Econ stat1
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This document discusses key concepts in economic statistics including: 1. The stages of the research process include problem identification, generating hypotheses, conducting research, statistical analysis, and drawing conclusions. 2. Descriptive statistics summarize and describe data while inferential statistics make inferences about a population based on a sample. 3. Data can be qualitative, quantitative, cross-sectional, or time-series. Common descriptive statistics include the mean, median, mode, standard deviation, and range.
Econ stat1
- 1. 1 Economic Statistics University of St. La Salle Bacolod City
- 2. 2 Stages of Research Process: 1.Problem Identification 2.Generating Hypothesis 3.Conducting the Research 4.Statistical Analysis (Descriptive and Inferential Statistics) 5. Drawing Conclusion
- 3. 3 Descriptive statistics Inferential statistics Mean t-test Median Analysis of variance (ANOVA) Mode Correlation Standard deviation Multiple regression Variance Factor analysis Range Discriminant analysis Chi square Repeated measures ANOVA
- 4. 4
- 5. Economic Statistics Statistics -are a collection of theory and methods applied for the purpose of understanding data. -art and science of collecting, analyzing, presenting, and interpreting data.
- 6. Why Study Econometrics? Economic theory makes statement or hypotheses Theories do not provide the necessary measure of strength of relationship (numerical estimate of the relationship) & the proper functional relationship between variables. Example: Law of Demand A reduction in price of a commodity is expected to increase the quantity demanded of that commodity. to provide empirical verification of theories
- 7. 7 Economic Statistics Data, Data Set, Elements, Variables and Observations Data are facts and figures that are collected, analyzed, and summarized for presentation and interpretation. Data set refers to all data collected in particular study. Elements are the entities on which data are collected. Variable is a characteristic of interest for the elements. Observation is a set of measurements obtained for a particular element.
- 8. 8 Economic Statistics Qualitative, Quantitative, Cross-section and time series Data Qualitative data are labels or names used to identity an attribute of each element. Quantitative data are numeric values that indicate how much or how many. Qualitative variable is a variable with qualitative data. Quantitative variable is a variable with quantitative data.
- 9. 9 Economic Statistics Cross-sectional data are data collected at the same or approximately the same point in time. Time series data are data collected over several time periods. Pooled data are data with elements of both cross- sectional and time series data. Panel data are data with the same cross-sectional unit, say, a family or firm, and is surveyed over time.
- 10. 10 Economic Statistics ITEM 1990 PHILIPPINES 9266287 CAR 165585 ILOCOS 847691 CAGAYAN VALLEY 1164758 CENTRAL LUZON 1910930 S. TAGALOG - A 904297 BICOL 686998 WESTERN VISAYAS 886732 CENTRAL VISAYAS 182940 EASTERN VISAYAS 337459 Western Mindanao 350313 NORTHERN MINDANAO 306069 Southern Mindanao 649812 Central Mindanao 443068 ARMM 203718 CARAGA 225917 Cross-sectional Data: Volume of Palay Production (000MT), Philippines, 1990.
- 11. 11 Economic Statistics YEAR REGION QRICE QCORN QSUGAR QCOCO QTOBAC QSPOT QCASVA 74 1 433810 33845 37367 68289 27694 57910 10351 75 1 603625 31285 38350 69646 25600 71097 12708 76 1 529395 19790 46503 73869 33852 88490 15817 77 1 596155 29155 50320 76316 31190 103983 18587 78 1 660195 30335 36143 81293 60845 107114 16640 79 1 676595 32945 38441 71253 68996 103589 16819 80 1 623640 40785 216334 74249 51149 100693 17928 81 1 730140 41600 218270 81387 54547 93099 19414 82 1 867890 54160 224672 85673 61888 90123 17728 83 1 733795 60200 253069 87104 64188 77201 16245 84 1 764640 63440 198391 88771 72296 62721 14633 85 1 840570 69680 189334 91436 56005 66949 12962 86 1 876690 64700 157268 99855 54370 65743 13686 87 1 729222 74500 136842 98082 60814 65053 13782 88 1 897616 78811 33005 88108 54469 66405 13863 89 1 837404 76686 32260 83421 57732 70148 13771 90 1 896563 97379 44446 84335 67617 38584 12891 91 1 947694 106248 31789 87010 65839 44800 13414 92 1 872891 78380 27402 83180 96460 44323 13140 93 1 886319 85992 30323 79650 84982 41443 14664 94 1 966001 121115 30170 78288 45593 44368 15064 95 1 939839 148501 17561 82704 49666 44488 14696 96 1 1052784 176874 15594 102112 50884 38025 14950 97 1 1137998 216994 13623 95459 49283 28326 15304 98 1 908964 232634 0 109484 49058 26772 15494 99 1 1151794 203720 0 102334 39466 27243 15794 00 1 1280443 195929 0 103868 38164 27320 16143 Time Series Data: Volume of Production (000 MT) of Selected Crops, Ilocos Region, Philippines 1974-2000.
