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Spain in PISA and in sports

7

Mathematics

Science

Reading

Country

Score

Country

Score

Country

Score

1 Singapore

573 1 Singapore

551 1 Singapore

542

2 Taiwan

560 2 Japan

547 2 Japan

538

3 South Korea 554 3 Finland

545 3 South Korea 536

4 Japan

536 4 Estonia

541 4 Finland

524

5 Liechtenstein 535 5 South Korea 538 5 Canada

523

. . .

. . .

. . .

28 Portugal

487 26 Spain

496 24 Italy

490

29 Italy

485 30 Italy

494 27 Portugal

488

30 Spain

484 33 Portugal

489 27 Spain

488

. . .

. . .

. . .

60 Qatar

376 60 Qatar

384 60 Kazakhstan 393

61 Indonesia

375 61 Indonesia

382 61 Qatar

388

62 Peru

368 62 Peru

373 62 Peru

384

OECD Mean: 494

OECD Mean: 501

OECD Mean: 498

Table 1: Results of the first five and last three countries in each item in the

PISA 2012 report, plus Italy, Portugal and Spain and the mean of the OECD.

In order to analyze the relation between the wealth of the countries and

their results in PISA, as stated in the introduction, I have chosen the P.GDP

as auxiliary variable. A graphical representation of the relation between those

variables appears in Figure

1

. In this figure Liechtenstein, Qatar and Vietnam

look as outliers; we will pay some attention to this fact in Subsection

2.1.1

.

The next step is to try to figure out the expected result for a country given

its P.GDP. This is usually done using a regression technique: if we call

X

the

P.GDP and

Y

the score in Mathematics, the goal is to find an expression like

Y

=

g

(

X

) +

ϵ,

(2.1)

where

g

is a function to be estimated and

ϵ

is a centered random variable inde-

pendent of

X

. Very often it is assumed that

g

belongs to a parametric family

and the problem is to estimate the parameters determining it.

It happens that there are big discrepancies in the P.GDP between countries

(see Table 2). When so large differences appear it is advisable to take logarithms

in the variable. Here we will use basis 10 logarithms, which reduce considerably

the differences as shown in Table 2. Therefore, we modify

(2.1)

to

Y

=

f

(log

10

(

X

)) +

ϵ.

(2.2)

There is no reason to assume any particular shape for

f

. Perhaps, the only

reliable assumption on

f

is to be increasing because an increment in the P.GDP