Adjustment of the protonorms to norms conversion - Old reports

Adjustment of the protonorms to norms conversion (females)

© 25 February 2013 Paul Cooijmans

The procedure

A sample of tests is chosen by these criteria:

The combined score distribution thus obtained is meant to be representative for candidates on serious high-range intelligence tests. A score distribution is derived, using all of the scores on the chosen tests. "Proportions outscored" are computed for each protonorm, within-sex, to the number of decimals needed to retain the information of the distribution. Sex-combined proportions outscored are computed by taking for each protonorm the weighted mean of the female and male proportions outscored (weighted by numbers that form a reduced fraction of the female/male ratio among candidates on this sample of tests; this ratio is 74/872, reduced to 37/436).

The Protonorms table shows the resulting norms. Below is the distribution.

Test-combined protonorm histogram with proportions outscored for females

In parenthesis the proportion outscored value for scores above the present score but lower than the next-higher score in the table.

Prot.Prop.# scores (* = 1 score)
00.007 (0.014) *
670.020 (0.027) *
700.034 (0.041) *
970.047 (0.054) *
1140.061 (0.068) *
1350.074 (0.081) *
2000.088 (0.095) *
2070.101 (0.108) *
2400.115 (0.122) *
2670.135 (0.149) **
2680.155 (0.162) *
2820.176 (0.189) **
2830.196 (0.203) *
2870.209 (0.216) *
2910.223 (0.230) *
2970.236 (0.243) *
3130.250 (0.257) *
3200.277 (0.297) ***
3280.304 (0.311) *
3330.351 (0.392) ******
3380.399 (0.405) *
3400.432 (0.459) ****
3470.466 (0.473) *
3570.480 (0.486) *
3600.493 (0.500) *
3670.507 (0.514) *
3800.527 (0.541) **
3810.547 (0.554) *
3920.561 (0.568) *
3970.574 (0.581) *
3980.588 (0.595) *
4000.608 (0.622) **
4040.635 (0.649) **
4120.655 (0.662) *
4140.669 (0.676) *
4170.682 (0.689) *
4330.703 (0.716) **
4350.723 (0.730) *
4400.743 (0.757) **
4430.764 (0.770) *
4530.784 (0.797) **
4570.804 (0.811) *
4580.818 (0.824) *
4600.831 (0.838) *
4730.845 (0.851) *
4840.858 (0.865) *
4860.872 (0.878) *
4870.892 (0.905) **
4910.912 (0.919) *
4930.926 (0.932) *
5000.939 (0.946) *
5130.953 (0.959) *
5200.966 (0.973) *
5590.980 (0.986) *
5690.993 (1.000) *

n = 74

Tests used

Adjustment of the protonorms to norms conversion

© 17 January 2011 Paul Cooijmans

The procedure

A sample of tests is chosen by these criteria:

The combined score distribution thus obtained is meant to be representative for candidates on serious high-range intelligence tests. Separate score distributions are derived for females and males, using all of the scores on the chosen tests. "Proportions outscored" are computed for each protonorm, within-sex, to the number of decimals needed to retain the information of the distribution. Sex-combined proportions outscored are computed by taking for each protonorm the weighted mean of the female and male proportions outscored (weighted by numbers that form a reduced fraction of the female/male ratio).

The Protonorms table shows the resulting norms. Below are the distributions.

Test-combined protonorm histogram with proportions outscored for females

In parenthesis the proportion outscored value for scores above the present score but lower than the next-higher score in the table.

Prot.Prop.# scores (* = 1 score)
00.009 (0.019) *
670.028 (0.037) *
970.046 (0.056) *
2000.065 (0.074) *
2070.083 (0.093) *
2400.102 (0.111) *
2670.130 (0.148) **
2820.157 (0.167) *
2830.176 (0.185) *
2870.194 (0.204) *
2910.213 (0.222) *
2970.231 (0.241) *
3130.250 (0.259) *
3200.287 (0.315) ***
3280.324 (0.333) *
3330.389 (0.444) ******
3400.481 (0.519) ****
3800.537 (0.556) **
3810.565 (0.574) *
3970.583 (0.593) *
4000.611 (0.630) **
4040.648 (0.667) **
4120.676 (0.685) *
4140.694 (0.704) *
4330.713 (0.722) *
4390.731 (0.741) *
4430.750 (0.759) *
4530.778 (0.796) **
4580.806 (0.815) *
4600.824 (0.833) *
4730.843 (0.852) *
4800.861 (0.870) *
4860.880 (0.889) *
4870.898 (0.907) *
4930.917 (0.926) *
5130.935 (0.944) *
5200.954 (0.963) *
5230.972 (0.981) *
5690.991 (1.000) *

n = 54

Test-combined protonorm histogram with proportions outscored for males

In parenthesis the proportion outscored value for scores above the present score but lower than the next-higher score in the table.

Prot.Prop.# scores (* = 1 score)
-130.0007 (0.0014) *
970.0021 (0.0027) *
1000.0034 (0.0041) *
1070.0055 (0.0068) **
1670.0075 (0.0082) *
1800.0096 (0.0110) **
1840.0116 (0.0123) *
1900.0130 (0.0137) *
1930.0144 (0.0151) *
1970.0158 (0.0164) *
2000.0185 (0.0205) ***
2060.0212 (0.0219) *
2130.0226 (0.0233) *
2240.0240 (0.0247) *
2330.0253 (0.0260) *
2530.0267 (0.0274) *
2670.0295 (0.0315) ***
2780.0322 (0.0329) *
2790.0336 (0.0342) *
2800.0349 (0.0356) *
2820.0370 (0.0384) **
2830.0397 (0.0411) **
2860.0418 (0.0425) *
2870.0432 (0.0438) *
2930.0452 (0.0466) **
2970.0479 (0.0493) **
3000.0514 (0.0534) ***
3010.0541 (0.0548) *
3020.0555 (0.0562) *
3060.0568 (0.0575) *
3070.0589 (0.0603) **
3080.0610 (0.0616) *
3100.0623 (0.0630) *
3120.0644 (0.0658) **
3130.0685 (0.0712) ****
3180.0733 (0.0753) ***
3190.0760 (0.0767) *
3200.0774 (0.0781) *
3220.0795 (0.0808) **
3230.0815 (0.0822) *
3270.0897 (0.0973) ***********
3290.0986 (0.1000) **
3300.1007 (0.1014) *
3310.1021 (0.1027) *
3330.1075 (0.1123) *******
3370.1137 (0.1151) **
3380.1158 (0.1164) *
3390.1185 (0.1205) ***
3400.1322 (0.1438) *****************
3410.1466 (0.1493) ****
3420.1514 (0.1534) ***
3430.1541 (0.1548) *
3440.1555 (0.1562) *
3450.1568 (0.1575) *
3470.1582 (0.1589) *
3500.1596 (0.1603) *
3520.1610 (0.1616) *
3530.1651 (0.1685) *****
3540.1699 (0.1712) **
3550.1726 (0.1740) **
3560.1747 (0.1753) *
3570.1767 (0.1781) **
3580.1788 (0.1795) *
3590.1815 (0.1836) ***
3600.1904 (0.1973) **********
3610.1979 (0.1986) *
3620.2021 (0.2055) *****
3640.2075 (0.2096) ***
3650.2110 (0.2123) **
3660.2137 (0.2151) **
3670.2301 (0.2452) **********************
3680.2473 (0.2493) ***
3690.2521 (0.2548) ****
3700.2568 (0.2589) ***
3710.2630 (0.2671) ******
3730.2685 (0.2699) **
3770.2726 (0.2753) ****
3790.2788 (0.2822) *****
3800.2938 (0.3055) *****************
3810.3068 (0.3082) **
3830.3103 (0.3123) ***
3850.3130 (0.3137) *
3860.3171 (0.3205) *****
3870.3404 (0.3603) *****************************
3880.3658 (0.3712) ********
3890.3740 (0.3767) ****
3900.3788 (0.3808) ***
3910.3829 (0.3849) ***
3920.3863 (0.3877) **
3930.3993 (0.4110) *****************
3940.4137 (0.4164) ****
3970.4178 (0.4192) **
4000.4260 (0.4329) **********
4010.4342 (0.4356) **
4020.4363 (0.4370) *
4030.4384 (0.4397) **
4040.4445 (0.4493) *******
4070.4541 (0.4589) *******
4090.4630 (0.4671) ******
4100.4678 (0.4685) *
4110.4692 (0.4699) *
4120.4733 (0.4767) *****
4130.4945 (0.5123) **************************
4140.5151 (0.5178) ****
4150.5192 (0.5205) **
4160.5212 (0.5219) *
4170.5253 (0.5288) *****
4180.5301 (0.5315) **
4200.5425 (0.5534) ****************
4210.5555 (0.5575) ***
4220.5582 (0.5589) *
4270.5623 (0.5658) *****
4280.5692 (0.5726) *****
4310.5733 (0.5740) *
4320.5760 (0.5781) ***
4330.5932 (0.6082) **********************
4340.6123 (0.6164) ******
4360.6178 (0.6192) **
4370.6205 (0.6219) **
4390.6240 (0.6260) ***
4400.6384 (0.6507) ******************
4410.6527 (0.6548) ***
4420.6555 (0.6562) *
4430.6582 (0.6603) ***
4440.6610 (0.6616) *
4470.6705 (0.6795) *************
4480.6836 (0.6877) ******
4520.6890 (0.6904) **
4530.7048 (0.7192) *********************
4540.7212 (0.7233) ***
4560.7253 (0.7274) ***
4570.7288 (0.7301) **
4590.7315 (0.7329) **
4600.7342 (0.7356) **
4610.7363 (0.7370) *
4640.7377 (0.7384) *
4670.7438 (0.7493) ********
4690.7500 (0.7507) *
4730.7527 (0.7548) ***
4760.7568 (0.7589) ***
4770.7596 (0.7603) *
4780.7623 (0.7644) ***
4790.7658 (0.7671) **
4800.7712 (0.7753) ******
4810.7760 (0.7767) *
4820.7808 (0.7849) ******
4830.7877 (0.7904) ****
4840.7911 (0.7918) *
4860.7925 (0.7932) *
4870.8048 (0.8164) *****************
4890.8171 (0.8178) *
4900.8185 (0.8192) *
4910.8212 (0.8233) ***
4920.8247 (0.8260) **
4930.8281 (0.8301) ***
4950.8308 (0.8315) *
4980.8322 (0.8329) *
5000.8384 (0.8438) ********
5030.8445 (0.8452) *
5040.8466 (0.8479) **
5050.8486 (0.8493) *
5070.8500 (0.8507) *
5100.8514 (0.8521) *
5130.8548 (0.8575) ****
5170.8582 (0.8589) *
5200.8616 (0.8644) ****
5250.8651 (0.8658) *
5270.8685 (0.8712) ****
5280.8719 (0.8726) *
5330.8760 (0.8795) *****
5340.8801 (0.8808) *
5360.8815 (0.8822) *
5370.8829 (0.8836) *
5400.8849 (0.8863) **
5430.8884 (0.8904) ***
5440.8911 (0.8918) *
5450.8925 (0.8932) *
5470.8945 (0.8959) **
5480.8979 (0.9000) ***
5500.9007 (0.9014) *
5510.9027 (0.9041) **
5520.9048 (0.9055) *
5530.9068 (0.9082) **
5540.9096 (0.9110) **
5570.9130 (0.9151) ***
5580.9164 (0.9178) **
5590.9192 (0.9205) **
5600.9226 (0.9247) ***
5610.9253 (0.9260) *
5620.9267 (0.9274) *
5650.9281 (0.9288) *
5670.9308 (0.9329) ***
5690.9336 (0.9342) *
5700.9349 (0.9356) *
5800.9377 (0.9397) ***
5840.9411 (0.9425) **
5870.9438 (0.9452) **
5930.9466 (0.9479) **
6000.9493 (0.9507) **
6030.9521 (0.9534) **
6050.9575 (0.9616) ******
6060.9623 (0.9630) *
6070.9637 (0.9644) *
6130.9651 (0.9658) *
6170.9664 (0.9671) *
6200.9678 (0.9685) *
6270.9705 (0.9726) ***
6320.9740 (0.9753) **
6330.9767 (0.9781) **
6400.9795 (0.9808) **
6490.9815 (0.9822) *
6500.9836 (0.9849) **
6530.9877 (0.9904) ****
6670.9911 (0.9918) *
6720.9925 (0.9932) *
6800.9952 (0.9973) ***
6810.9979 (0.9986) *
6870.9993 (1.0000) *

n = 730

Tests used

Adjustment of the relation between protonorms and I.Q.

A smoothening toward linearity has been applied on the local scale without significantly changing the norms on cardinal points or on the larger scale. As a result, the I.Q. distribution has regained its original irregularity (which had been retained by the protonorm distribution), with locally sharp differences between occurrence frequencies of neighbouring I.Q.s. In the 2009 adjustment these irregularities had been straightened out by forcing the distribution into a shape (the "10 000 model", which incidentally is not a "normal" distribution or "bell curve") with (above the mode) gradually declining numbers of scores per I.Q. slot. This approach has been discontinued in anticipation of a better way to norm protonorms to I.Q.s.

For clarity, note the paradox: A more linear relation between protonorm and I.Q. results in an irregular I.Q. distribution; a less linear relation between protonorm and I.Q. is needed to obtain a regular I.Q. distribution.

Adjustment of the protonorms to norms conversion

© January 2009 Paul Cooijmans

The procedure

A large sample of tests is chosen, such that the 1455 scores thereon are representative for the performance of candidates on true, serious high-range intelligence tests. Separate score distributions are derived for females and males, using all of the scores on the chosen tests. "Proportions outscored" are computed for each protonorm, within-sex, to four decimals to retain the full information of the distributions. Sex-combined proportions outscored are computed by taking for each protonorm the weighted mean of the female and male proportions outscored (weighted by the numbers of females [110] and males [1345]).

The sex-combined proportions outscored are linked to I.Q.s using the "10 000" model of the high-range population. General population proportions outscored are linked to the I.Q.s using the "normal distribution". Note that the I.Q.s obtained such are fully liberated from those of regular psychology, childhood tests, or other high-range test designers. They are normed directly on the high-range population, and any required correction to a score level needs to be made in one place only (the "10 000" model).

The Protonorms table shows the current norming.

Female distribution

67*
107*
135*
153*
179*
207**
212*
214*
227**
233*
234*
247*
260*
265**
266*
267**
279*
280**
282*
283*
287**
291*
294*
297*
300**
306***
307*
308***
313*
320**
323*
327*
335*
337**
340****
341*
357*
360**
361*
367*****
372*
380***
381**
392*
393*****
413*
414*
420*****
433***
435*
440***
453******
458*
459**
461*
485*
486*
487*
493**
500**
507*
547**
569*
573*
580*

Male score distribution

59*
61*
63*
65*
97*
107***
153*
187*
190*
193*
200**
207**
212**
220**
227**
228*
232*
234**
253*
255***
266*
267****
268**
270*
273*
278*
279*****
280*******
281****
282*
283**
286*
287***
289*
293*****
294*
296*
297*
300*********
302*
305*
306**
307*****
308**
309**
312***
313*******
314*
315**
316*
317*
318****
319**
320***
321*
322*
323***
324***
327***************************
331**
333*
335*****
337*
338**
339*****
340****************************
341***
342***
343***
345**
347***
349*
352*
353********
354*
355*
356*
357*
359****
360****************
361*****
362***
363*
364*
365*
366*********
367********************************************************************
368*******
369***
370*****
371**
372****
374**
376**
377*********
378***
379****************
380*********************************
381****
382******
383**
385**
386**
387***********************************************************************
388*****
389****
391****
392*****
393******************************
394**********************************
395**
400****************
401**
402*
403**
407**********
409*****
410*
411**
412*****
413**************************************************
414*******
415**
416*
417**
418*
420***************************
423*
424*
426**********
427*************
428****
431*
432***
433***********************************************************
434*********
435*******
436*
438*******
439******
440***********************
441*****
442*******
443*
446**
447*************************
448***
452****
453*****************************************
454*******
459********************
460******
461*********************
465***
467*****
472*
473**************
477*
478***
479**
480********
481*
482******
483****
484*
485**
486*
487***********************************************
489*
490**
493*******
496**
499**
500***********************
501**
506***
507************
508**
513********
520*********
525*
527*
533******
536*
537*
539*
540*****
544*
545*
547************************
551**
552**
553**********
554**
558**
559****
560*******
561**
562*
565*
567******************
568****
569*
570*
573*****
580*******
581*
584**
587************
589*
593*
600*
601**
605**
606*
607***
613**
620****
627******
632**
633*
640**
647*
653***
673**
680**

The tests used

Abbreviations are used to save space:

a_10, an1, an1_nl, ass, ass_nl, bonsai, cart, cit, cit2, dlt, fin, gat, giga, lcat, limit, ltfg, ltfg_f, ltfg_nl, nem, nit, num, odds, pigs1, pq, qmc4, reason, shock, sit, stfg, sth, tfg, tteat.

The March 2007 adjustment

Paul Cooijmans

From 1995 to early 2007, the IQ norms of my tests have been established by anchoring tests to each other and to other tests through rank equation of their shared score pairs. The statistical reports have always given full insight into that procedure. As a result, the IQs from the tests have become comparable across tests, and more or less in accordance with IQs from American high-range tests. I have no way to verify the absolute height of the IQs against any population.

Next to IQ norms, I have in more recent years provided high-range norms based directly on the group of high-range candidates, in the form of "proportion outscored". These are obtained by combining all of the scored IQs from a representative selection of my tests. The advantage of these norms over IQs is that they do indicate a score's standing among a population with certainty.

To allow for readjustment of the height of the IQ norms, I have in March 2007 invented protonorms as a step in between raw scores and norms. Protonorms are generalized raw scores that allow comparison between scores on all of my tests, but do not contain information as to where that score stands in any population; they are not standard scores or quantiles. From now on, tests will be normed to protonorms in exactly the same way as they have been normed to IQs before. The initial formula to derive protonorms from the IQs of the before-protonorm era is as follows:

Protonorm = ((IQ - 80)/15) × 100

Thanks to this, a readjustment of the protonorm-IQ relation requires change in one table only, and not in every singly test norm table that still contains IQs. To avoid confusion: the above formula will NOT change when the protonorm to IQ conversion table is adjusted.

Mid-March 2007 I have adjusted the relation between protonorms and IQs, and renormed the proportion of high-range candidates outscored. For this purpose I have combined the protonorms from a number of my tests, yielding exactly 1200 scores in total. The used tests are:

The proportion of high-range candidates outscored in the protonorms to norms conversion table is based directly on these 1200 scores. The number of decimals to which the proportion is expressed is always chosen so that each value is distinguished from the previous and next different values, and the proportions therefore form a true reflection of the score distribution.

The main features of the distribution are the mode (protonorm 367) and the top (673). Really there is a modal range from 367 to 387. The modal range corresponds to IQs 135 to 138. The top to IQ 181. These are old IQs, from before the present adjustment.

Also characteristic are certain near-vacuums, such as that from 513 to 547. I have been looking at those vacuums for years already, and they have remained stable. I have been wondering if they are a real phenomenon in the population or an artifact of the tests, and for the moment I am assuming they are an artifact. I am not going into the possible exact causes of such an artifact in this report.

Below the modal range there are clearly less scores in a steeply declining curve, as a result of the tests' being too difficult for people below a certain level. In the past I would have thought this could be due to people being preselected through Mensa membership, but nowadays with everything going via the Internet this does not apply any more, so it must be the test difficulty. The modal range is apparently where the high-range distribution "touches" the Bell curve of the total population, and starts declining gradually. The high-range distribution itself is NOT a Bell curve, but an asymmetric wedge, whereby the decline from mode to top is close to linear and not significantly concave or convex.

I adjusted the modal range downward by about three IQ points, so to 132-135. I cannot verify this with certainty but I believe this brings the IQs in line with the Western European population, which is a few points of IQ smarter than the US population, that (the latter) formed the basis of the norms of a lot of high-range tests in the past. Support for this three-point correction lies in the reported scores of candidates on the Nonverbal Cognitive Performance Examination - E, by Xavier Jouve. These scores show that my old IQs are about one to two points too high in the range below about IQ 146. I put it at three to compensate for the fact (observed by me) that candidates tend to withhold their lower scores and only report their higher scores. Naturally, this correction pushes the scores below the mode down by a few points as well.

Also, I adjusted the top IQs downward to 175. Again I have no way to verify this at this moment, but a consideration is that, as I have learnt over the past years, the distribution of intelligence over the world is uneven, and the world average IQ is not is not 100 but probably below 90. This means that the highest IQs to expect with norms based on a Western population are much lower than I thought in the past, and probably between 180 and 185. So the lowering to 175 is conservative, and if in say five years from now no one has scored above the current top scores, I will adjust them upward again to around 180. There is no other way of finding out the correct height of those top scores; I will have to wait and see if the current top scores are already from the smartest people, or if the real top dogs are still to come. After some time I will conclude that the then top scores are from the smartest people, and norm them accordingly. Until then I will keep the norms conservative.

Then, I smoothened the decline from mode to top. That is, I redistributed the protonorms over the IQs so that the number of scores in each IQ slot declines smoothly by about .9 score per IQ point. It could not be made perfectly smooth on the level of single IQs, but by balancing out adjacent IQs it is close to smooth on the level of two-IQ classes. As a result of this smoothening, IQs have not come down three points rigidly, and in fact in the range of roughly 140-169 they have at some points remained about the same or even got higher. The Glia Society admission level has stayed virtually the same (protonorm 442, IQ 147).

Above the top scores I kept the initial protonorm-IQ relation from IQ 190 on, and made a smooth path to that from 176 upward. So for the Giga and Grail Societies and Isis, all of the old IQ norms remain valid.

Below the mode I smoothened the decline similarly to that from mode to top, except that there are four different slopes there instead of one: there is a steep slope from IQ 132 to 121 (a drop of about three scores per IQ point); a gradual slope from 121 to 116 (about one score per point); a shallow slope from 116 to 106 (about .2 scores/point); and a very shallow slope from 106 to the low 90s (about .04 scores/point).

Below the actual distribution of 1200 scores in protonorms; protonorms with no scores are left out.

67****
106**
153**
178*
186*
200***
207***
211***
215*
220**
227***
231*
233***
235***
240*
247****
253**
258*
260**
265**
267*******
269***
273***
277*
278******
280********
282****
287********
293**
295*
300*********
304*
305******
307*********
308*****
309**
311***
313******
315**
316**
317**
318**
319*
320*********
321**
322***
326*
327****************************
331**
333**
335***
338*
339*****
340***********************
342*****
343**
351*
353*********
355*
356*
357***
360**********************
361*
362****
364*
365******
367***************************************************************
369***
370***
371*
372*****
373**
374*
375****
378************
379***
380***********************************
382*******
384******
385**
387************************************************************
388*
389*****
391****
392*
393************************************************
394******
395**
400******************
403*
405*
407*************
409*****
411****
412**
413**************************************************
414***
415****
416****
418**
420***********************
422*****
424*
425********
426*
427***********
429****
430*
431**
433******************************************************
435*******
436*
438****
440*************************
442*
445**
447***********************
449***
451**
453*********************************
455******
458*******************
459*
460*
461*
462****************
467*********
473************
475*
476***
477**
478******
480*******
481*
482*
483**
484******
485*
486**
487********************************************
488*
489**
493********
495*
498**
500****************************
502**
504***
507*********
509**
513******
520****
527*
533****
538**
540*****
547************************
551*
553******
557**
558******
560*******
562**
564**
567**************
569**
570**
573*
580***
582*
584**
587*************
589*
600*
605*
607*******
613*
620*
627******
631*
633*
640*
647*
653**
660*
673**

Finally, here is a model of the high-range population by IQ (not the actual current distribution):

80
81
82
83
84*
85
86
87
88
89
90*
91
92
93
94
95*
96
97
98
99*
100
101
102*
103
104*
105*
106*
107**
108**
109***
110***
111***
112****
113****
114****
115****
116****
117*****
118******
119*******
120********
121**********
122************
123***************
124******************
125*********************
126************************
127***************************
128******************************
129*********************************
130************************************
131***************************************
132******************************************
133*******************************************
134*******************************************
135******************************************
136****************************************
137***************************************
138**************************************
139*************************************
140************************************
141***********************************
142**********************************
143*********************************
144********************************
145*******************************
146******************************
147*****************************
148****************************
149***************************
150**************************
151*************************
152************************
153***********************
154**********************
155*********************
156********************
157*******************
158******************
159*****************
160****************
161***************
162**************
163*************
164************
165***********
166**********
167*********
168********
169*******
170******
171*****
172****
173***
174**
175*
176
177*
178
179
180*
181
182
183
184*
185

The "10 000" model of the high-range population

© February 2008 to November 2009 Paul Cooijmans

To show in more detail how the high-range distribution may lie relative to adult population IQ, here is a table of a hypothetical sample of 10 000 candidates representative for the high-range test-taking population. The columns show the number of candidates out of these 10 000 for each IQ, the cumulative number up to that IQ, and the (range of) "proportion outscored" that would belong to that IQ.

For better understanding: The purpose of a table like this is to aid in providing an IQ as additional information next to the proportion of high-range candidates outscored, which (the latter) I see now as the primary and most important score from my tests. These high-range proportions are now obtained directly from the protonorm distribution, independently of possible IQs from prior tests. The protonorms have been "liberated" from the concept of "IQ".

The high-range proportions obtained from the protonorm distribution can be looked up in the fourth column of a table as below to find an IQ for additional information, the exact relation between proportions and IQs being less important than the proportions themselves.

IQ (SD=15)#Cum.Prop.
81000
8211.0001
8312.0002
8413.0003
8514.0004
8615.0005
8716.0006
8817.0007
8918.0008
90210.0009-.0010
91212.0011-.0012
92214.0013-.0014
93216.0015-.0016
94218.0017-.0018
95220.0019-.0020
96222.0021-.0022
97224.0023-.0024
98327.0025-.0027
99330.0028-.0030
100333.0031-.0033
101437.0034-.0037
102643.0038-.0043
103750.0044-.0050
104858.0051-.0058
1051169.0059-.0069
1061483.0070-.0083
10717100.0084-.0100
10822122.0101-.0122
10924146.0123-.0146
11026172.0147-.0172
11128200.0173-.0200
11230230.0201-.0230
11340270.0231-.0270
11445315.0271-.0315
11555370.0316-.0370
11660430.0371-.0430
11770500.0431-.0500
11883583.0501-.0583
11983666.0584-.0666
12083749.0667-.0749
12183832.0750-.0832
12284916.0833-.0916
123841000.0917-.1000
124841084.1001-.1084
1251121196.1085-.1196
1261441340.1197-.1340
1271901530.1341-.1530
1282201750.1531-.1750
1292502000.1751-.2000
1302952295.2001-.2295
1312992594.2296-.2594
1323022896.2595-.2896
1333043200.2897-.3200
1343053505.3201-.3505
1353033808.3506-.3808
1363014109.3809-.4109
1372984407.4110-.4407
1382974704.4408-.4704
1392965000.4705-.5000
1402955295.5001-.5295
1412855580.5296-.5580
1422755855.5581-.5855
1432656120.5856-.6120
1442556375.6121-.6375
1452456620.6376-.6620
1462356855.6621-.6855
1472257080.6856-.7080
1482157295.7081-.7295
1492057500.7296-.7500
1501647664.7501-.7664
1511617825.7665-.7825
1521587983.7826-.7983
1531558138.7984-.8138
1541528290.8139-.8290
1551488438.8291-.8438
1561458583.8439-.8583
1571428725.8584-.8725
1581398864.8726-.8864
1591369000.8865-.9000
1601359135.9001-.9135
1611249259.9136-.9259
1621139372.9260-.9372
1631029474.9373-.9474
164919565.9475-.9565
165799644.9566-.9644
166689712.9645-.9712
167579769.9713-.9769
168469815.9770-.9815
169359850.9816-.9850
170229872.9851-.9872
171209892.9873-.9892
172189910.9893-.9910
173169926.9911-.9926
174149940.9927-.9940
175139953.9941-.9953
176119964.9954-.9964
17799973.9965-.9973
17879980.9974-.9980
17959985.9981-.9985
18049989.9986-.9989
18139992.9990-.9992
18239995.9993.-9995
18329997.9996-.9997
18429999.9998-.9999
1851100001