,000

Male Worker 2 $30,000

Male Worker 3 $30,000

What is the median of the variable Income?

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60 DATA ANALYSIS WITH SPSS SOFTWARE

Solution

Income now has 6 data values, so the median is the average of the third and fourth data values. That is, the median is the average of $40,000 and $35,000, which is $37,500. As we see the median is not a data val- ue, but three of the data values lie below $37,500 and three of the data values lie above $37,500. Thus the median splits the data in half. Let’s let SPSS find the median. Add the sixth data value to the data set you used for Example 2 or open chpt3example3.sav. Use Analyze � Descriptive Statistics � Frequencies, move the Income variable into the variable window, click on the Statistics tab, check the Median box, and click on Continue and OK. The median should read 37500.

Example 4a Find the median of these data values: 4, 8, 2, 3, 1, 6, 9, 11, 3.

4b. Find the median of these data values: 1, 3, 5, 8, 2, 4, 5, 6, 6, 7.

Solution

Try both of these examples by hand before using SPSS to make sure the concept is clear. To use SPSS open the data file chpt3example4.sav. Use the steps as outlined in the above examples; you can move both variables simul- taneously by highlighting both variables before moving them to the variable window. Just hold the shift key down as you click on the variables. The output should read the median for variable a is 4 and the median for varia- ble b is 5. If we rank the data, we can see 4 is the 5th data value in a and 5 is the average of the 5th and 6th data values in b.

a ranked: 1, 2, 3, 3, 4, 6, 8, 9, 11 b ranked: 1, 2, 3, 4, 5, 5, 6, 6, 7, 8

The median is that value where 50% of the data values are less than or equal to the median and 50% of the data values are greater than or equal to the median.

Medians are sometimes a more useful tool for describing the average value than the mean. This is because if the data have a few values called outliers (extreme values compared to the rest of the data values), then the mean will be pulled by the extreme values and may not reflect the “true” average of the data. This is particularly the case when discussing income or housing.

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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UNDERSTANDING CENTRAL TENDENCIES 61

Example 5 Refer back to the data in Example 2.

Gender Position Income Female President $100,000

Male Vice President $50,000

Male Manager $40,000

Female Worker 1 $35,000

Male Worker 2 $30,000

Find the mean and the median of Income for this data set.

Solution

Open the data set chpt1example2 and use Analyze � Descriptive Statis- tics � Frequencies, move income to the variables window, click on the tab Statistics, and check both boxes Mean and Median, Continue, and OK. The mean is 51000 and the median is 40000. Only two people make $51,000 or close or above $51,000. The rest make amounts closer to $40,000. Hence 40000 is actually a better average value for this data set.

In general the median is the most-used measure of central tendency when the data reflect income or the housing market, because most of the income values or house prices are grouped together, but always a few really extreme values shift the mean off the point where the majority of the data values lie.

The third measure of central tendency is the mode.

Mode

The mode is that data value that occurs most often. If two different data values have the same number of occurrences, more than once, then we say the data are bimodal. If no data value occurs more than once or if three or more data values have the same number of occurrences more than once, then we say the data do not have a mode. The data have only one mode, two modes, or no modes. SPSS will always give some value for the mode regardless if a mode exists or not. If all the data values occur only once, then SPSS will give the smallest data value and put a note on the bottom saying multiple modes exist. If one data value occurs more often than any other data value, then SPSS will give that mode. If

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62 DATA ANALYSIS WITH SPSS SOFTWARE

two or more different data values occur more than once and the same number of occurrences for each data value, then SPSS will give the smaller mode and put a note on the bottom saying multiple modes exist. In these cases the statistician needs to look at the data to see if the mode is significant.

Example 6 Find the modes for the following data sets, chpt3example6.sav.

