<p>I agree that the phrasing is confusing. Unfortunately, I’m beyond the 20 minute edit window, so I cannot modify the post. I meant the 1600th highest score and 400th highest score counting from the top, so there are 1600 students in the freshman class with scores above the listed score.</p>
<p>Your comparison is invalid, there is no available data outside of 25%-75% for comparison.</p>
<p>Comparable ones:</p>
<p>For Fall 2012 - Spring 2013 Enrolled freshman class: (smallest to biggest)</p>
<p>Freshman SAT 25-75%
Class
Size… R…M…W</p>
<hr>
<p>1 1098 670-780 680-780 680-790…Dartmouth
2 1140 670-770 740-800 680-780…MIT
3 1342 640-740 660-770 650-750…J.Hopkins
4 1356 700-800 710-790 710-800…Yale
5 1357 700-790 710-800 710-800…Princeton
6 1378 620-710 660-760 640-730…Emory
7 1413 700-780 700-790 700-790…Columbia
8 1511 700-780 700-790 --------- …U.Chicago
9 1539 660-760 660-770 670-780…Brown
10 1587 690-770 710-790 670-770…Vanderbilt
11 1668 700-770 720-790 --------- …WUSL
12 1724 700-800 710-790 710-800…Harvard
13 1733 660-750 690-780 670-770…Duke
14 1765 680-780 700-790 700-780…Stanford
15 2115 680-760 700-780 680-770…Northwestern
16 2461 660-760 690-780 680-770…Penn
17 3217 640-740 670-780 --------- …Cornell</p>
<p>Result of Student body SAT Strength comparison
(the starting SAT score of their top 800 students)
(NOT NECESSARILY bigger one with higher scores. i.e. Duke and Yale, Duke and WUSTL…)</p>
<p>SZ R#800 M#800 W#800 R+M</p>
<hr>
<p>17 740.00 780.00 -------- 1520 Cornell
12 757.19 755.75 761.47 1513 Harvard
16 744.99 766.49 756.49 1511 Penn
15 739.48 759.48 746.91 1499 Northwestern
11 737.85 757.85 -------- 1496 WUSTL
14 739.35 753.41 747.48 1493 Stanford
10 729.34 749.34 719.18 1479 Vanderbilt
8 735.29 739.70--------- 1475 U.Chicago
4 732.01 735.60 738.81 1468 Yale
5 728.88 738.88 738.88 1468 Princeton
7 729.41 733.09 733.09 1463 Columbia
13 711.91 741.91 727.67 1454 Duke
2 679.65 745.79 689.65 1425 MIT
9 706.04 710.64 720.64 1417 Brown
3 670.77 693.85 680.77 1365 J.Hopkins
1 674.71 684.28 684.71 1359 Dartmouth
6 650.50 693.89 670.50 1344 Emory </p>
<p>Completed my final explanation. 
I’ll move on to other topics.</p>
<p>Mind you, I’ve skimmed, but is there a reason posters haven’t included Tufts (21) or CMU (26) on your universities assessments/ number crunching? </p>
<p>Tufts student scores put them higher on the Business Insider list than JHU, Brown, Emory, Cornell…</p>
<p>
My data is no less accurate than yours. In many cases, it is more accurate since you are assuming a constant linear progression between the 25th and 75th percentile, ignoring the many intermediate points between and below this range listed in the CDS. I’d expect the actual scores form more of a bell curve than a linear progression, so using intermediate points can greatly improve accuracy. I’ll step through an example for you in more detail.</p>
<p>Harvard has freshman class size of 1657, so the 1600th best score out of 1657 would be the bottom 3.4%. The CDS lists the following SAT scores for the bottom few percent:</p>
<p>500 = bottom 0.13% Verbal
600 = bottom 3.95% Verbal (0.13% in 400s + 3.82% in 500s)</p>
<p>So we know the bottom 3.4% is somewhere between 500 and 600. If we assume a linear distribution (like you did in your 25 to 75% estimate), then the bottom 3.4% score would be approximately (3.4-0.13) / 3.82 ~= 86% * 100 + 500 = 586. I rounded to 590 in my earlier list. It’s silly to worry about this degree of linear distribution approximation inaccuracy when you consider how useful the result would be and how much that usefulness will change with slight inaccuracies.</p>
<p>I’ve added more precise values, this time rounding to the nearest multiple of 5. For the 400th worst, I used a different method this time, in which the linear approximation as if the rate continues between the percentile of the 700 SAT score and the 25th percentile score, which changed some values slightly from my earlier post.</p>
<p>Approximate SAT Scores of 1600th Best Student in Class
- University of Illinois – 795M, 690V
- University of Michigan – 760M, 700V
- Cal Berkeley – 745M, 690V
- Cornell – 720M, 680 V
- Stanford – 630M, 620V
- Harvard – 610M, 585V</p>
<p>1600th best SAT score is an arbitrary selection that was chosen to favor schools with ~1600x4 students in class, such as Illinois. </p>
<p>Approximate SAT Scores of 400th Best Student in Class
- Harvard – 800M, 790V
- Stanford – 795M, 785V
- University of Illinois – 800M, 770V
- Michigan – 800M, NA V
- Cal Berkeley – 800M, 750V
- Cornell – 790M, 755V</p>
<p>400th best SAT score is an arbitrary selection that was chosen to favor schools with ~400x4 students in class, such as Harvard and Stanford.</p>
<p>I already told you your comparison is invalid and it is TRULY invalid.</p>
<p>The top 400 is definitely outside of these schools’ 75 percentile students. You cannot provide data on enrolled freshman class profiles of these students.</p>
<p>(No way to compare those much bigger schools with the ones I listed, you should only compare within possible size with available data, not choosing arbitrarily.)</p>
<p>If you REALLY HAVE TO calculate mathematically, it becomes ridiculous </p>
<p>SZ…R…M
1724 803.60 792.88…Harvard
3217 765.13 807.65…Cornell
1765 784.67 794.21…Stanford</p>
<p>Harvard’s CR and Cornell’s Math are both over 800… invalid. See that? Stop comparing things the ways they shouldn’t be.</p>
<p>Done done on this one. If you want to know why 800 is picked (and only can be picked around that number for the group of generally top 20 ranked schools) go read my explanation in post #1 in “the link at the bottom of post #7 of this thread”. I am not going to spend time repeating. </p>
<p>For Tufts and CMU (Tufts common dataset can’t be found on their own web site but from other site online, the freshman class size 2013 is available)</p>
<p>Freshman SAT 25-75%
Class
Size… R…M…W</p>
<hr>
<p>1316 670-760 680-760 680-760…Tufts
1408 630-730 690-790 650-740…CMU</p>
<p>Result for top 800 students:</p>
<p>R#800 M#800 W#800 R+M</p>
<hr>
<p>695.58 702.74 702.74 1398.31…Tufts
666.36 726.36 682.73 1392.73…CMU </p>
<p>Tufts and CMU student body academic strength is about in between of Brown and Johns Hopkins.</p>
<p>
Let’s look specifically at the Harvard example you listed. 400th out of 1724 is the 76th percentile. Do you really believe that it’s impossible to accurately estimate 76th percentile SAT score, when you know the 75th percentile score? In Harvard’s case, the 75th percentile score is 800, so that means the 76th percentile is also 800, as well as the 80th percentile, 90th percentile, etc. </p>
<p>
My calculation was quite similar to yours. Both involved extrapolating based on a linear progression. I used the nearest available percentile listed in the CDS, while you limited yourself to just the 25th and 75th percentile listings and used those to extrapolate the wide range of percentiles in between. You REALLY HAVE TO calculate mathematically just as much so for your method, so as you said your method “becomes ridiculous.”</p>
<p>“if you know the 75th percentile score? In Harvard’s case, the 75th percentile score is 800, so that means the 77th percentile is also 800”</p>
<p>And top 400 of Cornell’s 3127 class? What is the score and where is the data? Your calculation is DIFFERENT from mine. I use data, you use imagination.</p>
<p>Let’s look at Cornell. We know the following:</p>
<p>Verbal – 53rd percentile = 700 and 75th percentile = 740.</p>
<p>The rate is 40 SAT points across 22% or 1.8 SAT points per percent. We want to estimate 12% above 75th percentile, so this gives 740 + 12% * 1.8 SAT points per % ~= 760.</p>
<p>While the progression is not expected to be a linear ideal, this methodology still seems more accurate than your method of extrapolating Harvard’s 800th best score assuming Harvard’s 800 math SAT score begins on the 75th percentile, when it actually may begin well below 75th, such as on the 68th percentile.</p>
<p>“We want to estimate 12% above 75th percentile”</p>
<p>This is my point exactly, no data above 75th percentile, no ESTIMATIOM/imagination is valid. The concentration of scores at the top are not exactly evenly distributed. You didn’t get it. Sigh.</p>
<p>Within 25-75 percentile, comparing to % of students scored 700 and above, it indicates the distribution is quite even. Above 75th percentile is an unknown distribution.</p>
<p>To show 25-75% is quite evenly distributed–</p>
<p>Get the data for the % of student scored 700 and above on Reading/M/W, calculate the score of that percentile, they are all pretty close to 700)</p>
<h2>R…M…W</h2>
<p>73, 76, 64 Vandy…694.8—705.2–694
69, 76, 76 Stanford…694-----698.2–700