- 12. 12 Economic Statistics YEAR REGION QRICE QCORN QSUGAR QCOCO QTOBAC QSPOT QCASVA 74 2 669500.00 255170.00 0.00 12850.98 25251.00 31132.60 1341.49 75 2 779430.00 300220.00 0.00 18883.58 17295.00 38221.79 1646.96 76 2 805030.00 287875.00 0.00 24577.35 14856.40 47572.31 2049.87 77 2 866540.00 330380.00 14488.21 32450.32 12500.00 55901.46 2408.77 78 2 844310.00 336665.00 94404.66 20446.00 12536.00 56975.00 2335.00 79 2 900815.00 338085.00 182568.24 19138.00 12700.00 55920.00 2349.00 80 2 775620.00 194415.00 201052.00 20039.00 13571.00 51298.12 2362.37 81 2 799815.00 278325.00 275132.00 21154.00 13063.00 49318.28 2293.15 82 2 814050.00 259075.00 364464.00 20146.00 13361.00 52396.00 2133.41 83 2 786420.00 230060.00 322158.00 19292.00 10951.00 48100.71 1884.06 84 2 947270.00 301080.00 243165.00 19896.00 11573.00 16789.82 1721.51 85 2 1144755.00 361240.00 231623.00 21103.71 7023.00 17336.10 1975.04 86 2 1158825.00 369180.00 124873.00 21714.92 9963.00 18569.20 2204.20 87 2 1112431.00 409935.00 82786.00 21517.85 11421.00 17938.50 2263.09 88 2 1221445.00 455325.00 66733.00 20192.33 11060.00 18593.50 2200.47 89 2 1206537.00 463414.00 62865.00 19159.40 11557.00 16684.04 2112.56 90 2 1281471.00 566449.00 55459.00 19395.22 6351.00 15684.00 1624.00 91 2 1137064.00 477085.00 68861.00 20490.27 4117.00 15911.00 1395.00 92 2 1198382.00 665865.00 56548.00 21739.50 10066.00 18380.00 1401.00 93 2 1010573.00 459298.00 89208.00 24049.79 9359.00 18140.00 1391.00 94 2 1379936.00 526872.00 93574.00 29282.45 7126.00 20132.00 1643.00 95 2 1489157.00 626654.00 91955.00 23125.95 9320.00 19842.00 1899.00 96 2 1590645.00 486497.00 156503.00 28391.39 9634.00 31565.00 13066.00 97 2 1702006.00 694466.00 241821.00 26691.46 11878.00 33527.00 20116.00 98 2 1224493.00 593341.00 193184.00 30377.32 9040.00 25886.00 16514.00 99 2 1860273.00 1073854.00 190147.00 28475.03 8914.00 30021.00 17987.00 00 2 1968404.00 1001836.00 192020.00 28867.16 7938.00 28161.00 17354.00 Time Series Data: Volume of Production (000 MT) of Selected Crops, Cagayan Valley, Philippines 1974-2000.
- 13. 13 Economic Statistics Scales of Measurement The nominal scale has no mathematical value. It is also called a categorical scale. Numbers are assigned to categories of nominal data/variables to facilitate data processing. An ordinal scale is a measure in which data or categories of a variables are ordered or ranked into two or more levels or degrees, such as from low to high or least to most. An interval scale has the characteristics of an ordinal scale, but in addition, the distance between points in interval scales is equal. A ratio scale is almost like the interval scale, except that the ratio scale has a real zero point.
