Mathematics AI SL
Sample Internal Assessment

Table of content
Rationale
Aim
Introduction
Data collection
Calculation of R2 for graph 1
Calculation of pearson’s correlation coefficient for graph 1
Conclusion
Bibliography

# Correlation between SAT Score and family income of score holder

6/7
0 Words
Candidate Name: N/A
Candidate Number: N/A
Session: N/A
Personal Code: N/A
Word count: 0

## Rationale

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.

## Aim

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.

## Introduction

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.

## Data collection

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:

Serial No.
Country
SAT Score
Annual Income (in lakh in INR)
1
900
3.5
2
Sri Lanka
906
3.5
3
India
907
3.5
4
910
3.5
5
India
916
3.5
6
Sri Lanka
919
4
7
928
4
8
India
939
4
9
Singapore
940
4
10
India
945
4
11
Sri Lanka
948
4.5
12
955
4.5
13
Germany
970
4.5
14
India
971
4.5
15
Sri Lanka
971
4.5
16
Austria
972
5
17
973
5
18
India
973
5
19
973
5
20
Sri Lanka
973
5
21
Austria
978
5.5
22
Singapore
979
5.5
23
980
5.5
24
India
985
5.5
25
986
5.5
26
United States
990
6
27
India
991
6
28
Sri Lanka
993
6
29
United Kingdom
996
6
30
996
6
31
India
997
6.5
32
United Kingdom
997
6.5
33
India
998
6.5
34
Sri Lanka
999
6.5
35
India
1001
6.5
36
1002
7
37
Australia
1005
7
38
Sri Lanka
1006
7
39
Singapore
1006
7
40
India
1006
7
41
Australia
1006
7.5
42
India
1009
7.5
43
1015
7.5
44
Germany
1016
7.5
45
Singapore
1019
7.5
46
India
1019
8
47
Austria
1019
8
48
1019
8
49
Germany
1020
8
50
Sri Lanka
1021
8
51
India
1025
8.5
52
India
1025
8.5
53
Sri Lanka
1025
8.5
54
Germany
1025
8.5
55
India
1025
8.5
56
United States
1025
9
57
India
1025
9
58
Hong Kong
1025
9
59
Germany
1039
9
60
Australia
1040
9
61
India
1042
9.5
62
Germany
1042
9.5
63
Australia
1043
9.5
64
Germany
1043
9.5
65
China
1043
9.5
66
India
1046
10
67
Australia
1050
10
68
Germany
1050
10
69
Austria
1050
10
70
Germany
1050
10
71
United States
1050
10.5
72
India
1050
10.5
73
United States
1050
10.5
74
Germany
1056
10.5
75
United States
1056
10.5
76
Singapore
1060
11
77
Singapore
1069
11
78
United States
1069
11
79
Germany
1070
11
80
India
1075
11
81
United States
1076
11.5
82
France
1079
11.5
83
Germany
1080
11.5
84
India
1081
11.5
85
United Kingdom
1085
11.5
86
Germany
1086
12
87
India
1087
12
88
United States
1088
12
89
1088
12
90
Singapore
1088
12
91
Germany
1100
12.5
92
Germany
1105
12.5
93
India
1109
12.5
94
United States
1120
12.5
95
France
1125
12.5
96
France
1126
13
97
1126
13
98
United Kingdom
1129
13
99
Singapore
1130
13
100
United States
1135
13
Figure 1 - Table On SAT Score Of Candidates Of Different Countries With Respect To Their Annual Family Income In INR

## Processed data

Figure 2 - Table On SAT Score With Family Income With Different Statistical Parametric Values Taking Groups Of Same Annual Family Income In INR

Sample Calculation:

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

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

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

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

## Graphical analysis

Figure 3 - Average SAT Score vs. Average Family Income In INR

## Calculation of R2 for graph 1

c
y
x2
y2
xy
3.5
907.8
12.25
824100.8
3177.3
4
934.2
16
872729.6
3736.8
4.5
963
20.25
927369
4333.5
5
972.8
25
946339.8
4864
5.5
981.6
30.25
963538.6
5398.8
6
993.2
36
986446.2
5959.2
6.5
998.4
42.25
996802.6
6489.6
7
1005
49
1010025
7035
7.5
1013
56.25
1026169
7597.5
8
1019.6
64
1039584
8156.8
8.5
1025
72.25
1050625
8712.5
9
1030.8
81
1062549
9277.2
9.5
1042.6
90.25
1087015
9904.7
10
1049.2
100
1100821
10492
10.5
1052.4
110.25
1107546
11050.2
11
1068.6
121
1141906
11754.6
11.5
1080.2
132.25
1166832
12422.3
12
1087.4
144
1182439
13048.8
12.5
1111.8
156.25
1236099
13897.5
13
1129.2
169
1275093
14679.6
∑x=165
∑y=20465.8
Σx2=1527.5
Σy2=21004028
Σxy=171987.9
Figure 4 - Table On Processed Data Table For Calculation Of R2 In Graph 1

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

=> r= 0.9662

## Calculation of pearson’s correlation coefficient for graph 1

x
y
$$x-\bar x$$
$$y\,-\bar{y}$$
$$(x-\bar{x})(y-\bar{y})$$
$$(x-\bar{x})^2$$
$$(y-\bar{y})^2$$
3.5
907.8
-4.75
-115.49
548.5775
22.5625
13337.94
4
934.2
-4.25
-89.09
378.6325
18.0625
7937.028
4.5
963
-3.75
-60.29
226.0875
14.0625
3634.884
5
972.8
-3.25
-50.49
164.0925
10.5625
2549.24
5.5
981.6
-2.75
-41.69
114.6475
7.5625
1738.056
6
993.2
-2.25
-30.09
67.7025
5.0625
905.4081
6.5
998.4
-1.75
-24.89
43.5575
3.0625
619.5121
7
1005
-1.25
-18.29
22.8625
1.5625
334.5241
7.5
1013
-0.75
-10.29
7.7175
0.5625
105.8841
8
1019.6
-0.25
-3.69
0.9225
0.0625
13.6161
8.5
1025
0.25
1.71
0.4275
0.0625
2.9241
9
1030.8
0.75
7.51
5.6325
0.5625
56.4001
9.5
1042.6
1.25
19.31
24.1375
1.5625
372.8761
10
1049.2
1.75
25.91
45.3425
3.0625
671.3281
10.5
1052.4
2.25
29.11
65.4975
5.0625
847.3921
11
1068.6
2.75
45.31
124.6025
7.5625
2052.996
11.5
1080.2
3.25
56.91
184.9575
10.5625
3238.748
12
1087.4
3.75
64.11
240.4125
14.0625
4110.092
12.5
1111.8
4.25
88.51
376.1675
18.0625
7834.02
13
1129.2
4.75
105.91
503.0725
22.5625
11216.93
Figure 5 - Table On Processed Data Table 1 For Calculation Of Pearson’s Correlation Coefficient In Graph 1

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

## Conclusion

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.