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Quantitativeanalysis

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Introduction to Quantitative Analysis

Introduction to Quantitative Analysis

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  • 1. What is a Variable? • any entity that can take on different values • not always 'quantitative' or numerical, but we can assign numerical values • attribute = a specific value of a variable Examples: • gender: 1=female; 2=male • attitudes: 1 = strongly disagree; 2 = disagree; 3 = neutral; 4 = agree; 5 = strongly agree
  • 2. Coding in a data matrix Variables Cases Gender Age Political Orientation Social class Person 1 Male 36 years Progressive Working Person 2 Male 19 years Moderate Lower middle Person 3 Female 30 years Progressive Upper working Person 4 Male 55 years Traditionalist Upper middle Person 5 Female 42 years Traditionalist Middle
  • 3. Gender: Male = 1; Female=2 Political Orientation: Traditionalist=1; Moderate=2; Progressive=3 Social Class: Working=1; Upper working=2; Lower middle=3; Middle=4; Upper middle=5 Variables Cases Gender Age Political Orientation Social class Person 1 1 36 3 1 Person 2 1 19 2 3 Person 3 2 30 3 2 Person 4 1 55 1 5 Person 5 2 42 1 4 Coding in a data matrix
  • 4. Levels of Measurement • different kinds of variables (1) Nominal (2) Ordinal (3) Interval and Ratio
  • 5. Nominal Variable • used to classify things • represents equivalence (=) • adding, subtracting, multiplying or dividing nominal numbers is meaningless • tells you how many categories there are in the scheme
  • 6. Ordinal Variable • ordering or ranking of the variable • the relationship between numbered items • ‘higher’, ‘lower’, ‘easier’, ‘faster’, ‘more often’ • equivalence (=) and relative size (greater than) and < (less than)
  • 7. Interval (and Ratio) Variable • All arithmetical operations are allowed • intervals between each step are of equal size • Examples: - length, weight, elapsed time, speed, temperature
  • 8. Women’s Shoe Sizes British 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 European 34 35 35.5 36 37 37.5 38 38.5 39 39.5 40 41 42 American 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 Japanese (cm) 21.5 22 22.5 23 23 23.5 24 24 24.5 25 25.5 26 26.5
  • 9. Levels of measurement Level are names have an inherent order from more to less or higher to lower are numbers with equal intervals between them Nominal level  Ordinal level   Interval level   
  • 10. Frequency distributions • count number of occurrences that fall into each category of each variable • allow you to compare information between groups of individuals • also allow you to see what are the highest and lowest values and the value at which most scores cluster • variables of any level of measurement can be displayed in a frequency table
  • 11. Frequency table Subject studied Frequency of boys Total Science LLLLLLLLLLLLLLLLLLLLLLLLLL 26 Arts/Social sciences JJJJJJJJJJJJJJJJJ 17 Total 43 Subject studied Frequency of girls Total Science LLLLLLLLLLLLLLLLLLLLLLL 23 Arts/Social sciences JJJJJJJJJJJJJJJJJJJJJJJJJJ JJJJJJJJJJJJJJJJJJ 44 Total 67
  • 12. Percentages • number of cases belonging to particular category divided by the total number of cases and multiplied by 100. • the total of percentages in any particular group equals 100 per cent. A-level subject Boys Girls Frequency (f) Percentages Frequency (f) Percentages Science 26 5.60100 43 26 =· 23 3.34100 67 23 =· Arts/Social sciences 17 5.39100 43 17 =· 44 7.65100 67 44 =· Total 43 100 67 100 100%  N f
  • 13. Graphical presentation • Pie charts • Barcharts • Line graphs • Histograms
  • 14. Pie chart • illustrates the frequency (or percentage) of each individual category of a variable relative to the total. • Pie charts are not appropriate for displaying quantitative data. Gender of Sociology Students 81% 19% Female Male Sociology Students Female 26 Male 6