- 14. 14 Economic Statistics Scale Description Example Nominal Categories do not have mathematical values. One is not higher or lower than the other. Sex: male, female Color: red, white, yellow Civil Status: single, married Ordinal Categories can be ranked. The difference between the first and the second rank is not the same as the difference between the second and the third ranks. Degree of malnutrition: 1st degree, 2nd degree, 3rd degree Honor roll: 1st, 2nd, 3rd Level of anger: not angry, very angry. Interval The data have numerical value. The distance between two points is the same, but there is no zero point or it may be arbitrary. Body temperature in Fahrenheit: 30 degrees, 40 degrees, 50 degrees Business capital (PhP): 1m, 2m, 3m Ratio The same as interval data but the zero point is fixed. No. of children: 0,1,2,3,4 Hrs. spent in studying: 0, 5,10 Descriptions and Examples of the Four Scales of measurement
- 15. 15 Economic Statistics Data Qualitative Data Quantitative Data Tabular Methods Graphical Methods Tabular Methods Graphical Method Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Bar Graph Pie Chart Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Cumulative Frequency Distribution Cumulative Relative Frequency Distribution Cumulative Percent Frequency Distribution Histogram Scatter Diagram
- 16. 16 Economic Statistics Frequency Distribution: Qualitative Data A Frequency Distribution is a tabular summary of data showing the number (frequency) of items in each of several nonoverlapping classes
- 17. 17 Economic Statistics Coke Classic Sprite Pepsi-Cola Diet Coke Coke Classic Coke Classic Pepsi-Cola Diet Coke Coke Classic Diet Coke Coke Classic Coke Classic Coke Classic Diet Coke Pepsi-Cola Coke Classic Coke Classic Dr. Pepper Dr. Pepper Sprite Coke Classic Diet Coke Pepsi-Cola Diet Coke Pepsi-Cola Coke Classic Pepsi-Cola Pepsi-Cola Coke Classic Pepsi-Cola Coke Classic Coke Classic Pepsi-Cola Dr. Pepper Pepsi-Cola Pepsi-Cola Sprite Coke Classic Coke Classic Coke Classic Sprite Dr. Pepper Diet Coke Dr. Pepper Pepsi-Cola Coke Classic Pepsi-Cola Sprite Coke Classic Diet Coke Data From a Sample of 50 Soft Drink Purchases
- 18. 18 Economic Statistics Frequency Distribution of Softdrink Purchases Softdrink Frequency Coke Classic 19 Diet Coke 8 Dr. Pepper 5 Pepsi-Cola 13 Sprite 5 Total 50
- 19. 19 Economic Statistics Relative Frequency Distribution A Relative Frequency distribution is tabular summary of data showing the relative frequency for each class Relative Frequency = Frequency of the Class n n = number of observations Percent Frequency Distribution A percent frequency distribution is a tabular summary of data showing the percent frequency for each class.
- 20. 20 Economic Statistics Frequency Distribution of Softdrink Purchases Relative Percent Softdrink Frequency Frequency Coke Classic 0.38 38 Diet Coke 0.16 16 Dr. Pepper 0.10 10 Pepsi-Cola 0.26 26 Sprite 0.10 10 Total 1.00 100 n = 50
- 21. 21 Economic Statistics A bar graph is a graphical device depicting data that have been summarized in a frequency, relative frequency, or percent frequency distribution. The pie chart is a graphical device for presenting relative frequency and percent frequency distributions.
- 22. 22 Economic Statistics 0 5 10 15 20 25 30 35 40 Frequency Coke Classic Diet Coke Dr. Pepper Pepsi- Cola Sprite Soft Drinks Bar Graph of Soft Drink Purchases
- 23. 23 Economic Statistics Pie Chart of Soft Drink Purchases 38% 16%10% 26% 10% Coke Classic Diet Coke Dr. Pepper Pepsi-Cola Sprite
- 24. 24 Economic Statistics Sex Number Percent Male 45 39.13 Female 70 60.87 Total 115 100 Frequency Distribution of Students According to Sex
- 25. 25 Economic Statistics Nutritional status Number Percent Normal 30 40 1 st degree malnourished 20 26.7 2nd degree malnourished 15 20 3 rd degree malnourished 10 13.3 Total 75 100 Frequency Distribution of Children by Nutritional Status
- 26. 26 Economic Statistics Frequency Distribution: Quantitative Data 1. Determine the number of nonoverlapping classes. 2. Determine the width of each class. 3. Determine the class limits.
