Correlation between SAT Score and family income of score holder

I have grown up aspiring to take up degree courses in abroad in the top colleges and have been preparing accordingly. Higher studies in abroad is a big deal for everyone in my family as no-one has done it before. Despite initial dilemma, everyone quite encouraged and supported me.

Recently I came across a statement while going through multiple reviews. The statement read that family income is a factor which determines the SAT score.

SAT is an entrance examination which is necessary for a bachelor's degree in abroad.

So, I started my research regarding the same and tried to find the correlation between the SAT score of an aspirant and his or her family income. This IA is based on this correlation.

The main motive of this IA is to cite a relationship between SAT score and average family income of score holder. In addition to that, a regression model will be prepared in this IA on same topic.

The SAT is an entrance exam used by most colleges and universities to make admissions decisions. The SAT is a multiple-choice, pencil-and-paper test created and administered by the College Board. The purpose of the SAT is to measure a high school student's readiness for college, and provide colleges with one common data point that can be used to compare all applicants. College admissions officers will review standardized test scores alongside your high school GPA, the classes you took in high school, letters of recommendation from teachers or mentors, extracurricular activities, admissions interviews, and personal essays. How important SAT scores are in the college application varies.

Overall, the higher you score on the SAT and/or ACT, the more options for attending and paying for college will be available to you.

SAT is considered as one of the most expensive exams to take because of several reasons. Firstly, SAT is an internationally acclaimed examination. Thus, the registration cost is in USD. Currently, the registration fee of SAT is $40 or $60. Secondly, the syllabus of SAT differs completely from the syllabus of school academics in India. Thus, separate coaching is very necessary for SAT which in turn is very expensive in almost every country.

A survey has been carried on in which SAT score and average income of score holders are noted. The surveyed data is shown below. All the data is shown in ascending order for better understanding:

Sample Calculation:

\(Mean = \frac{y_1+y_2+y_3+y_4+y_5}{5}\)

\(\text{Mean Score of Group} 1 = \frac{900+906+907+910+916}{5}=907.8\)

\(\text{Standard Deviation} = \sqrt{\frac{(\bar y-y_1)^2+(\bar y-y_2)^2+(\bar y-y_3)^2+(\bar y-y_4)^2+(\bar y-y_5)^2}{5}}\)

\(\text{SD of Group} 1 = \sqrt{\frac{(907.8-900)^2+(907.8-906)^2+(907.8-907)^2+(907.8-910)^2+(907.8-916)^2}{5}}=5.23\)

Mode = 1025 and 1050

\(r =\frac{n\big(∑ xy\big)-(∑ x)(∑ y)}{\sqrt{[n∑ x^2-\big(∑ x\big)^2][n∑ y^2-\big(∑ y\big)^2]}}\)

\(=> r =\frac{20(171987.9)-(165)(20465.8)}{\sqrt{[20×1527.5-(165)^2][20×21004028-(20465.8)^2]}}\)

=> r = 0.9829

=> r² = 0.9662

Calculation

\(\bar{x}=\frac{\sum x}{20}=\frac{165}{20}= 8.25\)

\(\bar y=\frac{\sum y}{20}=\frac{20465.8}{20} =1023.29\)

\(\sum(x-\bar x)(y-\bar y) =3145.05\)

\(\sum(x-\bar x)^2 =166.25\)

\(\sum(y-\bar y)^2 =61579.8\)

Let, the Pearson’s Correlation Coefficient be ℜ.

\(R= \frac{\sum(x-\bar x)(y-\bar y)}{\sqrt{\sum(x-\bar x)^2×\sum (y-\bar y)^2}}\)

\(R= \frac{3145.05}{\sqrt{166.25×61579.8}}=\frac{3145.05}{\sqrt{10237641.75}}=\frac{3145.05}{3199.63}\)

R = 0.982

In this IA, I have deduced a relationship between SAT Scores and annual income of score holder. From the background information study, we have found that, SAT is one of the few entrance examinations that requires the SAT aspirant to be financially stable. From the collected data, we have concluded that, with increase in annual family income of candidates, the aberration in marks achieved amongst the candidates of each income group tends to decrease though there are some exceptions. The exceptions in getting a high range of marks secured by the candidates of higher income group may get nullified by taking considerably large data sheet. Though, in some cases, with increased family income. Nowadays, tendency of securing in-depth knowledge on any topic seems to decrease amongst the students belonging to such groups. But with increase in family income, usually, the range of marks achieved is decreasing and often in some groups, the score of all the candidates is same because of getting almost same intensity of tutorial or guidance from several institutes as well as study materials. Furthermore, in low income groups, standard deviation is more because of lack of availability of traditional guidance required for SAT examination. The median of each of the income groups lie close to the mean value of SAT score which signifies that the marks secured by the candidates of each group are very close to each other. On the other hand, in the survey of 100 candidates, 7 candidates have secured 1025 and 1050 score. Thus, it can be stated that the frequency of these two scores is maximum and most of the candidates are likely to secure a score which is equal to 1025 and 1050 or close to it. Thus, the mode of the data sheet is 1025 and 1050. From the above survey, we have concluded the graph that shows a positive increasing relationship. Initially, I have derived a linear relationship using the collected data. The equation of the relationship is given by:

y = 0.0507x + 43.6

R² = 0.9661

From this data, we can clearly say that, with increase in family income, the candidates are being able to get more efficient and professional tutorials as well as study materials which helps the candidates in boosting their SAT score. The correlation co-efficient is also 0.9661 which is very close to 1, which validates our conclusion.

In addition, we have found the Pearson’s Correlation coefficient to establish another correlation analysis giving more validation to this IA. In Pearson’s Correlation, we know that the coefficient lies between 1 and -1 where 1 positive side signifies direct relationship between the two variables and negative side signifies inverse or indirect relationship between the two variables. In this correlation, zero signifies no relationship. In this IA, the value of Pearson’s correlation constant has come out to be 0.982 which is very close to 1 signifying a positive relationship between SAT score and the average family income of the candidates with a strength of very close to 1. Thus, it proves that, the correlation is also linear in nature.

• Bagamery, Bruce D., John J. Lasik, and Don R. Nixon. "Determinants of success on the ETS Business Major Field Exam for students in an undergraduate multisite regional university business program." Journal of Education for Business 81.1 (2005): 55-63.
• https://collegereadiness.collegeboard.org/sat/register/fees
• https://www.theatlantic.com/politics/archive/2014/03/the-real-problem-with-the-sat/453804/
• Benesty, Jacob, et al. "Pearson correlation coefficient." Noise reduction in speech processing. Springer, Berlin, Heidelberg, 2009. 1-4.