  • 15. 15 Barcharts • the height of the bar is proportional to the category of the variable - easy to compare • used for Nominal or Ordinal level variables (or discrete interval/ratio level variables with relatively few categories) Marital Status f % Adjusted % Married 140 28 33 Living as married 60 12 14 Single 85 17 20 Divorced 75 15 18 Separated 30 6 7 Widowed 35 7 8 Missing 75 15 - Totals 500 100% 100% Marital Status 140 60 85 75 30 35 75 0 20 40 60 80 100 120 140 160 Married Living as married Single Divorced Separated Widow ed Missing
  • 16. Multiple barchart Marital Status 1995 2000 Married 140 122 Living as married 60 67 Single 85 99 Divorced 75 85 Separated 30 30 Widowed 35 20 Totals 425 423 Marital status 0 20 40 60 80 100 120 140 160 Married Living as married Single Divorced Separated Widow ed Frequencies 1995 2000
  • 17. Compound or Component barchart Sociology Students 2001 2002 Female 41 42 Male 26 39 Total 67 81 SociologyStudents 41 42 26 39 0 20 40 60 80 100 2001 2002Year Frequency Male Female
  • 18. Line graphs • interval/ratio level variables that are discrete • need to arrange the values in order Beer production year 000 of gallons/ per week 1991 154 1992 167 1993 132 1994 145 1995 154 1996 145 1997 113 1998 156 1999 154 2000 123 2001 144 YEAR 20012000199919981997199619951994199319921991 ValuePRODUCT 170 160 150 140 130 120 110
  • 19. Histograms • represents continuous quantitative data • The height of the bars corresponds to the frequency or percentage of cases in the class. • The width of the bars represents the size of the intervals of the variable • The horizontal axis is marked out using the mid points of class intervals
  • 20. Example: Histogram MARKS 85.0 80.0 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 30 20 10 0 Std. Dev = 11.83 Mean = 55.4 N = 100.00
  • 21. Graphs have the capacity to distort YEAR 20012000199919981997199619951994199319921991 170 160 150 140 130 120 110 YEAR 200120001999199819971996199519941993199219911990 ValuePRODUCT 200 100 0
  • 22. Measures of Central Tendency • describe sets of numbers briefly, yet accurately • describe groups of numbers by means of other, but fewer numbers • Three main measures: • mean • median • mode
  • 23. The Mean X = Xå n • most common type of average that is computed. · The letter X with a line above it (also called “X bar”) is the mean value of the group of scores or the mean. · The Greek letter sigma is the summation sign, which tells you to add together whatever follows it. · The X is each individual score in the group of scores. · The n is the size of the sample from which you are computing the mean.
  • 24. When to use the Mean • When values in a particular group cluster closely around a central value, the mean is a good way of indicating the ‘typical’ score, i.e. it is truly representative of the numbers. • If the values are very widely spread, are very unevenly distributed, or clustered around extreme values, than the mean can be misleading, and other measures of central tendency should be used instead.
  • 25. The Median • Also an average, but of different kind. • It is defined as the midpoint in a set of scores. It is the point at which one-half, or 50% of the scores fall above and one-half, or 50%, fell below. • Computing the Median: (1) List the scores in order, either from highest to lowest or lowest to highest. (2) Find the middle score. That’s the median.
  • 26. The Median: Pros and Cons • time-consuming • if one of the numbers near the middle of the distribution moves even slightly, than the median would alter, unlike the mean, which is relatively unaffected by a change in one of the central numbers • if one of the extreme values changes, than the median remains unaltered. - 2, 80, 100, 120, 130, 140, 160, 200, 3150 • single scores which are quite clearly ‘deviant’ when compared with others, are known as outliers – 2 and 3150
  • 27. The Mode • the value in any set of scores that occurs most often • example 1: – 5, 6, 7, 8, 8, 8, 9, 10, 10, 12 – the mode = 8 • example 2: – 5, 6, 7, 8, 8, 8, 9, 10, 10, 10, 12 –two modes: 8 and 10 – bimodal • very unstable figure – 1,1,6,7,8,10 – mode = 1 – 1,6,7,8,10,10 – mode = 10
  • 28. When to Use What? • depends on the type of data that you are describing – for nominal data - only the mode – for ordinal data - mode and median – for interval data - all of them • but, for extreme scores - use the median