6a. 11, 13, 12, 10, 10, 15, 17, 11, 12, 15, 15, 13, 18, 11, 15 6b. 11, 13, 12, 10, 10, 15, 17, 11, 12, 15, 15, 13, 18, 11, 15, 11 6c. 11, 13, 12, 10, 15, 17, 18

Solution

We can see from the data sets that a has a mode of 15 (15 occurs four times and no other data value occurs four times), b has two modes 11 and 15 (now both 11 and 15 occur four times) and c has no mode (because all data values only occur once). With SPSS use Analyze � Descriptive Statistics � Frequencies, move all three variables into the variable window, click on the Statistics tab, check the Mode box, click Continue, and hit OK. You should get the following output (Figure 5.5)). SPSS gave a mode for all the variables a, b, and c, but with b and c there’s a footnote attached that says, “a. Multiple modes exist. The smallest value is shown.”

Statistics

a b c N Valid 15 16 7

Missing 2 1 10

Mode 15.00 11.00a 10.00a

a. Multiple modes exist. The smallest value is shown

Figure 5.5

The mode is not used often for giving the central value of a data set, but if the data set has a mode, it is a number that will stand out in the data set because it the data value that occurs most often. It can be used as a quick measure of the average value without doing any ranking or any summing of the data values. When the data are categorical, nominal data, then the mode is the only measure of any significance. Categorical data can’t have a mean or median because they can’t be summed or ranked.

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UNDERSTANDING CENTRAL TENDENCIES 63

Example 7 Find the mode for the data set PartyMembershipsUS- sen2012(1).

Solution

Open the data set PartyMembershipsUSsen2012.sav. Find the sum for the three different parties. Use Analyze � Descriptive Statistics � Fre- quencies, move all three variables, Democrats, Republicans, and Inde- pendents into the variable window. Click on Statistics, check the Sum box, and hit Continue and OK. The output is given in Figure 5.6. We see the Democrats have a sum of 51, the Republicans sum is 47, and the Independents sum is 2. The mode is Democrats. (Modes are data values, not frequency values.) In 2012, the senate had more Democrats than any other party, so the “average” senator was a Democrat. (The number 50 in the top row means all 50 states are accounted for in each party.)

Statistics

Democrats Republicans Independents N Valid 50 50 50

Missing 0 0 0

Sum 51 47 2

Figure 5.6

Mean, Median and Mode of Weighted Data

Weighted data means each data value has a weight, or frequency attached to it. For example the data set 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6 can be tabulated as such:

x value 1 2 3 4 5 6

Frequency 4 2 3 2 3 6

Thus the given x values are weighted by the frequency values, so to find the sum we multiply the x values by the frequency. Thus the sum of the x values is 1 × 4 + 2 × 2 + 3 × 3 + 4 × 2 + 5 × 3 + 6 × 6 = 76. The frequency values are given because 1 occurs four times, hence the frequency is 4; 2 occurs twice, so the frequency is 2; 3, three times; 4,

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64 DATA ANALYSIS WITH SPSS SOFTWARE

twice; 5, three times; and 6 occurs six times. To find the mean, median, and mode we need to weight cases as will be shown in the next example.

Example 8 Find the mean, median, and mode of the weighted data set:

X value 1 2 3 4 5 6

Frequency 4 2 3 2 3 6

Solution

Enter the values into two columns as shown in Figure 5.7 or open the data set chpt3example7.sav. In the tool bar, click on the tab labeled Data and scroll down to weight cases. Click on that. A new window will appear, see Figure 5.8 check the circle next to “Weight cases by.” The Frequency varia- ble window will now be active, so move the freq variable into that window. Click on freq, then the move � button. Hit OK. SPSS has now attached all variables in the data editor to the freq variable. To find the mean, medi- an, and mode use Analyze � Descriptive Statistics � Frequencies, move the x variable into the variable window, click on Statistics, check the boxes Mean, Median, and Mode, and hit Continue and OK. The mean median and mode are given in Figure 5.9. To avoid clutter in the output always deselect Display frequency tables unless they’re needed.

Figure 5.7

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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UNDERSTANDING CENTRAL TENDENCIES 65

Figure 5.8

Statistics x

N Valid 20

Missing 0

Mean 3.80

Median 4.00

Mode 6

Figure 5.9

You may recall that all the descriptive values were given in the descriptive output under Analyze � Descriptive Statistics � Explore functions, so we don’t need to use Analyze � Descriptive Statistics � Frequencies if we want an overview of the data set; Explore will cover everything for us. However, if we want to find the sum or just one particular statistic, we can use the above procedure.

