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SAT

Getting the lowest score possible

Colin Fahey

1. Introduction

This document describes my effort to get the lowest score possible on the SAT exam.

2. SAT

The SAT is a test designed to predict how well a high school student would perform as a freshman at a college or university in the United States of America (USA).

More than 1000 (80%) of colleges and universities in the United States of America (USA) accept or require SAT scores from people applying for admission.

The SAT was first administered on 1926.6.23 to 8040 people.

In the 2003-2004 SAT testing year, 1419007 high school seniors took the SAT.

On 2005.3.12, a new version of the SAT was administered for the first time to approximately 300000 people.

3. Trust only the College Board for information about the SAT

The College Board creates all versions of the SAT, and is the only authority on all aspects of the SAT.

The College Board Internet site describes the content of the SAT and various testing administration conditions and policies.

http://www.collegeboard.com

4. SAT question types

This section describes all question types on the SAT, for each of the three divisions of the SAT: mathematics (M), critical reading (CR), and writing (W).

All sample questions shown here appeared on the SAT version with form code BWBA that was administered on 2005.3.12.

Most of the sample questions shown here have a difficulty rating of "5", the highest difficulty rating on a scale from 1 through 5.

Answers to the sample questions appear at the end of this section.

4.1 Math (M)

This section describes questions types appearing in the math (M) division of the SAT.

4.1.1 General instructions

This section shows the general math instructions, and the instructions for "student-produced response" grids.

4.1.2 Number and operations

The following question is an example of the "number and operations" type of math question.

4.1.3 Algebra and functions

The following question is an example of the "algebra and functions" type of math question.

4.1.4 Geometry and measurement

The following question is an example of the "geometry and measurement" type of math question.

4.1.5 Data analysis, statistics, and probability

The following question is an example of the "data analysis, statistics, and probability" type of math question.

4.2 Critical reading (CR)

This section describes questions types appearing in the critical reading (CR) division of the SAT.

4.2.1 Passage-based reading

The following question is an example of the "passage-based reading" type of critical reading question.

4.2.2 Sentence completion

The following question is an example of the "sentence completion" type of critical reading question.

4.3 Writing (W)

This section describes questions types appearing in the writing (W) division of the SAT.

4.3.1 Identifying sentence errors

The following question is an example of the "identifying sentence errors" type of writing question.

4.3.2 Improving sentences

The following question is an example of the "improving sentences" type of writing question.

4.3.3 Improving paragraphs

The following question is an example of the "improving paragraphs" type of writing question.

4.3.4 Essay

The following question is an example of the "essay" type of writing question.

4.4 Answers to the sample questions

math (M)

--------

  number and operations                        answer: C

  algebra and functions                        answer: 5/2  or  2.5

  geometry and measurement                     answer: A

  data analysis, statistics, and probability   answer: C

critical reading (CR)

---------------------

  passage-based reading                        answer: B

  sentence completion                          answer: E

writing (W)

-----------

  identifying sentence errors                  answer: A

  improving sentences                          answer: D

  improving paragraphs                         answer: C

  essay                                        answer: Yes

5. SAT structure

5.1 SAT question raw score points by format

question format	raw points if wrong	raw points if omitted	raw points if correct
5-choice	(-1/4)	0	+1
12700-choice	0	0	+1
essay	0	0	+2 ... +12

5.2 SAT score structure by division

division	question format	total questions	minimum raw score	maximum raw score
math (M)	5-choice	44	-11	+44
math (M)	12700-choice	10	0	+10
critical reading (CR)	5-choice	67	-17	+67
writing (W)	5-choice	49	-12	+49
writing (W)	essay	1	0	+12

5.3 SAT question totals by format

question format	total questions
5-choice	160
12700-choice	10
essay	1

5.4 SAT question subjects by division

division	question subjects		questions		division questions
math (M)	number and operations		11 ... 13		54
	algebra and functions		19 ... 21
	geometry and measurement		14 ... 16
	data analysis, statistics, and probability		6 ... 7
critical reading (CR)	passage-based reading	extended reasoning	36 ... 40	48	67
		literal comprehension	4 ... 6
		vocabulary in context	4 ... 6
	sentence completion		19
writing (W)	improving sentence errors		18		49
	improving sentences		25
	improving paragraphs		6
	essay		1		1

5.5 SAT chronological structure

(1) There are ten, independently-timed sections, with the following sequence of durations in minutes: {25,25,25,25,25,25,25,20,20,10}, for a total testing duration of 225 minutes (3 hours, 45 minutes).