- 27. 27 Economic Statistics Number of Classes: Five or six classes Width of the Classes Approximate Class Width = Largest Data Value – Smallest Data Value Number of Classes Class Limits: The lower class limit identifies the smallest possible data value assigned to the class. The upper class limit identifies the largest possible data value assigned to the class.
- 28. 28 Economic Statistics 12 14 19 18 15 15 18 17 20 27 22 23 22 21 33 28 14 18 16 13 Audit Times (In Days)
- 29. 29 Economic Statistics Audit Time Frequency 10-14 4 15-19 8 20-24 5 25-29 2 30-34 1 Total 20 Frequency Distribution for the Audit-time Data
- 30. 30 Economic Statistics Histogram for Audit-Time Data 0 1 2 3 4 5 6 7 8 9 10-14 15-19 20-24 25-29 30-34 Audit Time in Days Frequency
- 31. 31 Economic Statistics Audits Time (days) Relative Percentage Frequency 10-14 .20 20 15-19 .40 40 20-24 .25 25 25-29 .10 10 30-34 .05 5 Total 1.00 100 Relative and Percent Frequency Distributions for the Audit-Time Data n = 20
- 32. 32 Economic Statistics Cumulative Frequency Distribution shows the number of data items with values less than or equal to the upper class limit of each class. Cumulative Relative Frequency distribution shows the proportion of data items with values less than or equal to the upper class limit of each class. Cumulative Percent Frequency distribution shows the percentage of data items with values less than or equal to the upper class limit of each class.
- 33. 33 Economic Statistics Cumulative Frequency Distribution Audits Time (days) Cumulative Cumulative Relative Cumulative Percent Frequency Frequency Frequency Less than or equal to 14 4 0.20 20 Less than or equal to 19 12 0.60 60 Less than or equal to 24 17 0.85 85 Less than or equal to 29 19 0.95 95 Less than or equal to 34 20 1.00 100 Cumulative Frequency, Cumulative Relative Frequency, and Cumulative Percent Frequency Distributions for the Audit-Time Data
- 34. 34 Economic Statistics Scatter Diagram – is a graphical presentation of the relationship between two quantitative variables.
- 35. 35 Economic Statistics Week Number of commercial Sales ($100s) x y 1 2 50 2 5 57 3 1 41 4 3 54 5 4 54 6 1 38 7 5 63 8 3 48 9 4 59 10 2 46 Sample Data for the Stereo and Sound Equipment Store
- 36. 36 Economic Statistics Scartter Diagram for the Stereo and Sound Equiptment Store 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 No. of Commercials SalesVolume
- 38. Summation Notation S = sum of; X is a variable such as family income Then total family income across N observations is = = N i Ni XXXX1 21 ...
- 39. Summation Notation Summation of a constant times a variable is equal to the constant times the summation of that variable: = = N i Ni kXkXkXXk 1 21 ...
- 40. Summation Notation Summation of the sum of observations on two variables is equal to the sum of their summations: === = N i i N i i N i ii YXYX 111 )(
- 41. Summation Notation Summation of a constant over N observations equals the product of the constant and N: kNk N i= =1
- 42. 42 Economic Statistics Measures of Central Tendency: Mean, Median and Mode The mean is the average of all values. It is useful in analyzing interval and ratio data. The mean is derived by adding all the values and dividing the sum by the number of cases. Example: Achievement can be measured by a score in a 100 item test. Scores of 15 students in the test 82 83 85 87 87 88 90 91 93 93 94 95 95 95 96 Mean = Sum of 82 + 83 + 85 + 87…96 = 1266/15 = 84.4
- 43. 43 Economic Statistics The median is the value in the middle when the data are arranged from highest to lowest. For example: Scores: 82 83 85 87 87 88 90 91 93 93 94 95 95 95 96 Note: For an odd number of observations, the median is the middle value. For an even number of observations, the median s the average of the two middle values. Scores: 82 83 85 87 87 88 90 91 93 93 94 95 95 95 96 98
- 44. 44 Economic Statistics The mode is the most frequently occurring in a set of figures or value that occurs with greatest frequency. Example. 82 83 85 87 87 88 90 90 90 91 93 93 96 97 97
- 45. 45 Economic Statistics Describing the Variance in the data (Univariate) The range is a simple measure of variation calculated as the highest value in a distribution, minus the lowest value plus 1. Example: 82 83 85 87 87 88 90 90 90 91 93 93 96 97 97 Range = highest value – Lowest value 97 - 82 = 15
- 46. 46 Economic Statistics Variance The variance is a measure of variability that utilizes all the data. The variance is based on the difference between the value of each observation (xi) and the mean. The difference between each xi and the mean (x for a sample , u for a population) is called a deviation about the mean.