62, 73, 68 Penn…698----695.4-- 692.6</p>
<p>
So the concentration of scores between the 25th and 75th percentile is evenly distributed, but the concentration of scores slightly outside of this range is not? If a college has an 800 in their 75th percentile, such as Harvard and Yale, how can you assume there is a linear distribution of scores between the 65th and 75th percentile? Even without the 800 issue, the scores are not expected to be linearly distributed in your range. Instead scores are expected to form a bell curve, with possible skews or subcurves within that for multiple groups/colleges. For example, there might be one bell curve for the engineering school summed with a bell curve for the humanities school.</p>
<p>When you account for a non-even distribution by using a mean and standard deviation bell curve model, it has little effect on the Cornell socres. For example, in the Cornell example above, the CDS data 25th and 75th percentile values you are so fond of result in an approximate mean SAT score of ~690 with 1 standard deviation of ~60. The percentile of the 400th best student at Cornell corresponds to mean + ~1.15 SDs = 690 +60*1.15 = 759. Recall the linear distribution method approximated a score of 760, nearly the same value. So both the linear model and a normal distribution model to account for non-even distribution give a nearly identical result of 760 for 400th best student at Cornell.</p>
<h2>
Note that the majority of your estimations use the 75th percentile to estimate the 73-76th percentile. Obviously this is too small an offset to show a significant error. Your worst listed error is 7.4 off when measuring 7% away from 75th. If this same error rate continued when getting further away from the 75th percentile, it’s easy to get errors of +/- 10 to 20. You can get even further off in the situations I discussed earlier, such as the 800 issue. With this degree of accuracy it’s silly to list results down to hundredths place precision, as you have been doing. </p>
<p>Along the same lines, the error rate doesn’t blow up to >20 if you measure at the 80th percentile instead of 68th. It’s still a rough estimate, getting less accurate the further you get away from the available data. Sure the rate of error increases at a greater rate for more than 75th percentile than less than 75th percentile (until hitting 800), but it’s still accurate enough to generate an approximation and ranking order.</p>
<p>Sigh. Going in circles! Get in range of data! End of discussion.</p>
<p>Instead of SAT scores, let’s consider a coin flip. A coin is flipped 100 times repeatedly, resulting in 25th and 75th percentile number of heads are 46 and 54. Would you say that it’s impossible to accurately estimate what the number of heads at 80th percentile or 85th percentile are because they are out of the 25-75th percentile range? Or would you use a standard binomial distribution approach to estimate the results, like countless calculators and equations listed on the web?</p>
<p>In this thread, I’ve mentioned that I didn’t think choosing the SAT score 400th/800th/1600th best student in the freshman class while excluding a large number of colleges was a useful measure. Rather than just complain, I’ve added a list that I think is more interesting. The tables rank colleges based on both the estimated number of perfect 800 SAT scores and the estimated chance of a student having an 800 score. Estimates were done by cut and pasting the data from the thread at <a href=“http://talk.collegeconfidential.com/college-search-selection/1524929-selectivity-schools-based-2012-13-standardized-test-scores.html[/url]”>http://talk.collegeconfidential.com/college-search-selection/1524929-selectivity-schools-based-2012-13-standardized-test-scores.html</a> into a spreadsheet and doing a normal distribution approximation using OpenOffice’s stat functions. When 800s occurred below 75th percentile a different method was used, which increases expected rate of error, particularly for Caltech, MIT, and Mudd. The number of students is based on multiplying chance of 800 by size of freshman class, as if all students submitted SAT and none ACT. Rankings are ordered first by math, then by verbal.</p>
<p>Colleges With Highest Percentage of Students Getting 800 on SAT
- Caltech – 53% Math, 20% Verbal
- MIT – 41% Math, 14% Verbal
- Harvey Mudd – 41% Math, 13% Verbal
4 (Tie). Princeton – 25% Math, 20% Verbal
4 (Tie). Yale – 25% Math, 20% Verbal - Carnegie Mellon – 21% Math, 5% Verbal
- University of Illinois – 21% Math, 3% Verbal
- Harvard – 20% Math, 25% Verbal
- Stanford – 20% Math, 17% Verbal
10 (Tie). Columbia – 20% Math, 16% Verbal
10 (Tie). Chicago – 20% Math, 16% Verbal - Vanderbilt – 20% Math, 12% Verbal
- WUSTL – 19% Math, 11% Verbal
- Cornell – 18% Math, 7% Verbal
- Dartmouth – 17% Math, 18% Verbal
- University of Pennsylvania – 16% Math, 11% Verbal
- Northwestern – 16% Math, 9% Verbal
- Duke – 16% Math, 8% Verbal
- Rice – 16% Math, 8% Verbal
- Cal Berkeley – 16% Math, 6% Verbal</p>
<p>Colleges With Highest Number of Students Getting 800 on SAT
- University of Illinois – 1470 Math, 190 Verbal
- University of Michigan – 750 Math, 90 Verbal
- Cal Berkeley – 650 Math, 240 Verbal
- Cornell – 570 Math, 220 Verbal
- USC – 560 Math, 180 Verbal
- UCLA – 550 Math, 190 Verbal
- University of Wisconsin – 520 Math, 100 Verbal
- University of Minnesota – 490 Math, 260 Verbal
- MIT – 470 Math, 160 Verbal
- University of Pennsylvania – 410 Math, 280 Verbal</p>
<p>This is really interesting. Nice to see University of Minnesota on one of the lists.</p>
<p>I’ve listed the data sorted by verbal instead of math. This changes the results significantly. Illinois was in a class by itself on number of top math scores (class size of ~7000 with 790 75th percentile), but did not reach top 10 in verbal.</p>
<p>Colleges With Highest Percentage of Students Getting 800 on Verbal SAT
- Harvard – 25% Verbal, 20% Math,
- Caltech – 20% Verbal, 53% Math,
3 (Tie). Princeton – 20% Verbal, 25% Math
3 (Tie). Yale – 20% Verbal, 25% Math - Dartmouth – 18% Verbal, 17% Math
- Stanford – 17% Verbal, 20% Math,
- Swarthmore – 17% Verbal, 14% Math
8 (Tie). Columbia – 16% Verbal, 20% Math
8 (Tie). Chicago – 16% Verbal, 20% Math - Williams – 15% Verbal, 12% Math</p>
<p>Colleges With Highest Number of Students Getting 800 on Verbal SAT
- Harvard – 430 Verbal, 340 Math
- Stanford – 300 Verbal, 360 Math
3 (Tie). Princeton – 280 Verbal, 340 Math
3 (Tie). Yale – 280 Verbal, 340 Math - University of Pennsylvania – 280 Verbal, 410 Math
- University of Minnesota – 260 Verbal, 490 Math
- Cal Berkeley – 240 Verbal, 650 Math
- Chicago – 230 Verbal, 310 Math
- Columbia – 220 Verbal, 290 Math
- Cornell – 220 Verbal, 570 Math</p>
<p>
</p>
<p>I think you may want to rerun your stats. There seem to be obvious errors. I don’t see how Cornell would have higher percentage when it has lower range than, say, Duke.</p>
<p>The same goes for Cal Berkeley. Seems like the percentages were overstated relative to others.</p>
<p>
Cornell and Duke both have a 75th percentile of 780 for math. Cornell has a slightly higher estimated 800 due a larger standard deviation of scores than Duke. This may relate to a wider range of scores between Cornell’s schools. For example, Cornell’s engineering school may be packed with 780-800s math, while the hotel school may not. In verbal Duke is slightly higher than Cornell, consistent with slightly higher stats. Berkeley has a 770 in 75th percentile, so it’s expected to be slightly below 780s with similar SD, as it is. Note that the range you quoted only has a 2% variation in estimated chance, from highest to lowest. The printed values are approximations, not exact measures. Beyond the methodology approximations, even the CDS rounding (for example 774 rounded to 770 vs 776 rounded to 780) could cause a 2% variation between similar data. As such the actual order may vary when results are this close to one another.</p>
<p>Why would the CDS use rounding? They report scores using … Percentiles, and there is no such thing as a 774 or 776 75th percentile. All scores are in multiples of ten.</p>
<p>I meant that the approximation treats scores as a continuous distribution, rather than incrementing in steps of 10, like the CDS data. A reported CDS score of 780 may be right on the border of the percentile on which the 780 score changes to a 770, in which case an ideal continuous distribution should model it as ~775 instead of 780. The difference in predicted chance between the continuous dist. 775 with a CDS reported 780 and a continuous dist. 785 with a CDS reported 780 is a greater number of percentage points than the range of results in the quoted post.</p>