Research Methods Chapter 8 Data Analysis

Two Types of Statistics Descriptive –Allows you to describe relationships between variables Inferential –Allows one to test hypotheses & see if results are generalizable

Descriptive Statistics Often begins with univariate analysis –Displays the variation of a variable –Several ways to display variation Bar Chart, Frequency Polygram, Histogram, etc.

Rates of Church Affiliation, U.S., Percent of Church Membership Year

Frequency Polygon

–3 features of the shape of variation are important: Central Tendency: The most common value or the value around which cases tend to center around –a.k.a averages like mean, median, mode Variability: the degree to which cases are spread out or clustered together Skewness –The extent to which cases are clustered more at one or the other end of a distribution »Can be either non, positive, or negative

Negative Skew: Test to Easy Freq. 0 Score100

Positive Skew: Test to Hard Freq. 0 Score100

Frequency Distribution of Voting in 1992 Presidential Election ValueFrequencyValid Percent Voted 1, % Did not vote Not eligible Refused Don’t know No answer Total 2, %

Ungroup and Grouped Age Distributions Ungrouped Grouped AgePercentAge Percent 180.2% And so on…...

Calculating The Mean X = The Sum of Scores / # of Scores So if you had the following test scores (5, 10, 15, 10, 5, 10, 5, 15, 15, 10) What would be the mean? Answer: 10! (100/10)

Calculating the Mode Mode = The most frequent value in a distribution So if you had the following test scores: (10, 5, 10, 15, 10, 10, 5, 10, 5, 15, 15, 10) What would be the mean? Answer: 10! (There are more 10’s than any other number)

Calculating the Median Median = The value in the middle of a distribution Example: (22, 25, 34, 35, 41, 41, 46, 46, 46, 47, 49, 54, 54, 59, 60) Several Steps to calculate the Median –Arrange all observations in order of size, from smallest to largest –Determine the number of values in the distribution (N) N in this case = 15

–Plug N into the following formula (N+1)/2 = (15+1)/2 = 16/2= 8 –If you get a whole number (in this case you got an “8”) then count up that number in the distribution (22, 25, 34, 35, 41, 41, 46, 46, 46, 47, 49, 54, 54, 59, 60) Thus, the median is “46”

If you don’t get a whole number then you have to add a step Example: 8, 13, 14, 16, 23, 26, 28, 33, 39, 61 Find the N (In this case, the N is “10” (N + 1)/2 = (10+1)/2 = 5.5. Thus, counting up 5.5 gets you to the point between “23” & “26” The extra step…. (N1 + N2)/2 = ( )/2 = 49/2 = 24.5 Thus, the Median in this case is 24.5

Determine the Mean, Median and Mode 2, 2, 2, 2, 2 1,2,2,2,5,5,10,10,15,25 17, 18, 9, 9, 5 7, 7, 14, 3, 11, 27, , 67, 43, 2, 2, 2, 6

Answers 2, 2, 2, 2, 2 –Mean = 10/5 = 2 –Median =(5 + 1)/2 = 6/2 = 3 Then: count up 3 spaces to get to “2” –Mode = 2 1,2,2,2,5,5,10,10,15,25 –Mean = 77/10 = 7.7 –Median = (10 + 1)/2 = 11/2 =5.5 Then: (5 + 5)/2 = 10/2= 5 –Mode = 2

17, 18, 9, 9, 5 –Mean = 58/5 = 11.6 –Median = (5 + 1)/2 = 3 Then: = 9 –Mode = 9 7, 7, 14, 3, 11, 27, 498 –Mean = 567/7 = 81 –Median =(7 + 1)/2 = 4 Then: = 11 –Mode = 7 11, 67, 43, 2, 2, 2, 6 –Mean = 133/7 = 19 –Median = (7 + 1)/2 = 4 Then: = 6 –Mode = 2

Suppose You Had the Following 1 person making $45,000 1 person making $15,000 2 People making $10,000 1 Person making $5,700 3 people making $5,000 4 people making $3,700 1 person making $3, people making $2,000

What did you Get? Mean = –$142,500 / 25 = $5,700 Median = –$3,000 (there are 12 above you and 12 below you Mode = –$2,000 (occurs the most frequently)

Mean Vs. Median Vs. Mode Generally use the mean for interval or ratio levels of measurement –E.g. Fahrenheit temperatures, Age, Income Look at shape of distribution first, however –If there are lot’s of outliers, the median might be preferable Income if including Bill Gates Use the mode for nominal levels of measurement –Gender

Measures of Variation Central tendency (mean, median, mode) although valuable, only shows us a small piece of the picture –Relying only on central tendency may give us an incomplete and misleading picture Three towns may have the same mean and median income but be very different in social character –One may be mostly middle class with a few rich and many poor –One may have an euqal number of rich, middle class, & poor Looking at measures of variation can help us see past the limitations of central tendency

The Four Popular Measures of Variation 1Range –Calculated by taking the highest value in a distribution and subtracting the lowest value, and then adding 1 –Shows us the range of possible values that may be encountered –Weakness: The range can be drastically altered by just one exceptionally high or low value (known as an “outlier”).

2Interquartile Range –Avoids the problem created by outliers –Quartiles are the points in a distribution corresponding to the first 25%, the first 50%, and the first 75% of the cases. The second quartile (50%) is the median 3Variance –The average of the squared deviations from the mean

Variance __ __ XX-X (X - X)2 X Total __ X = 9

4Standard Deviation –Gives an “average distance” between all scores and the mean –Calculated by squaring the variance

Crosstabulation Family Income $17,500-$35,000- Voting<$17,500$34,999$59,999$60,000+ Voted 60% 73% 75% 84% Did not 40% 27% 25% 16% Total 100% 100% 100% 100% (n) (424) (550) (541)(433)

Crosstabulating Variables Crosstabulations reveal 4 aspects of the association between 2 variables: –Existence: is there a correlation? –Strength: How strong does the correlation appear to be? –Direction: Positive or negative correlation? –Pattern: Are changes in the percentage distribution of the dependent variable fairly regular (simply increasing or decreasing), or do they vary?

Evaluating Association Inferential Stats are used to determine the likelihood that an association exists in the larger pop. From which the sample is drawn Thus, researchers often calculate probability levels that determine the probability of chance –E.g. p<.05 means that the probability that the association is due to chance is less than 5 out of 100, or 5% Generally looking for at least.05, but some want.01 or.001

Controlling for a Third Variable Associations, however, do not necessary mean causation Use elaboration analysis to determine whether an association is due to a causal relationship or to another variable Three types…. Intervening, extraneous, and specification...

Intervening Variables Income Perceived Efficacy Voting

Extraneous Variables IncomeVoting Education

Findings The 3 criteria –Time Order Asked the following questions: – How long have they been attended church? Used only those who had attended for over a year or more – Eight questions about their deviant acts WITHIN THE PAST YEAR!! –CorrelationCorrelation The data indicated a correlation between the two variables (church attendance and delinquency) –Spuriousness Could another variable be the determining factor for delinquency instead of church attendance? (Elaboration Analysis) – Race Race – School School – Grade – Gender Gender

Church Attendance And Delinquency Church Attendance FrequentInfrequent Delinquent 22% 38% Not Delinquent 78% 62% % 100%

Control for Race Church Attendance Frequent Infrequent Whites Delinquent Not Delinquent % 100% African Americans Delinquent Not Delinquent % 100%

Control for School Church Attendance Frequent Infrequent High School #1 Delinquent Not Delinquent % 100% High School #2 Delinquent Not Delinquent % 100%

Control for Sex Church Attendance Frequent Infrequent Boys Delinquent Not Delinquent % 100% Girls Delinquent Not Delinquent % 100%

Findings The hypothesis was not supported! The correlation between church attendance and delinquency is spurious –The third variable of gender appears to be an extraneous variable