- 47. 47 Economic Statistics Population Variance Sample Variance 2 2 N xi = 1 2 = n xx s i 2
- 48. 48 Economic Statistics Number of Students Mean Class Size Deviation About Squared Deviation in Class the Mean About the Mean 46 44 2 4 54 44 10 100 42 44 -2 4 46 44 2 4 32 44 -12 144 0 256 Computation of Deviations and Squared Deviations About the Mean for the Class-Size Data 64 4 256 1 2 2 == = n xx s i xxi 2 xxi
- 49. 49 Economic Statistics Standard Deviation The standard deviation is defined as the positive square root of the variance .The standard deviation is easier to interpret than the variance because standard deviation is measured in the same units as the data. 2 ss = 2 = Sample Standard Deviation Population Standard Deviation
- 50. 50 Economic Statistics Number of Students Mean Class Size Deviation About Squared Deviation in Class the Mean About the Mean 46 44 2 4 54 44 10 100 42 44 -2 4 46 44 2 4 32 44 -12 144 0 256 64 4 256 1 2 2 == = n xx s i xxi 2 xxi 864 ==s
- 51. 51 Economic Statistics The coefficient of variation is a relative measure of variability; it measures the standard deviation relative to the mean. It is computed as follows 100 Mean DeviationStandard x 100x x s
- 52. 52 Economic Statistics Number of Students Mean Class Size Deviation About Squared Deviation in Class the Mean About the Mean 46 44 2 4 54 44 10 100 42 44 -2 4 46 44 2 4 32 44 -12 144 0 256 64 4 256 1 2 2 == = n xx s i xxi 2 xxi 2.18100 44 8 100 == xx x s
- 53. 53 Economic Statistics The z-score is often called the standardized value. The standardized value or z-score, zi can be interpreted as the number of standard deviation xi is from the mean x. The z-score for any observation can be interpreted as a measure of the relative location of the observation in a data set. s xx z i i =
- 54. 54 Economic Statistics Z-Scores for the Class-Size Data Number of Students in Class Deviation about the Mean z-score 46 2 2/8 = 0 .25 54 10 10/8 = 1.25 42 -2 -2/8 = -0.25 46 2 2/8 = 0.25 32 -12 -12/8 = -1.50 s xx z i i =
- 55. Economic Statistics 12 14 19 18 15 15 18 17 20 27 22 23 22 21 33 28 14 18 16 13 Audit Times (In Days) n x xi= = 19.3
- 56. Economic Statistics Audit Time Frequency 10-14 4 15-19 8 20-24 5 25-29 2 30-34 1 Total 20 Frequency Distribution for the Audit-time Data
- 57. Economic Statistics Sample Mean for Grouped Data n Mf x ii= Mi = the midpoint for class i fi = the frequency for class i n = Sfi = the sample size
- 58. Economic Statistics Audit Time Class Midpoint Frequency (Days) Mi fi fiMi 10-14 12 4 48 15-19 17 8 136 20-24 22 5 110 25-29 27 2 54 30-34 32 1 32 Total 20 380 days19 20 380 === n Mf x ii
- 59. Economic Statistics Sample Variance for Grouped Data 1 2 2 = n xMf s ii
- 60. Economic Statistics Among the measures of central tendency discussed, the mean is by far the most widely used. The mean is not appropriate for highly skewed distributions and is less efficient than other measures of central tendency when extreme scores are possible. The geometric mean is a viable alternative if all the scores are positive and the distribution has a positive skew.
