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

37. Summation Notation

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

66. Economic Statistics
Mean,
Median,
Mode
Mean Mean
MedianMedian Mode
Mode

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: