1. Analysis of Relationships among Variables
Joel D. Sherman, Ph.D.
Secretariat of Public Education – Mexico
UNESCO Regional Office for Latin America and the Caribbean
Nassau, Bahamas
9-10 December 2008

2. Why Analyze Relationships?
Understand factors that may contribute to differences across countries.
Identify potential “causes” of differences across countries.
“Predict” future values on key measures of interest based on past and current values and factors that contribute to them.
Determine consistency of indicators that reflect a similar concept.

3. Presenting Relationships among Variables: Cross-Tabulations
Non-statistical relationships (presented in tables).
Generally shows relationships between only two variables.
Stratification on one variable, breakdown into groups such as quartiles or quintiles, averages for each variable calculated and compared.
Upward progression on averages for both variables suggests a positive relationship between them.
Upward progression on one variable with downward progression on the second suggests a negative relationship between them.
No common pattern suggests “no relationship” between the variables.

4. Cross-Tabulations Suggesting a Positive Relationship Between Variables
Cross-Tabulation Suggesting a Positive Relationships Between Variables
Quartiles
Variable A
Variable B
Quartile 1 (lowest)
25.3
16.8
Quartile 2
28.5
19.2
Quartile 3
33.7
21.4
Quartile 4
35.4

5. Cross-Tabulations Suggesting a Negative Relationship Between Variables
Cross-Tabulation Suggesting a Negative Relationships Between Variables
Quartiles
Variable A
Variable B
Quartile 1 (lowest)
25.3
Quartile 2
28.5
21.4
Quartile 3
33.7
19.2
Quartile 4
35.4
16.8

6. Cross-Tabulations Suggesting “No Relationship” Between Variables
Quartiles
Variable A
Variable B
Quartile 1 (lowest)
25.3
21.4
Quartile 2
28.5
Quartile 3
33.7
16.8
Quartile 4
35.4
19.2

7. Presenting Relationships among Variables: Correlation Coefficients
Measure s linear relationships between two variables.
Range is from +1.0 to -1.0.
Correlation of +1.0 shows a one-to-one positive correspondence between two variables; as one variable increases in value, there is a corresponding increase in the second variable.
Correlation of -1.0 shows a one-to-one negative correspondence between two variables; as one variable increases in value, there is a corresponding decrease in the second variable.
Correlation of 0.0 shows “no relationship” between two variables.

8. Strength of Correlations
Correlation Coefficients
Positive
Negative
No Relationship
+0.01 to 0.24
-0.01 to -0.24
Weak
+0.25 to +0.49
-0.25 to -0.49
Moderate
+0.50 to +0.69
-0.50 to -0.69
Strong
+0.70 to +1.00
-0.70 to -1.00

9. Correlation of +1.0 Between Variables
Country
Variable A
Variable B
Country A
43.0
21.2
Country B
57.3
23.5
Country C
62.5
24.8
Country D
72.5
27.9
Country E
73.2
29.2
Country F
86.4
30.7
Country G
89.1
38.5
Country H
95.1
40.2

10. Correlation of -1.0 Between Variables
Country
Variable A
Variable B
Country A
43.0
40.2
Country B
57.3
38.5
Country C
62.5
30.7
Country D
72.5
29.2
Country E
73.2
27.9
Country F
86.4
24.8
Country G
89.1
23.5
Country H
95.1
21.1

11. Graphing Relationships Between Two Variables: Positive Correlation

12. Graphing Relationships Between Two Variables: Negative Correlation

13. Limitations of Cross-Tabulations and Correlation Analysis
Show relationships but not “causation”.
Not always clear which is the independent variable.
Doesn’t take into account other factors that may influence variable of interest.
Results could be influenced by one or two units, which could be different if these units were removed.
Doesn’t examine “how much” an increment in one unit relates to an increment in the other unit (percentage change).

14. Relationship Between Graduation Ratios and Educational Expectancy
Upper Secondary Graduation Ratios and Educational Expectancy
Country
Ratio
Expectancy
Argentina
43.0
17.5
Germany
99.7
17.4
Brazil
72.5
17.0
Hungary
84.3
17.8
Chile
73.2
15.2
Indonesia
43.5
12.1
China
57.3
12.5
Ireland
90.7
Czech Republic
89.1
17.2
Italy
81.6
17.1
Denmark
86.4
19.1
Jordan
72.8
14.0
Egypt
62.5
13.2
Malaysia
86.6
13.5
Finland
95.1
20.6
Mexico
40.4
13.8

15. First Steps in Organizing the Data
Upper Secondary Graduation Ratios and Educational Expectancy
WEI Countries
Ratio
Expectancy
OECD Countries
Ratios
Argentina
43.0
17.5
Czech Republic
89.1
17.2
Brazil
72.5
17.0
Denmark
86.4
19.1
Chile
73.2
15.2
Finland
95.1
20.6
China
57.3
12.5
Germany
99.7
17.4
Egypt
62.5
13.2
Hungary
84.3
17.8
Indonesia
43.5
12.1
Ireland
90.7
Jordan
72.8
14.0
Italy
81.6
17.1
Malaysia
86.6
13.5
Mexico
40.4
13.8
WEI Mean (All)
61.4
14.2
OECD Mean (All)
81.1
17.7

16. Making the Data More Useful
Upper Secondary Graduation Ratios and Educational Expectancy
Country
Ratio
Expectancy
Argentina
43.0
17.5
Mexico
40.4
13.8
Indonesia
43.5
12.1
Italy
81.6
17.1
China
57.3
12.5
OECD Mean (All)
81.1
17.7
WEI Mean (All)
61.4
14.2
Hungary
84.3
17.8
Egypt
62.5
13.2
Denmark
86.4
19.1
Brazil
72.5
17.0
Czech Republic
89.1
17.2
Jordan
72.8
14.0
Ireland
90.7
17.4
Chile
73.2
15.2
Finland
95.1
20.6
Malaysia
86.6
13.5
Germany
99.7

17. Final Steps in Organizing the Data
Upper Secondary Graduation Ratios and Educational Expectancy – 2005
WEI Countries
Ratio
Expectancy
OECD Countries
Argentina
43.0
17.5
Mexico
40.4
13.8
Indonesia
43.4
12.1
OECD Mean (All)
81.6
17.7
China
57.3
12.5
Italy
17.1
WEI Mean (All)
61.4
14.2
Hungary
84.3
17.8
Egypt
62.5
13.2
Denmark
86.4
19.1
Brazil
72.5
17.0
Czech Republic
89.1
17.2
Jordan
72.8
14.0
Ireland
90.7
17.4
Chile
73.2
15.2
Finland
95.1
20.6
Malaysia
86.6
13.5
Germany
99.7

18. Graphing Relationship Between Ratios and Expectancy

19. Graphing Relationship Between Ratios and Expectancy (2)

20. Finding “Outliers”
Upper Secondary Graduation Ratios and Educational Expectancy
Upper Secondary Ratios (Mean 72.7)
Educational Expectancy
(Mean 16.2)
Below Mean
Above Mean
China
Egypt
Indonesia
Mexico
Argentina
Brazil
Chile
Jordan
Malaysia
Czech Republic
Denmark
Finland
Germany
Hungary
Ireland

21. What Can We Say?: General Findings
WEI countries have a lower mean upper secondary graduation ratio and educational expectancy than OECD countries. Excluding Greece and India, the mean graduation ratio in WEI countries is 20 points lower than the OECD mean; educational expectancy is 3 years lower in WEI countries than in OECD countries.
The ranges in WEI countries are narrower on both measures than in OECD countries. In WEI countries, the range in upper secondary graduation ratios is about 45 points; in OECD countries, it is almost 60 points. Similarly, the range in educational expectancy is about 5 years in WEI countries, but nearly 9 years in OECD countries.

22. What Can We Say?: General Findings (continued)
Overall, there is a moderate correspondence between countries’ upper secondary graduation ratios and their total years of expected education. However, the correspondence is primarily in OECD countries (correlation +0.60) and not in WEI countries (correlation +0.06).
In OECD countries with a mean upper secondary graduation ratio below 50, the mean educational expectancy is only 13 years; in countries with a mean ratio between 90 and 100, educational expectancy averages just over 18 years.

23. What Can We Say?: General Findings (continued)
In WEI countries, countries with both low and high upper secondary graduation ratios have similar years of educational expectancy – 14.4 years in countries with a mean graduation ratio under 50 and 13.3 years in countries with a mean ratio between 80 and 89.
A large majority of countries with available data demonstrate a consistency in their upper secondary graduation ratios and their levels of educational expectancy. Twelve countries fall below the median on both measures and another 12 are above the median on both measures.

24. What Can We Say?: General Findings (continued)
Several countries show less consistency in their upper secondary graduation ratios and levels of educational expectancy. Slovakia, Malaysia, Switzerland and the Republic of Korea all are above the median in their graduation ratios, but below the median in their levels of educational expectancy. In contrast, Argentina, New Zealand and Spain fall above the median on the first measure, but below on the second.

25. Country Focus: Argentina
In Argentina, there are significant differences between the upper secondary graduation ratio and the years of expected education. Argentina’s upper secondary graduation ratio (43.0) is the third-lowest among WEI and OECD countries with available data, while its level of educational expectancy (17.5 years) is more consistent with those found in OECD than WEI countries.

26. Country Focus: Malaysia
In Malaysia, there are significant differences between the upper secondary graduation ratio and the years of expected education. Malaysia’s upper secondary graduation ratio 86.6 is well above the country mean for all countries and is more consistent with those found in OECD than WEI countries. However, its level of educational expectancy (13.5 years) is the sixth-lowest among countries with available data.

27. Country Focus: Thailand
Thailand is typical of countries in their relationship between upper secondary graduation ratios and years of expected education. The upper secondary graduation ratio of 65.4 and the 16.1 years of expected education both fall below the median of WEI and OECD countries with available data.