1. IB Math Studies Internal Assessment:
What is the Relationship between SAT Scores and Family Income of the Test
Takers around the World?
Exam Session: May 2012
School name: International School Bangkok
Teacher: Mrs.Goghar
Date: November 8th, 2011
Course: IB Math Studies SL
Word Count: 1,832
Name: Abel T. Koster

2. What is the Relationship between SAT Scores and Family Income of the Test Takers
around the World?
Introduction
The SAT examination is mostly in today’s world of academics, a requirement of
getting accepted into collage. Not only is it enough to take the examination but the
student has to pass with an average score or above to even have his/her application be
considered. Many students around the world recognize this and therefore apply to prep
schools for the SAT or their parents send them to a higher educational institution for
that purpose. The prep schools such as Princeton are not cheap however as it helps give
advice on how to best tackle the SAT examination, neither are higher educational
institutions. Also it can be considered a luxury service by some middle class and low
class societies in the world to be able to attend either one. This being said, the SAT prep
course and higher educational institutions are,as a result, aimed at the high class
societies in the world or those who can afford it. If this is true, it would putfamilies with
a higher income at an advantage for their childrento get accepted into collage compared
to families who cannot afford for their children to take the course or school fee and
learn the advice of how to pass the SAT examination with a high score.Are the
collageswhich students aim to be accepted into for a better education really based on
which families can afford for their children to take the SAT prep course or learn at a
higher educational institution? The data collected from Collage Board in year2007 was
analyzed to determine whether there is a relationship between SAT scores and family
income of the test takers around the world (Rampell).
Statement of Task
The main purpose of this investigation is to determine whether there is a
relationship between SAT scores and family income of the test takers around the world.
The type of data that will be collected is the SAT scores and family income of the two-
thirds of test takers who voluntarily reported it to collage board when signing up for the
SAT examination worldwide.The SAT scores are used to determine how high of a score
the test taker got and family income to determine the possibility to send their children
to SAT prep schools or better educational institutions.The data used to generate the
data breaks down the average score for ten different income groups of $20,000 range.

3. Plan of investigation
I am investigating the relationship of SAT scores and family income of the test takers
around the world. I have collected data on SAT scores and family income of the test takers
around the world. With the collection of data that I have acquired, a number of mathematical
processes were used to analyze the data: a scatter plot of the data, calculation of the least
squares regression line and correlation coefficient. I am going to do a χ2 test on the data to show
the dependence of SAT scores and family income of the test takers around the world.

4. Mathematical Investigation
Collected Data
Table 1: Mean SAT scores per section categorized in family income of test taker in 2007
Family income of test takers Percentage of test takers
within each family income Critical
group reading Math Writing
Less than $10,000 4% 427 451 423 1301
$10,000–$20,000 8% 453 472 446 1371
$20,000–$30,000 6% 454 465 444 1363
$30,000–$40,000 9% 476 485 466 1427
$40,000–$50,000 8% 489 496 477 1462
$50,000–$60,000 8% 497 504 486 1487
$60,000–$70,000 8% 504 511 493 1508
$70,000–$80,000 9% 508 516 498 1522
$80,000–$100,000 14% 520 529 510 1559
More than $100,000 26% 544 556 537 1637
This bottom row, the “More than $100,000” I am going to consider as an outlier
therefore excluded in all calculations as it goes from $100,000 up to the millions of

5. dollar of income which is too wide of a range to include into the calculations of this
assessment.
Graph 1: Average SAT Score Vs. Family Income
1600
1559
Overall Averaged SAT Score (top score 2400)
1550 1522
1508
1487
1500
1462
1450 1427
1400
1371 Average SAT score
1363
1350
1301
1300
1250
0 20 40 60 80 100
Family Income of SAT Takers ($ in Thousands)
Graph 1 shows the average SAT score Vs. family income of test taker. As of now, there
seems to be very strong positive correlation. It does appear that the SAT scores improve
as the family income increases. (Graph was generated through Microsoft Excel)

6. Calculation of the Least Squares Regression
The Least Square regression identifies the relationship between the independent
variable, x, and the dependent variable, y. It is given by the following formula:
where and
Table 2: Values of Least Squares Regression
x y xy x2
15000 1301 19515000 225000000
25000 1371 34275000 625000000
35000 1363 47705000 1225000000
45000 1427 64215000 2025000000
55000 1462 80410000 3025000000
65000 1487 96655000 4225000000
75000 1508 113100000 5625000000
85000 1522 129370000 7225000000
95000 1559 148105000 9025000000
∑ = 495000 ∑ = 13000 ∑ = 733350000 ∑ = 33225000000
= 55000 = 1444. = 79444444.44 = 3691666667
These are the calculated values used in finding the Least Squares Regression
2

7. Calculation of Pearson’s Correlation Coefficient
Pearson’s Correlation Coefficient indicates the strength of the relationship
between the two variables (SAT scores and family income of test taker). It is given by the
following formula:
where , and
Table 3: Values of Pearson’s Correlation Coefficient
x y
15000 1301 1600000000 20576.30864
25000 1371 900000000 5394.08642
35000 1363 400000000 6633.197531
45000 1427 100000000 304.308642
55000 1462 0 308.1975309
65000 1487 100000000 1810.975309
75000 1508 400000000 4039.308642
85000 1522 900000000 6014.864198
95000 1559 1600000000 13122.97531
∑ = 495000 ∑ = 13000 ∑ = 6000000000 ∑ = 58204.22222
= 55000 = 1444.
These are the calculated values used in finding the Correlation Coefficient.