- 61. Economic Statistics A distribution is skewed if one of its tails is longer than the other. This distribution has a positive skew. This means that it has a long tail in the positive direction. Distributions with positive skew are sometimes called "skewed to the right”.
- 62. Economic Statistics The distribution below has a negative skew since it has a long tail in the negative directions,so it is “skewed to the left.
- 63. Economic Statistics The third distribution is symmetric and has no skew.
- 64. Economic Statistics Personian Coefficient of Skewness deviationstandard 3 medianmean SK =
- 65. Economic Statistics X Y 595 68 520 55 715 65 405 42 680 64 490 45 565 56 580 59 615 56 435 42 440 38 515 50 380 37 510 42 565 53 534=x 53.96=xs Median = 520 47.51=y 11.10=ys Median = 53 deviationstandard 3 medianmean SK = SK = 0.43 SK = -0.45
- 67. Economic Statistics Measures of Association Between Two Variables Covariance Correlation Coefficient
- 68. 68 Economic Statistics Week Number of commercial Sales ($100s) x y 1 2 50 2 5 57 3 1 41 4 3 54 5 4 54 6 1 38 7 5 63 8 3 48 9 4 59 10 2 46 Sample Data for the Stereo and Sound Equipment Store
- 69. 69 Economic Statistics Scartter Diagram for the Stereo and Sound Equiptment Store 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 No. of Commercials SalesVolume
- 70. 70 Economic Statistics Sample Covariance 1 = n yyxx s ii xy
- 71. 71 Economic Statistics 2 50 -1 -1 1 5 57 2 6 12 1 41 -2 -10 20 3 54 0 3 0 4 54 1 3 3 1 38 -2 -13 26 5 63 2 12 24 3 48 0 -3 0 4 59 1 8 8 2 46 -1 -5 5 30 510 0 0 99 iyix xxi yyi yyxx ii Calculations for the Sample Covariance 11 110 99 1 = = = n yyxx s ii xy
- 72. 72 Economic Statistics Scartter Diagram for the Stereo and Sound Equiptment Store 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 No. of Commercials SalesVolume II I III IV 3 51
- 73. Economic Statistics Correlation Coefficient yx xy xy ss s r = rxy = sample correlation coefficient sxy = sample covariance sx = sample standard deviation of x sy = sample standard deviation of y
- 74. Economic Statistics 5 10 10 30 15 50 xi yi Scatter Diagram Depicting a Perfect Linear Relationship 0 10 20 30 40 50 60 5 10 15 x y
- 75. Economic Statistics Pearson r (Pearson product-moment correlation coefficient) 1 = n zz r yx xy
- 76. Economic Statistics 595 68 0.63 1.64 1.03 520 55 -0.15 0.35 -0.05 715 65 1.88 1.34 2.51 405 42 -1.34 -0.94 1.25 680 64 1.51 1.24 1.87 490 45 -0.46 -0.64 0.29 565 56 0.32 0.45 0.14 580 59 0.48 0.74 0.35 615 56 0.84 0.45 0.38 435 42 -1.03 -0.94 0.96 440 38 -0.97 -1.33 1.30 515 50 -0.20 -0.15 0.03 380 37 -1.60 -1.43 2.28 510 42 -0.25 -0.94 0.23 565 53 0.32 0.15 0.05 8010 772 0.00 0.00 12.64 ZxZyZyZxYX 534=x 53.96=xs 47.51=y 11.10=ys 63.0 53.96 534595 = =xz 64.1 11.10 47.5168 = =yz 90.0 14 67.12 ==xyr 1 = n zz r yx xy Data for Calculating the Pearson Product-Moment Correlation Coefficient
- 77. Economic Statistics Spearman rho (p) Applicable to some research studies in which the data consist of ranks or the raw scores can be converted to ranking. Spearman rho is a special case of the Pearson r because rankings are ordinal data. rankspairedebetween thdifferenced rankspairedofnumbern where 1 6 1 2 2 = = = nn d