(2) There is a five-minute break (leaving the room to go to the bathroom is allowed) after section #2, and a one-minute "stretching break" (leaving the room is not allowed) after section #4, and another five-minute break (leaving the room to go to the bathroom is allowed) after section #6.

(3) Section #1 is always the essay section of the Writing (W) division.

(4) Section #10 is always a 14-question section of the Writing (W) division.

(5) Sections #8 and #9 always include a 16-question section of the Math (M) division, and a 19-question section of the Critical Reading (CR) division, but in either of the two possible orderings.

(6) Sections {2,3,4,5,6,7} always include: two 24-question sections from the Critical Reading (CR) division, one 20-question section from the Math (M) division, one 18-question section from the Math (M) division, one 35-question section from the Writing (W) division, and one "variable" section that has the same format as one of the other sections in the set of these six sections.

The order of the section kinds is "random", and the identity of the "variable" section is intended to not be discovered while taking the test.

I took the SAT on 2005.3.12.

The following was the chronology of my particular test day experience:

section	duration (minutes)	division	total questions	comments
1	25 min	Writing (W)	1 (essay)	essay is always first
2	25 min	Math (M)	18	8(5-choice);10(12700-choice)
(BREAK)	5 min	----	----	long/bathroom break
3	25 min	Writing (W)	35	sent. errors, imp. paragraphs
4	25 min	Critical Reading (CR)	23	passages and sentence comp.
(BREAK)	1 min	----	----	short/stretch break
5	25 min	Math (M)	20	----
6	25 min	Critical Reading (CR)	25	long reading passage!
(BREAK)	5 min	----	----	long/bathroom break
7	25 min	*VARIABLE*	????	----
8	20 min	Math (M)	16	geometry; number and op.
9	20 min	Critical Reading (CR)	19	(1-6;7-19)
10	10 min	Writing (W)	14	always last; improve sentences

Here are the chronologies of those eight practice SATs:

practice SAT index	section number
practice SAT index	#1	#2	#3	#4	#5	#6	#7	#8	#9	#10
#1	WE	CR24	M20	VAR	CR24	M18	W35	CR19	M16	W14
#2	WE	CR24	M20	VAR	CR24	M18	W35	CR19	M16	W14
#3	WE	M20	CR24	M18	VAR	W35	CR24	M16	CR19	W14
#4	WE	M20	CR24	M18	VAR	W35	CR24	M16	CR19	W14
#5	WE	CR24	M18	W35	CR24	VAR	M20	CR19	M16	W14
#6	WE	CR24	M18	W35	CR24	VAR	M20	CR19	M16	W14
#7	WE	M18	W35	CR24	M20	CR24	VAR	M16	CR19	W14
#8	WE	M18	W35	CR24	M20	CR24	VAR	M16	CR19	W14

The chronologies of these eight practice SATs only illustrate possible orders of the sections, given the constraints.

One should not try to form other conclusions based on these chronologies of the practice tests; this sample size is very small relative to the large number of possible chronologies, and the College Board has no incentive to describe any additional constraints they might use to form an acceptable chronology.

For example, although the variable section appears in sections {4,5,6,7} in the practice tests listed above, there is no basis for concluding that it is not just as likely that the variable section can appear in section #2 or section #3.

Also, there is no basis for concluding that sections within the same division won't ever appear in consecutive sections in the chronology.

For example, there might be a version of the SAT with two consecutive sections in the Math (M) division.

5.6 Determining the "variable" section while taking the SAT

One of the sections {2,3,4,5,6,7} will be for research purposes only and will not be given a score.

The section used for research purposes is named the "variable" section.

Consider the non-variable sections that must appear in the set of sections {2,3,4,5,6,7}:

division              questions       total by division

-------------------------------------------------------

Math (M)                 20              38

Math (M)                 18

-------------------------------------------------------

Critical Reading (CR)    24 (+\-1)       48

Critical Reading (CR)    24 (+\-1)

-------------------------------------------------------

Writing (W)              35              35

-------------------------------------------------------

Therefore, the division with the variable section well be known as soon as one encounters:

(1) a third section in the Math (M) division;

(2) a third section in the Critical Reading (CR) division;

or, (3) a second section in the Writing (W) division.

This will happen while taking the SAT.

Therefore, one will know, before starting work on a section that matches one of the three cases, that there is a (1/3) chance (cases 1 and 2) or a (1/2) chance (case 3) that the current section is the variable section and will not be given a score.

