Chapter 2 Organizing the Data

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Presentation transcript:

Chapter 2 Organizing the Data

Introduction Learn how to show variable relationship through diagrams Thematically cover graphs and maps Understand the importance of using appropriate data in representing variables Become comfortable in applying graphical representation within various types of analyses (e.g., bivariate and multivariate)

Frequency Distributions of Nominal Data Formulas and statistical techniques used by social researchers to: Organize raw data Test hypotheses Raw data is often difficult to synthesize Most common types of distributions are: Frequency Percentage Combination

Conventions for Building Tables Title of distribution explains contents Variable(s) in table Shows data distribution Use column headings Relevant columns are totaled Footnotes added if needed

Nominal Data and Distributions Frequency distribution of nominal data consists of two columns: Left column has characteristics (e.g., Response of Child) Right column has frequency (f) Responses of Young Boys to Removal of Toy Response of Child f Cry 25 Express Anger 15 Withdraw 5 Ply with another toy N=50

Comparing Distributions Comparisons clarify and add information Response to Removal of Toy by Gender of Child Gender of Child Response of Child Male Female Cry 25 14 Express Anger 15 1 Withdraw 5 2 Play with another toy 8 Total 50

Proportions and Percentages Proportions - Compares the number of cases in a given category with the total size of the distribution Most prefer percentages to show relative size. Percentage – The frequency per 100 cases Formula for proportion Formula for percentage

Illustration: Gender of Students Majoring in CJ(f) Criminal Justice Majors Gender College A College B Male 879 119 Female 473 64 Total 1,352 183

Illustration: Gender of Students Majoring in CJ (f and %) Criminal Justice Majors College A College B Gender f % Male 879 65 119 Female 473 35 64 Total 1,352 100 183

Rates Rates usually preferred by social researchers Rate – comparison between actual and potential cases Base terms in rates may vary

Some Common Rate Calculations Suppose 500 births occur among 4,000 women of childbearing age. This would be a rate of 125 live births for every 1,000 women of childbearing age. Suppose 562 suicides occur in a state with 4.6 million residents. The suicide rate would be 12.2 suicides per 100,000 residents.

Rate of Change Compare the same population at two points in time time 2f – time1f time 1f Year Theft Rate1 % Change 2005 120.3 2006 127.4 5.9% 2007 116.8 -8.3% 2008 107.4 -8.0% 2009 98.7 -8.1% 2010 94.6 -4.2% (100)* A negative sign signifies a reduction A positive sign signifies an increase 1Source: National Crime Victimization Survey

Ordinal/Interval Data and Distributions Attitudes Toward Televised Trials F Slightly Favorable 9 Somewhat Unfavorable 7 Strongly Favorable 10 Slightly Unfavorable 6 Strongly Unfavorable 12 Somewhat Favorable 21 Total 65 Incorrect Attitudes Toward Televised Trials F Strongly Favorable 10 Somewhat Favorable 21 Slightly Favorable 9 Slightly Unfavorable 6 Somewhat Unfavorable 7 Strongly Unfavorable 12 Total 65 Correct Must go from highest (at the top) to lowest (at the bottom).

Frequency Distribution of Final-Examination Grades for 71 Students 99 85 2 71 4 57 98 1 84 70 9 56 97 83 69 3 55 96 82 68 5 54 95 81 67 53 94 80 66 52 93 79 8 65 51 92 78 64 50 91 77 63 N = 71 90 76 62 89 75 61 88 74 60 87 73 59 86 72 58 Hard to see any patterns.

Grouped Frequency Distributions of Interval Data Grouped frequency distribution used to clarify presentation of data. Categories or groups referred to a class intervals Class interval size determined by the number of values

Grouped Frequency Distributions of Interval Data Grouped Frequency Distribution of Final-Examination Grades for 71 Students Class Interval f % 95-99 3 4.23 90-94 2 2.82 85-89 4 5.63 80-84 7 9.86 75-79 12 16.90 70-74 17 23.94 65-69 60-64 5 7.04 55-59 50-54 71 100 Results are more easily shown/displayed

Constructing Class Intervals Categories must be mutually exclusive and exhaustive Designed to reveal or emphasize patterns Possible to have too few or too many groups – blurs the data Class intervals have a midpoint Dealing with decimal data M exclusive – Every case can only be placed in one, and only one, category M exhaustive – All cases are able to be put into a category

Flexible Class Intervals Income Category F % $100,000 and above 16,886 21.9 $75,000-$99,999 10,471 13.5 $50,000-$74,000 15,754 20.3 $40,000-$49,999 7488 9.7 $30,000-$39,999 7996 10.3 $20,000-$29,999 8169 10.6 $15,000-$19,999 3709 4.8 $10,000-$14,999 2890 3.7 $5000-$9999 2024 2.6 Under $5000 2031 N = 77688

Cumulative Distributions Cumulative frequencies involve the total number of cases having a given score or a score that is lower Cumulative frequency shown as cf cf obtained by the sum of frequencies in that category plus all lower category frequencies Cumulative percentage – percentage of cases having any score or a lower score Only find CF/C% IF data is at least ordinal. Cannot do it for nominal data.

Grouped Frequency Distributions of Interval Data Grouped Frequency Distribution of Final-Examination Grades for 71 Students Class Interval f % 95-99 3 4.23 90-94 2 2.82 85-89 4 5.63 80-84 7 9.86 75-79 12 16.90 70-74 17 23.94 65-69 60-64 5 7.04 55-59 50-54 71 100 Results are more easily shown/displayed

Grouped Frequency Distributions of Interval Data Grouped Frequency Distribution of Final-Examination Grades for 71 Students Class Interval f Cf % C% 95-99 3 71 4.23 100 90-94 2 68 2.82 95.76 85-89 4 66 5.63 92.94 80-84 7 62 9.86 87.31 75-79 12 55 16.90 77.45 70-74 17 43 23.94 60.55 65-69 26 36.31 60-64 5 14 7.04 19.71 55-59 9 12.67 50-54 Results are more easily shown/displayed

Cross-Tabulations Frequency distributions are limited Sometimes we want to know how is one variable (usually the dependent variable) distributed across another (usually the independent variable) Cross-tabulations meet this need as they allow us to consider two or more dimensions of data.

Cross-tab Cross-Tabulation of Seat Belt Use by Gender Frequency Distribution of Seat Belt Use Use of Seat Belts f % All the time 499 50.1 Most of the time 176 17.7 Some of the time 124 12.4 Seldom 83 8.3 Never 115 11.5 Total 997 100 Cross-Tabulation of Seat Belt Use by Gender Gender of Respondents Use of Seat Belts Male Female Total All the time 144 355 499 Most of the time 66 110 176 Some of the time 58 124 Seldom 39 44 83 Never 60 55 115 367 630 997

What Type to Choose? There are three sets of percentages Total Row Column All are correct, mathematically speaking Total percentages may be misleading Row and column percentages come down to which is more relevant to the purpose of the analysis

Cross-tab Formulas Formula for total percents Formula for row percents Formula for column percents

Victim-Offender Relationship Cross Tabulations – Victim-Offender Relationship by Gender of Victim for Homicides in US for 2005 (With Row%) Victim-Offender Relationship Gender Intimate Intimate % Family Family % Other Other % Total Total % Male 617 1,310 11,235 13,161 Female 1,470 639 1,421 3,531 2,087 1,949 12,656 16,692

Victim-Offender Relationship Cross Tabulations – Victim-Offender Relationship by Gender of Victim for Homicides in US for 2005 (With Row%) Victim-Offender Relationship Gender Intimate Intimate % Family Family % Other Other % Total Total % Male 617 4.7% 1,310 10.0% 11,235 85.4% 13,161 100% Female 1,470 41.6% 639 18.1% 1,421 40.2% 3,531 2,087 12.5% 1,949 11.7% 12,656 75.8% 16,692

Cross Tabulations – Victim-Offender Relationship by Gender of Victim for Homicides in US for 2005 (With Column%) Victim-Offender Relationship Male Female Total Intimate 617 1,470 2,087 Family 1,310 639 1,949 Acquaintance 7,237 998 8,235 Stranger 3,998 423 4,421 13,161 3,531 16,692

Cross Tabulations – Victim-Offender Relationship by Gender of Victim for Homicides in US for 2005 (With Column%) Victim-Offender Relationship Male Female Total Intimate 617 1,470 2,087 4.7% 41.6% 12.5% Family 1,310 639 1,949 10.0% 18.1% 11.7% Acquaintance 7,237 998 8,235 55.0% 28.3% 49.3% Stranger 3,998 423 4,421 30.4% 12.0% 26.5% 13,161 3,531 16,692 100%

Graphic Presentations Graphs are useful tools to emphasize certain aspects of data. Many prefer graphs to tables. Types of graphs include: Pie charts, bar graphs, frequency polygons, line charts, and maps

Pie Charts Pie chart – a circular chart whose pieces add up to 100%. Especially good for nominal data. Possible to highlight or “explode” certain pieces for emphasis

Exploded Pie Chart

Bar Graphs and Histograms Represent frequency distribution plot of: Categories/variables on one axis Responses as bars on another axis Bar length represents category frequency Bar graphs used primarily for discrete variables Histograms used to show continuity along a scale

Bar Graph

Histogram of Distribution of Children in Little Rock Community Survey Y axis represents the percentage of number of respondents X axis represnts number of children of respondents ie, 35% of respondents have 0 or 1 children, 15% have 2 or 3 children, etc

Frequency Polygons Best suited to emphasize continuity rather than differences Frequency distribution of a single variable Used for: Continuous data Interval data Ratio data Continuous data = count data, # of arrests. Typically, has no limit

Frequency Polygon Example Commonly done with homicide rates, etc

Line Charts Generally show change (trends) temporally Show trends in: One variable Plotting two or more variables Similar to polygons, but not enclosed on the right margin

Number of Adolescents (< 18 y/o) Using for the First Time by Month

Maps Growing in popularity due to geo-coding and geo-mapping Unparalleled method for exploring geographical patterns in data For instance, a map of the U.S. Helps show which area has more or less points

Example of mapping within criminal justice research

Shape of a Distribution Kurtosis Leptokurtic Platykurtic Mesokurtic Skewness Negative Positive Normal Curve

Kurtosis Leptokurtic Platykurtic Mesokurtic Some Variation in Kurtosis among Symmetrical Distributions

Skewness Negatively skewed Positively skewed Symmetrical (Normal) Three Distributions Representing Direction of Skewness

Summary Organizing raw data is critical Data can be summarized using frequency distributions. Comparisons of groups possible through proportions, percentages and rates. Cross-tabs allow dimensional (and more) analysis Graphic presentations: help to emphasize findings make data more accessible to consumers of research help researchers identify trends