8. 0.9819360378
The calculation suggests that the strength of the
association of the data is very strong since 0.90 r2<1.
I compared this value of with the standard table of coefficient of
determinations which places it in the “very strong” category (Whiffen).
Graph 2: Average SAT Score Vs. Family Income
Linear Fit line
1600
Overall Averaged SAT Score (top score 2400)
1559
1550 1522
1508
1487
1500
1462
1450 1427
1400 1371 1363 Average SAT score
Linear (Average SAT score)
1350
1301
1300
1250
0 20 40 60 80 100
Family Income of SAT Takers ($ in Thousands)
Graph 2 indicates that there is a strong positive linear correlation. This is also indicated
through the value of correlation coefficient, 0.96.(the graph was generated through Microsoft
Excel )
2
Calculation of a test

9. The 2 test is used to measure whether two classifications or factors from the
same sample are independent of each other– if the occurrence of one of them does not
affect the occurrence of the other.
Observed Values:
B1 B2 Total
A1 A B A+B
A2 C D C+D
Total A+C B+D N
Calculations of Expected Values:
B1 B2 Total
A1 A+B
A2 C+D
Total A+C B+D N
Degrees of freedom measure the number of values in the final calculation that are free
to vary:
Null (H0) Hypothesis: SAT scores and family income are independent from each other.
Alternative (H1) Hypothesis: SAT scores and family income are dependent from each
other.

10. Table 4: Observation Values
Score
Income($) 1300-1430 1431-1561 Total
15000 – 55000 4 1 5
56000 – 96000 - 4 4
Total 4 5 9
Table 2 shows the observed values for SAT score Vs. family income. The data pieces have
been put into ranges that represent the income of the families of the test takers.
Table 5: Calculations for the Expected Values
Score
Income($) 1300-1430 1300-1430 Total
15000 – 55000 4+1
56000 – 96000 0+4
Total 4+0 1+4 9
Table 3 shows the individual calculations for each of the expected values.
Table 6: Expected Values
Score
Income($) 1300-1430 1300-1430 Total
15000 – 55000 2.22222 2.77777 5
56000 – 96000 1.77777 2.22222 4
Total 4 5 9
Table 6 shows the expected values retrieved by the calculations in table 4

11. The 2 critical value at 5% significance with 1 degree of freedom is 3.841. As the 2 value
is greater than the critical value, 5.760 3.841, the null hypothesis is rejected and SAT
score is assumeddependent from family income.
Discussion/Validity
Limitations
Throughout the investigation between the correlation of SAT scores and family
income, various limitations may have affected the outcome of the results.
One limitationof the data collected could be that it only reflects on the people
who filled in the family income section before signing up for the SAT. There is no
evidence that the data reflects everyone who has taken the SAT score as there may be
people who did not fill that section.
Another limitation could be that not everyone in the world decide to take the
SAT, people who cannot afford it or take alternative tests are being neglected. Also the
data does not confirm of how many SAT takers are being considered. The data can be
proved insufficient and inaccurate for those reasons.
There is also a limitation in the data as it states income of “$100,000 and above”.
That could mean that the data goes on unto family incomes of millions which is not
proportionate to the other ranges of family income given. Due to this however, that
piece of data was left out in the calculations.
Continuing, there might be a limitation to the recording of the data itself as SAT
takers are to take a survey where they mention family income when signing up for SAT.
This might cause a problem as many SAT takers, mostly in ages 15-17, do not know the
actual income of their family therefore wrong data may be entered.
Then there could be a limitation to the data due to culture and race. The data
does not mention culture and race which might affect the data as there might have

12. been more American surveys who mentioned family income compared to Asian who
answered the survey.
2
Another limitation is that the table of expected values in the test has all
values less than 5 which reduces its validity.
Adding on to that, there might be a limitation to the amount of data that was
collected as 9 pieces of data may not prove to be sufficient enough to reflect the
correlation between SAT scores and family income in a world perspective.
Lastly, there may be many other factors taking place when considering the
correlation between SAT scores and family income such as reasons for having a high
family income and IQ of SAT test takers.
Conclusion
Despite of the previously mentioned limitations, the found 2 value, 5.760,
rejects the null hypothesis that SAT scores are independent from family income and
accepts the alternative hypothesis that SAT scores are dependent from family income.
Furthermore,the investigation clearly shows that there is a strong and positive
correlation between SAT score and family income as it can be an assumed dependence
from each other.
Work Cited
Rampell, Catherine. "SAT Scores and Family Income - NYTimes.com." The Economy
and the Economics of Everyday Life - Economix Blog - NYTimes.com. 28 Aug.
2009. Web. 01 Nov. 2010.<http://economix.blogs.nytimes.com/2009/08/27/sat-
scores-and-family-income/>.
Downey, Joel. "SAT Scores Rise with Family Income." Cleveland OH Local News,
Breaking News,Sports & Weather - Cleveland.com. 10 Apr. 2008. Web. 01 Nov.
2010.<http://www.cleveland.com/pdgraphics/index.ssf/2008/04/sat_scores_rise_
with_family_in.html>.
Whiffen, Glen, John Owen, Robert Haese, Sandra Haese, and Mark Bruce. "Two

13. Variable Statistics." Mathematics for the International Student: Mathematical
Studies SL. By Mal Coad. [S.l.]: Haese And Harris Pub, 2010. 581-82. Print.