Regardless of the sequence of sections, when a test taker encounters a section that proves which division has the variable section, the test taker has a (1/3) chance (cases 1 and 2) or a (1/2) chance (case 3) that the current section is the variable section.

Before gaining this information, the probability was only (1/6).

Also, the probability for all subsequent sections becomes zero.

Furthermore, if a math section with 20 questions has been encountered, and then another math section with 20 questions is encountered, then one knows, before starting work on the second math section with 20 questions, that the variable section is in the math division, and also that there is a (1/2) chance that the current section is the variable section and will not be given a score.

Similarly, encountering a math section with 18 questions, and later encountering another math section with 18 questions, leads to the same conclusions.

The probability of any subsequent section being the variable section becomes zero.

Okay, now consider this information leak from the perspective of the authors of the SAT.

The goal of the authors of the SAT is to have test takers work on the variable section with the same concern and effort given to all other sections of the SAT, giving the authors of the SAT a method of linking performance on the particular version of the SAT to performances on versions administered throughout the long history of the SAT.

Therefore, the authors of the SAT want to minimize the chance that a person taking the test will determine that a particular section is the variable section.

For example, the authors of the SAT probably would avoid having sections #2 and #3 be math sections with exactly 20 questions each, because the test taker would know, at the very beginning of section #3, that either section #2 or section #3 must be the variable section.

Also, the test taker would know that all subsequent sections will be scored.

In this hypothetical situation, the test taker gets information about the variable section at the earliest possible time in the sequence of sections in the SAT.

The test taker can use this information in a few ways.

If the test taker is consciously or tacitly "taking advantage" of the general (1/6) chance that each section, of the sections {2,3,4,5,6,7}, is the variable section, putting in only (5/6) of maximum personal effort to avoid wasting a full effort on the variable section, then the test taker would change this conservative strategy after learning that the sections {4,5,6,7} will be graded, investing full effort in to those sections.

The test taker might also take advantage of the (1/2) chance that section #3 is the variable section, putting in less effort, or skipping the section entirely and instead resting or doing work on another section (against SAT rules; don't cheat!).

It is my guess that the authors of the SAT delay conclusive evidence of the division containing the variable section until section #7.

This minimizes any advantage to the test taker.

Section #7 itself need not be the variable section, but I believe that delaying conclusive evidence of the division having the variable section until section #7 is best for the test authors.

( Note: On the 2005.3.12 administration of the SAT, and form code BWBA, section #7 itself happened to be variable section. )

In conclusion, there is a way to be certain which division has the variable section while taking the SAT -- and this information might offer, at the very least, some psychological relief (closure, or SATisfaction of morbid curiosity) for a person taking the SAT.

6. Calculating SAT scores

This section describes how to convert total numbers of correct and incorrect responses to scaled scores, for each of the three divisions of the SAT: mathematics (M), critical reading (CR), and writing (W).

6.1 Math (M)

This section describes the procedure to compute the raw and scaled scores for the math (M) division of the SAT.

6.1.1 Calculations

// INPUTS

// [Answers left blank are neither counted as correct nor counted as wrong.]

// Number of math multiple-choice questions answered correctly

// [an integer from 0 through 44]:

int mathMCCorrect;

// Number of math multiple-choice questions answered incorrectly

// [an integer from 0 through 44]:

int mathMCWrong;

// Number of correct math "student-produced responses"

// [an integer from 0 through 10]:

int mathSPRCorrect;

// CALCULATIONS

// Overall number of correct answers

// [an integer from 0 through 54]:

int mathCorrect   =  ( mathMCCorrect + mathSPRCorrect );

// Overall number of incorrect answers (such that blank answers are ignored)

// [an integer from 0 through 44]:

int mathWrong     =  ( mathMCWrong );

// Raw score with fractional part

// [a decimal number from -11.0 through +54.0]

decimal mathRawFractional =

             (decimal) mathCorrect - ( (decimal) mathWrong / (decimal) 4 );

// Raw score rounded to the nearest integer

// [an integer from -11 through +54]:

int mathRawScore = Nearest( mathRawFractional );

// Scaled score

// [an integer, multiple of 10, from 200 through 800]:

// (The MathRawToScaledScore() function is shown below as a graph.)

int mathScaledScore =  MathRawToScaledScore( mathRawScore );

6.1.2 Graph

The following graph shows the conversion from a multiple-choice raw score (-11 through +54) to a scaled score (200 through 800, in multiples of 10) for the Math (M) division of the SAT.

6.2 Critical reading (CR)

This section describes the procedure to compute the raw and scaled scores for the critical reading (CR) division of the SAT.

6.2.1 Calculations

// INPUTS

// [Answers left blank are neither counted as correct nor counted as wrong.]

// Number of critical reading multiple-choice questions answered correctly

// [an integer from 0 through 67]:

int criticalReadingCorrect;

// Number of critical reading multiple-choice questions answered

// incorrectly [an integer from 0 through 67]:

int criticalReadingWrong;

// CALCULATIONS

// Raw score with fractional part

// [a decimal number from -(67/4) = -16.75 through +67.0]:

decimal criticalReadingRawFractional =

                 (decimal) criticalReadingCorrect

              - ((decimal) criticalReadingWrong / (decimal) 4);

// Raw score rounded to the nearest integer

// [an integer from -17 through +67]:

int criticalReadingRawScore = Nearest( criticalReadingRawFractional );

// Scaled score

// [an integer, multiple of 10, from 200 through 800]:

// (The CriticalReadingRawToScaledScore() function is shown below

// as a graph.)

criticalReadingScaledScore =

              CriticalReadingRawToScaledScore( criticalReadingRawScore );

6.2.2 Graph

The following graph shows the conversion from a multiple-choice raw score (-17 through +67) to a scaled score (200 through 800, in multiples of 10) for the critical reading (CR) division of the SAT.

6.3 Writing (W)

This section describes the procedure to compute the raw and scaled scores for the writing (W) division of the SAT.

6.3.1 Calculations

// INPUTS

// [Answers left blank are neither counted as correct nor counted as wrong.]

// Number of writing multiple-choice questions answered correctly

// [an integer from 0 through 49]:

int writingMCCorrect;

// Number of writing multiple-choice questions answered incorrectly

// [an integer from 0 through 49]:

int writingMCWrong;

// Essay score

// [an integer; zero, or, 2 through 12; { 0, 2..12 }]:

int writingEssayScore;

// CALCULATIONS

// Raw score with fractional part

// [a decimal number from -(49/4) = -12.25 through +49]:

decimal writingMCRawFractional =

               (decimal) writingMCCorrect

            - ((decimal) writingMCWrong / (decimal) 4);

// Raw score rounded to the nearest integer

// [an integer from -12 through +49]:

int writingMCRawScore = Nearest( writingMCRawFractional );

// Scaled score

// [an integer, from 20 through 80]:

// (The writingMCRawToScaledScore() function is shown below, as a graph.)

int writingMCScaledScore = writingMCRawToScaledScore( writingMCRawScore );

// Combined score

// [an integer, multiple of 10, from 200 through 800]:

// (The writingCSRawToScaledScore() function is shown below as a graph.)

int writingCSScaledScore =

        writingCSRawToScaledScore( writingMCRawScore, writingEssayScore );

6.3.2 Graphs

The following graph shows the conversion from a multiple-choice raw score (-12 through +49) to a scaled score (20 through 80) for the writing (W) division of the SAT.

The following graph shows the conversion from a multiple-choice raw score (-12 through +49), and the essay raw score (0, +2 ... +12), to a combined scaled score (200 through 800, in multiples of 10) for the Writing (W) division of the SAT.

Notice that the graph is a family of curves.

Thus, one finds the proper horizontal coordinate using the multiple-choice raw score, and then selects the proper curve using the essay score.

The point on that curve at the proper horizontal coordinate is the composite scaled score.

The graph is missing data for a small region of score combinations, marked by the question mark ("?") on the graph.

The College Board did not provide data for this region in the table in the Question and Answer Service (QAS) report.

I suppose the College Board doesn't think there will be many people who write competent essays (with a pair of scores adding to "6" or higher), and, at the same time, get a multiple-choice raw score less than "-2".

But, hey, it could happen.

7. SAT scaled score distributions from 2004

Although the following graphs pertain to an old SAT format, the distributions of scaled scores are likely to be maintained for the new SAT format.

The following graphs show the percentages of graduating seniors in 2004 whose scaled scores were within particular ranges, for the math and verbal divisions of the old SAT format (prior to the introduction of the new SAT on 2005.3.12).

These distributions were designed by the College Board, and achieved by computing, and using, appropriate "raw score to scaled score" conversion curves.

For the 2004 SAT testing year, the average math score was 518, with a standard deviation of 114.

For the 2004 SAT testing year, the average verbal score was 508, with a standard deviation of 112.

8. Analysis of "student-produced response" grid encodings

8.1 Introduction

The following image shows the format of the "student-produced response" grid as it appears on the SAT answer sheet.

Various consequences of this response grid format are also shown.

8.2 Fractions may always be avoided

It is always possible to encode a correct response value using a decimal format on the SAT.

Fractions may always be avoided.

There are encodable fractions whose exact decimal format encodings cannot fit in the space provided, but it is always acceptable to encode the values in a decimal format, as long as the decimal encoding is as precise as possible, given the limited space.

It is acceptable to "truncate" the decimal encoding, which involves simply stopping the writing of digits beyond the most-significant digits that fit in the space provided.

It is also acceptable to "round" the value, choosing the decimal encoding, among all decimal encodings that will fit in the space provided, that has a value that is closest to the exact value of the correct result.

The following image shows how a fractional value without a corresponding exact decimal format encoding may nonetheless be correctly encoded on the SAT in a decimal format after a process of either truncation or rounding.

8.3 All 22308 encodings

8.3.1 Introduction

The symbols available in the four columns of the "student-produced response" grid imply a total of ((11)*(13)*(13)*(12)) = 22308 encodings.

This section describes various classifications of the encodings.

8.3.2 Hierarchy of classifications for all 22308 encodings

The following chart shows a hierarchy of classifications of subsets of all possible encodings.

All 22308	Valid 17936	Blank (No Response; Skipped; Omitted) 1
		No Fraction 15568 (implies Encoded Value is Exact)	No Decimal (implies Integer Value) 11229
			With Decimal 4339	Integer Value 1243
				Non-integer Value 3096
		With Fraction 2367	Integer Value 474
			Non-integer Value 1893	With Exact Encoding 546
				Without Exact Encoding 1347	Truncated is Closest 718
					Truncated is not Closest 629
	Invalid 4372	Syntax Error 4199
		Undefined Value 173	Division-by-zero 171
			Zero-over-zero 2

8.3.3 Blank

If the entire response is blank, the response is considered "omitted"; the test-taker decided not to respond to the corresponding question.

The response is given a score of zero points.

( An incorrect "student-produced response" is also given a score of zero points. )

Blank response:

     ( blank, blank, blank, blank )     "    "

8.3.4 Invalid syntax

Encoding the values of numbers requires a grammar, or syntax, so that a reader can unambiguously interpret the encoding as a specific numeric value.

The existence of a grammar implies the existence of encodings that are inconsistent with that grammar.

[ This generalization is not true for the all-inclusive, non-constraining, trivial "non-grammar grammar", often exploited in advertising, online chat, and spam messages. ]

The following encodings are examples of encodings that violate the implicit grammar of the "student-produced responses grid".

Examples of invalid syntax:

    Involving only punctuation and blanks:

        ( blank, blank, blank, point }     "   ."

        ( point, point, point, point )     "...."

        ( blank, slash, blank, blank )     " /  "

        ( blank, slash, slash, blank )     " // "

        ( point, slash, slash, point )     ".//."

        ( blank, slash, point, blank )     " /. "

          -->  News for nerds: slash-dot will never be

               a correct response on the SAT!

    Involving only digits and blanks:

        ( blank,     0, blank,     0 )     " 1 2"

        (     1, blank, blank,     2 )     "1  2"

        (     1,     2, blank,     3 )     "12 3"

    Involving only digits and points:

        (     1, point, point,     2 )     "1..2"

        (     1, point,     2, point )     "1.2."

        ( point,     1,     2, point )     ".12."

    Involving only digits, slashes, and blanks:

        ( blank, slash,     2, blank )     " /2 "

        ( blank, slash, blank,     2 )     " / 2"

        ( blank,     1, slash, blank )     " 1/ "

    Involving only digits, slashes, points, and blanks:

        ( point, slash, blank,     2 )     "./ 2"

        ( blank, slash,     2, point )     " /2."

8.3.5 Division-by-zero error

Some encodings express the idea of dividing a number by zero.

The result of dividing a number by zero cannot be defined without contradicting one or more axioms of mathematics.

Therefore, encodings which express the idea of dividing a number by zero cannot be interpreted as having any particular numerical value.

I wondered what would happen if I wrote " 1/0" as a "student-produced response" on the SAT.

Would I sabotage the SAT-grading computer software?

I wanted to try this out, but I didn't want to jeopardize getting my score report.

Example of the division-by-zero error:

     ( blank,