- 78. Economic Statistics X Y X rank Y rank d 595 68 4.0 1.0 3.0 9.00 520 55 8.0 7.0 1.0 1.00 715 65 1.0 2.0 -1.0 1.00 405 42 14.0 12.0 2.0 4.00 680 64 2.0 3.0 -1.0 1.00 490 45 11.0 10.0 1.0 1.00 565 56 6.5 5.5 1.0 1.00 580 59 5.0 4.0 1.0 1.00 615 56 3.0 5.5 -2.5 6.25 435 42 13.0 12.0 1.0 1.00 440 38 12.0 14.0 -2.0 4.00 515 50 9.0 9.0 0.0 0.00 380 37 15.0 15.0 0.0 0.00 510 42 10.0 12.0 -2.0 4.00 565 53 6.5 8.0 -1.5 2.25 8010 772 0 36.50 D2 122515 50.366 1 = 93.0 07.01 = =
- 79. Economic Statistics Size of Correlation Interpretation 0.90 to 1.00 (-0.90 to -1.00) Very high positive (negative) correlation 0.70 to 0.90 (-0.70 to -0.90) High positive (negative) correlation 0.50 to 0.70 (-0.50 to -0.70) Moderate positive (negative) correlation 0.30 to 0.50 (-0.30 to -0.50) Low positive (negative) correlation 0.00 to 0.30 (-0.00 to -0.30) Little if any correlation Rule of Thumb for Interpreting the Size of a Correlation Coefficient A correlation coefficient can take on values between –1.0 and +1.0, inclusive. The sign indicates the direction of the relationship. A plus indicates that the relationship is positive; a minus sign indicates that the relationship is negative. The absolute value of the coefficient indicates the magnitude of the relationship.
- 80. Economic Statistics Variable X Variable Y Pearson r Interval/Ratio Number of Commercial Salary Interval/Ratio Sales Years of Schooling Spearman (p) Ordinal (Ranking) Ordinal (Ranking) Point-Biserial Nominal (Dichotomous) Gender Interval/Ratio Test Scores Phi (Φ) Nominal (Dichotomous) Gender Gender Nominal (Dichotomous) Political Party Affiliation Issues Rank-Biserial Nominal (Dichotomous) Marital Status Ordinal Socio-economic Status Lambda (λ) Nominal (more than two classification levels) Level of Education Nominal (more than two classification levels) Occupational Choice Matrix Showing Correlation Coefficients Appropriate for Scales of Measurement for Variable X and Variable Y
- 81. Economic Statistics Student IQ Ranked Dichotomy IQ 1 103 5 1 2 94 7 0 3 117 1 1 4 112 2 1 5 89 9 0 6 93 8 0 7 99 6 0 8 107 4 1 9 87 10 0 10 110 3 1
- 82. Economic Statistics Subject Item Score Test Score (X) (Y) A 1 10 B 1 12 C 1 16 D 1 10 E 1 11 F 0 7 G 0 6 H 0 11 I 0 8 J 0 5 5 96 X = nominal data with two classification levels (a dichotomous variable). Assignment of value 1 to correct response to item 1 of the 20-item test and value 0 to an incorrect response. Y = data on the total test scores for ten students Need to correlate success on one item of a test (the dichotomy—either right or wrong) with total score on the test. Data for Calculating the Point-Biserial Correlation Coefficient
- 83. Economic StatisticsThe point-biserial correlation coefficient =1Y mean of the Y scores for those individuals with X scores equal to 1 0Y = mean of the Y scores for those individuals with X scores equal to 0 ys = standard deviation of all Y scores p = proportion of individuals with an X score of 1 q = proportion of individuals with an X score of 0 pq s YY r y pb 01 = The resulting correlation coefficient is the index of the relationship between performance on one test item and performance on the test as a whole.
- 84. Economic Statistics 50.050.0 07.3 40.780.11 =pbr = 0.716 Subjects scoring high on the total test tended to answer item 1 correctly and those with lower scores tended to answer the item 1 incorrectly.