Quartile

In Explore we also got a chart called a box plot. The old texts used to call it the box-and-whiskers plot, see Figure 5.10, because the middle is a box and the two ends are straight lines called the whiskers. (This is the box plot for Example 9.) The box represents the middle 50% of the data, and the two lines at the end represent the minimum and maxi- mum values of the data. The graph is divided into four sections called

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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66 DATA ANALYSIS WITH SPSS SOFTWARE

quartiles, which are divided as such: Quartile 1, Q1 = 25, the 25th percentile, 25 % of the data values are less than or equal to Q1; Quartile 2, Q2 = 50, the 50th percentile, 50% of the data values are less than or equal to Q2 (Q2 is the median); and Quartile 3, Q3 = 75, the 75th percentile, 75% of the data values are less than or equal to Q3.

Figure 5.10

Example 9 Input the numbers from 1 to 100 in an SPSS data editor and find the Quartile and box plot. Chpt3example9.

Solution

To find the quartile use Analyze � Descriptive Statistics � Frequen- cies, move the variable of interest into the variable window, click on the Statistics tab, and check the box marked Quartiles; see Figure 5.11. The output will be as in Figure 5.12. To find the quartiles without the computer, just find the medians of the first half and the second half of the data. That is, treat the first half as a set of data and use the methods described above to find the median, which will be Q1. Likewise, treat the second half as a set of data and find the median, which will be Q3. Note: SPSS uses a different formula to find the quar- tiles, but the method described above works for most cases. Also the boxplots drawn by SPSS don’t always align with the quartiles.

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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UNDERSTANDING CENTRAL TENDENCIES 67

SPSS method:

Q1 = (1−a)xk + axk+1 where [ ]1 ( 1)mod 4 4

a n= + and k equals the

integer part of ( 1)

; 4

n +

Q3 = (1 − a)xk + axk+1 where [ ]1 (3 3)mod 4 4

a n= + and k equals the

integer part of (3 3)

, 4

n n

+ is the number of data values.

Figure 5.11

Statistics

Numbers from 1 to 100

N Valid 100

Missing 0

Percentiles 25 25.250

50 50.500

75 75.750

Figure 5.12

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68 DATA ANALYSIS WITH SPSS SOFTWARE

The interquartile range is the difference between the third quartile and the first quartile, which is IInterquartile Range = Q3 − Q1.

Box Plot

A box plot (also called a box-and-whiskers plot) illustrates the first three quartiles and the middle 50% of the data; see Figure 5a. Since the first quartile is the 25th percentile, then 25% of the data lie at or below that value. The third quartile is the 75th percentile, so 75% of the data lie at or below that value. Thus 50% of the data lie between Q1 and Q3. This is represented by the box in the box plot. The whiskers are the 25% of the data that lie at or below Q1 and at or above Q3. As we expect the mean to lie somewhere in the middle, the box plot gives a quick range of values for the mean. Input the quartiles into the boxplot by using Options � Text Box in the Chart Editor. The textbox can be moved around inside the chart editor using the cursor.

Example 10 Use the data set chpt3example4.sav to draw two box plots, and find the quartile and interquartile range. Show that the answers are almost the same by finding the medians of the first half and the second half.

Solution

Let’s first draw the box plots. Use Graphs � Legacy Dialog � Boxplot. Check the box marked Summaries of separate variables and leave Simple as default. Click on Define. Move the two variables a and b into the win- dow Boxes Represent and click OK. The graphs should be as in Figure 5.13. To find the quartiles use Analyze � Descriptive Statistics � Fre- quencies, put a and b in the Variable window, and click on Statistics. Check the box Quartiles, then Continue and OK. The output should be as in Figure 5.14.

For variable a, the minimum value is 1, the first quartile Q1 is 2.5 (the average of 2 and 3, the second and third data values), the median of Q2 is 4, and the third quartile Q3 is 8.5 (the average of 8 and 9, the sev- enth and eighth data values). The interquartile range is 6, the difference of 8.5 and 2.5. The maximum value is 11.

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UNDERSTANDING CENTRAL TENDENCIES 69

For variable b, the minimum value is 1, the first quartile, Q1 is 2.75 (the first quartile by the method described above would be 3), the median of Q2 = 5, and the third quartile Q3 is 6.25 (6 by the method described above, that is, find the median of the second half). The inter- quartile range is 3.5 (6.25 – 2.75) but we can use 3, which is the differ- ence between 6 and 3. The maximum value is 8.

We can find the interquartile range without evaluating Q3 − Q1. Use Analyze � Descriptive Statistics � Explore, click on Statistics, and check the box Descriptives. (It should already be checked as it is the default case.) The output is given in Figure 5.15. The interquartile range is highlighted.

Figure 5.13

Statistics

a b N Valid 9 10

Missing 1 0

Percentiles 25 2.5000 2.7500

50 4.0000 5.0000

75 8.5000 6.2500

Figure 5.14

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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70 DATA ANALYSIS WITH SPSS SOFTWARE

Descriptives

Statistic Std. Error a Mean 5.2222 1.15202

95% Confidence Interval for Mean Lower Bound

2.5656

Upper Bound 7.8788

5% Trimmed Mean 5.1358

Median 4.0000

Variance 11.944

Std. Deviation 3.45607

Minimum 1.00

Maximum 11.00

Range 10.00

Interquartile Range 6.00

Skewness .530 .717

Kurtosis −1.089 1.400

b Mean 4.4444 .72860

95% Confidence Interval for Mean Lower Bound

2.7643

Upper Bound 6.1246

5% Trimmed Mean 4.4383

Median 5.0000

Variance 4.778

Std. Deviation 2.18581

Minimum 1.00

Maximum 8.00

Range 7.00

Interquartile Range 3.50

Skewness −.089 .717

Kurtosis −.434 1.400

Figure 5.15

Because this text is using SPSS for all the concepts, we’ll use the quartile found by SPSS and not concern ourselves with finding the quartile by hand. The interested reader can look up the definition for modular arithmetic (basically a mod b is the remainder when a is divid- ed by b) and check if the above formulas for SPSS are indeed accurate. (They are, we checked.)

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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UNDERSTANDING CENTRAL TENDENCIES 71

Percentiles

The formulas for percentiles are also quite involved so we’ll skip them and simply use the SPSS output to find percentiles. Percentiles have the same definition as quartiles; for example the 30th percentile is that value (which may or may not be a data value) such that 30 percent of the data values in the data set are at or below that value. As a short reference to find percentiles without SPSS, simply multiply the total number of data values by the percentile (as a decimal value, not a percent) and find that data value in the ranked data set.

Example 11 Open the data set StateSATscores1990.sav. Find the 10, 20, 30, …, 90, 100 percentiles for both the math and verbal scores. (2)

Solution

Use Analyze � Descriptive Statistics � Frequencies, move both variables into the variable window, and click on the tab Statistics. In the Frequen- cies: Statistics window, check the box Cut points for 10 equal groups (see Figure 5.16). If you need to add other percentiles (such as 68 and 75), then put a check mark in the Percentiles box and add the Percentile you want by putting the number in the box next to the Percentile(s) line and click Add; see Figure 5.16 After you added all the percentiles of interest, click Continue and OK. Your output should look like Figure 5.17.

Figure 5.16

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72 DATA ANALYSIS WITH SPSS SOFTWARE

Statistics

Mean SAT I Verbal by State

Mean SAT I Math by State

N Valid 51 51

Missing 0 0

Percentiles 10 408.2000 459.4000

20 418.4000 466.0000

30 425.8000 472.2000

40 433.8000 483.4000

50 443.0000 490.0000

60 459.2000 513.2000

68 470.0000 519.0000

70 471.2000 519.8000

75 476.0000 523.0000

80 477.0000 526.2000

90 490.4000 545.4000

Figure 5.17

Davis, S., & Davis, E. (2015). Data analysis with spss software : Data types, graphs, and measurement tendencies. Momentum Press. Created from ncent-ebooks on 2022-06-17 19:10:03.

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