- 85. Economic Statistics Person Gender Political Affiliation (X) (Y) A 1 1 B 1 1 C 1 0 D 1 1 E 1 1 F 0 0 G 0 1 H 0 1 I 0 0 J 0 0 5 6 1 = FEMALE 1 = PRO-ADMIN 0 = MALE 0 = ANTI-ADMIN Data for Calculating the Phi (Φ) Coefficient X and Y are nominal dichotomous variables
- 86. Economic Statistics Gender Male (0) Female (1) Totals Political affiliation Pro-Admin (1) 2 4 6 Anti-Admin (0) 3 1 4 Totals 5 5 10 Variable X 0 1 Totals Variable Y 1 A B A + B 0 C D C + D Totals A + C B + D N DBCADCBA ADBC = •Phi (Φ) coefficient 2x2 Contingency Table for Computing the Phi (Φ) Coefficient
- 87. Economic Statistics 14321342 1234 = = 0.408 This coefficient indicates that there is a low positive relationship between gender and political affiliation. Females tend to be pro-admin and males tend to be anti-admin. This direction is evidenced by the positive correlation, which indicates that scores of 1 tend to be associated with scores of 1 (1 = female, pro-admin) and zeros (0 = male, anti-admin)
- 88. Economic Statistics Less HS Some College Graduate Total than HS Graduate College Graduate Degree Laborer/Farmers 347 128 84 37 5 601 Skilled Crafts 164 277 103 43 36 623 Sales/Clerical 30 77 217 147 80 551 Professional/Managerial 2 34 82 198 267 583 Total 543 516 486 425 388 2358 Data for Determining the Relationship Between Level of Education and Occupational Choice Lambda (λ) coefficient mm j j I I mmimmj nnn nnnn = = = 2 1 1 nmj = largest frequency in the jth column nim = largest frequency in the ith row nm+ = largest marginal row total n+m = largest marginal column total n = number of observation
- 89. Economic Statistics = == j j mjn 1 1306267198217277347 = == j j imn 1 1108267217277347 nm+ = 623 n+m = 543 n = 2358 543623)2358(2 54362311081306 = = 0.394 There is a moderate relationship between level of education and occupational choice. Based on the data, those individuals with more education tend to have sales/clerical or professional/ managerial positions, where as those with less education tend to have laborer/farmer or skilled-crafts positions.
- 90. Economic Statistics Person Immigrating Rank of Socio- Generation (X) economic Status (Y) A 1 1 B 1 2 C 1 3 D 0 4 E 0 5 F 1 6 G 1 7 H 0 8 I 1 9 J 0 10 K 0 11 L 0 12 Data for Calculating the Rank-Biserial Correlation Coefficient Need to know the relationship between the fact that an individual is at least a second-generation American (X) and socio-economic status (Y). The X variable (immigration status) is considered a nominal dichotomy ( 0 = less than second generation; 1 = second generation or greater). The data for the Y variable (socio-economic status) are ranked with 1 = highest value; 2 = next highest status; and so on.
- 91. Economic Statistics Rank-Biserial Correlation Coefficient 01 2 YY n rrb = n = number of observations 1Y = mean rank for individuals with X scores equal to 1 2Y = mean rank for individuals with scores equal to 0
- 92. Economic Statistics = 6 50 6 28 12 2 rbr 333.8667.4 6 1 =rbr = -0.611 This negative coefficient indicates that those who are at least second-generation Americans tend to have higher socioeconomic.
- 93. Economic Statistics Aside from Spearman rank correlation, there are correlations that are applied to two ordinal kinds of variables. These correlation coefficients are distribution free and are usually applied to the ranks of the two variables. Examples are the Gamma and the Kendal.
- 94. Economic Statistics Goodman and Kruskal Gamma The Gamma is a simple symmetric correlation. It does not correct for tied ranks. It is one of many indicators of monotonicity that may be applied. Monotonicity is measured by the proportion of concordant changes from one value in one variable to paired values in the other variable. Concordance (C)--when the change in one variable is positive and the corresponding change in the other variable is also positive. Discordance (D) --when the change in one variable is positive and the corresponding change in the other variable is negative.
- 95. Economic Statistics Kendall's Tau a The number of concordances minus the number of discordances is compared to the total number of pairs, n(n-1)/2.
- 96. Economic Statistics Kendall's Tau b (the Kendall's Tau b statistic controls for tied ranks)
- 97. Economic Statistics With specific Y and X as dependent variables, respectively: Asymmetric Somer's D
- 98. Economic Statistics 1. Tetrachoric correlation 2. Biserial correlation 3. Polychoric correlation 4. Polyserial correlations 5. Partial Correlation 6. Multiple Correlation Other correlations: