Statistical Distribution of Childhood IQ Scores

By John Scoville

I got the idea for the log-normal distribution from Robert Dick, who pointed out that this distribution would eliminate the possibility of negative scores inherent in the deviation system.  I worked on this a bit and found that the log-normal distribution was inadequate for scores less than 100 - the falloff is simply too much to account for the observed scores.  It did, however, match the semi-empirical data of Vernon Sare very well.  I feel that it warrants further investigation.

Background

IQ, or intelligence quotient, is generally used to refer to the ratio of a child's mental age to chronological age.  The formula (MA/CA)*100 yields the IQ score.  For example, an eight-year-old that correctly answers as many problems on a test as an average ten-year-old would receive an IQ score of (10/8)*100=125.  The distribution of IQ scores was found to have a standard deviation of about 16.  Score distributions seemed to fit the normal curve, so it was assumed that the distribution was normal.  A score of 116 was one standard deviation from the norm, corresponding to the 84th percentile, and a score of 132 was about two standard deviations from the norm, corresponding to the 98th percentile in the general population.  Abnormal behavior becames evident, however, beyond this point.  Many more extreme scores were observed than would be predicted by a normal curve.  For a chart of deviation IQs based on the normal curve, look here.Observed rarities of high scores were between the normal distribution (too rare) and logarithmic distribution (too common).   A logical candidate for the 'true' distribution would be the log-normal distribution.

Log-normal distribution

In a log-normal distribution of IQ scores, the logarithm of the IQ scores would be normally distributed.  ln(MA/CA) is now a normal random variable.   Equivalently, ln(MA)-ln(CA) is normally distributed.  If the log-normal distribution is valid, the latter may have some interesting effects on our concept of IQ and mental development in general.  As we will see in a moment, it might be more valid to say that the difference of the logarithms of the mental age and chronological age are normally distributed (abs[ln(MA)-ln(CA)] is normally distributed).  Near the mean, the log-normal distribution is very close to the normal distribution, explaining why most scores are normally distributed.  Far from the mean, however, rarities differ wildly from what would be expected from a normal curve.  These scores are so rare, however, they have little bearing on the standard deviation of the population.

Sare's Findings

In his 1951 master's thesis at the University of London, Vernon Sare predicted, based on empirical data, that ratio scores should be distributed as thus:

 Ratio IQ Sare's predicted rarity Normal (s.d. 16) rarity 210 1/2,730,000 1/278,000,000,000 200 1/532,000 1/5,870,000,000 190 1/109,000 1/93,300,000 180 1/72,000 1/3,490,000 170 1/5,040 1/167,000 160 1/1,170 1/11,500 150 1/286 1/1,140 140 1/80 1/161

Producing the log-normal standard deviation

To each ratio IQ in the previous table, I assigned an approximate sigma (# of standard deviations from the mean) value to correspond to Sare's predicted rarity.  They are 4.94, 4.63, 4.3, 4.19, 3.56, 3.13, 2.69, and 2.25, respectively.  To determine the standard deviation of the log-normal distribution, I divided the natural logarithm of each ratio (IQ/100) by its corresponding sigma score.  The results are as follows:

 Ratio ln(ratio) ln(ratio)/sigma 2.1 0.742 0.1501 2.0 0.693 0.1497 1.9 0.642 0.1493 1.8 0.587 0.1402* 1.7 0.531 0.1491 1.6 0.470 0.1501 1.5 0.405 0.1507 1.4 0.336 0.1495

All the results, with the exception of the 180 IQ* data point, fall in the surprisingly narrow range of 0.149-0.151!  With such consistency among the other points, it appears that Sare's estimate for the rarity of the 180 IQ point could be in error.   Taking 0.15 as the standard deviation of ln(MA/CA), we can produce the table at the bottom of this document.  The first column is the Ratio IQ.  The second column is MA/CA, or Ratio IQ/100.  The third column is the natural logarithm of the MA/CA.   The fourth column divides the ln(MA/CA) by 0.15 to produce a sigma score (z-score).   The final column performs 100+16*sigma to produce a corresponding 16-point S.D. deviation IQ (DIQ) for each ratio IQ.  The deviation IQ's, which would be predicted from the normal curve, match the ratio scores perfectly until IQ 120.  A noticable departure does not occur until the 140s.  Thus, less than one percent of the population would be noticably affected by the deviation-ratio IQ gap, explaining why it has been largely ignored over the years.  The study of this gap has many applications, particularly in the study of adult IQs, which are almost always produced by mapping one's percentile in the population to a (possibly misrepresentative) deviation IQ.   It also offers a partial explanation for the radical discrepancy in childhood IQ scores as reported by ratio tests (such as the Stanford-Binet L-M) and deviation tests (such as the WISC-R).

Shortcomings of the log-normal distribution of IQ scores

The most serious shortcoming of the log-normal distribution comes below the mean IQ of 100.  The falloff to negative infinity occurs much too rapidly - an IQ of 50 would have the same predicted rarity as an IQ of 200!  The distribution may produce accurate results under normal circumstances, but factors such as chromosomal abnormalities in the general population severely affect the left half of the curve.  A rough approximation could be made by mirroring the curve by normally distributing abs(ln[MA/CA]) or abs(ln[MA]-ln[CA]) as mentioned above.  In reality, however, the left end of the curve may have an entirely different distribution.

Table of Ratio IQ Scores on a log-normal distribution

 Ratio IQ MA/CA ln(MA/CA) sigma DIQ 100 1 0 0 100 101 1.01 0.00995 0.06633 101.0 102 1.02 0.01980 0.13201 102.1 103 1.03 0.02955 0.19705 103.1 104 1.04 0.03922 0.26147 104.1 105 1.05 0.04879 0.32526 105.2 106 1.06 0.05826 0.38845 106.2 107 1.07 0.06765 0.45105 107.2 108 1.08 0.07696 0.51307 108.2 109 1.09 0.08617 0.57451 109.1 110 1.1 0.09531 0.63540 110.1 111 1.11 0.10436 0.69573 111.1 112 1.12 0.11332 0.75552 112.0 113 1.13 0.12221 0.81478 113.0 114 1.14 0.13102 0.87352 113.9 115 1.15 0.13976 0.93174 114.9 116 1.16 0.14842 0.98946 115.8 117 1.17 0.15700 1.04669 116.7 118 1.18 0.16551 1.10342 117.6 119 1.19 0.17395 1.15968 118.5 120 1.2 0.18232 1.21547 119.4 121 1.21 0.19062 1.27080 120.3 122 1.22 0.19885 1.32567 121.2 123 1.23 0.20701 1.38009 122.0 124 1.24 0.21511 1.43407 122.9 125 1.25 0.22314 1.48762 123.8 126 1.26 0.23111 1.54074 124.6 127 1.27 0.23901 1.59344 125.4 128 1.28 0.24686 1.64573 126.3 129 1.29 0.25464 1.69761 127.1 130 1.3 0.26236 1.74909 127.9 131 1.31 0.27002 1.80018 128.8 132 1.32 0.27763 1.85087 129.6 133 1.33 0.28517 1.90119 130.4 134 1.34 0.29266 1.95113 131.2 135 1.35 0.30010 2.00069 132.0 136 1.36 0.30748 2.04989 132.7 137 1.37 0.31481 2.09873 133.5 138 1.38 0.32208 2.14722 134.3 139 1.39 0.32930 2.19535 135.1 140 1.4 0.33647 2.24314 135.8 141 1.41 0.34358 2.29059 136.6 142 1.42 0.35065 2.33771 137.4 143 1.43 0.35767 2.38449 138.1 144 1.44 0.36464 2.43095 138.8 145 1.45 0.37156 2.47709 139.6 146 1.46 0.37843 2.52290 140.3 147 1.47 0.38526 2.56841 141.0 148 1.48 0.39204 2.61361 141.8 149 1.49 0.39877 2.65850 142.5 150 1.5 0.40546 2.70310 143.2 151 1.51 0.41210 2.74739 143.9 152 1.52 0.41871 2.79140 144.6 153 1.53 0.42526 2.83511 145.3 154 1.54 0.43178 2.87854 146.0 155 1.55 0.43825 2.92169 146.7 156 1.56 0.44468 2.96457 147.4 157 1.57 0.45107 3.00717 148.1 158 1.58 0.45742 3.04949 148.7 159 1.59 0.46373 3.09156 149.4 160 1.6 0.47000 3.13335 150.1 161 1.61 0.47623 3.17489 150.7 162 1.62 0.48242 3.21617 151.4 163 1.63 0.48858 3.25720 152.1 164 1.64 0.49469 3.29797 152.7 165 1.65 0.50077 3.33850 153.4 166 1.66 0.50681 3.37878 154.0 167 1.67 0.51282 3.41882 154.7 168 1.68 0.51879 3.45862 155.3 169 1.69 0.52472 3.49819 155.9 170 1.7 0.53062 3.53752 156.6 171 1.71 0.53649 3.57662 157.2 172 1.72 0.54232 3.61549 157.8 173 1.73 0.54812 3.65414 158.4 174 1.74 0.55388 3.69256 159.0 175 1.75 0.55961 3.73077 159.6 176 1.76 0.56531 3.76875 160.3 177 1.77 0.57097 3.80653 160.9 178 1.78 0.57661 3.84408 161.5 179 1.79 0.58221 3.88143 162.1 180 1.8 0.58778 3.91857 162.6 181 1.81 0.59332 3.95551 163.2 182 1.82 0.59883 3.99224 163.8 183 1.83 0.60431 4.02877 164.4 184 1.84 0.60976 4.06510 165.0 185 1.85 0.61518 4.10123 165.6 186 1.86 0.62057 4.13717 166.1 187 1.87 0.62593 4.17292 166.7 188 1.88 0.63127 4.20847 167.3 189 1.89 0.63657 4.24384 167.9 190 1.9 0.64185 4.27902 168.4 191 1.91 0.64710 4.31402 169.0 192 1.92 0.65232 4.34883 169.5 193 1.93 0.65752 4.38346 170.1 194 1.94 0.66268 4.41791 170.6 195 1.95 0.66782 4.45219 171.2 196 1.96 0.67294 4.48629 171.7 197 1.97 0.67803 4.52022 172.3 198 1.98 0.68309 4.55397 172.8 199 1.99 0.68813 4.58756 173.4 200 2 0.69314 4.62098 173.9 201 2.01 0.69813 4.65423 174.4 202 2.02 0.70309 4.68731 174.9 203 2.03 0.70803 4.72023 175.5 204 2.04 0.71294 4.75299 176.0 205 2.05 0.71783 4.78559 176.5 206 2.06 0.72270 4.81803 177.0 207 2.07 0.72754 4.85032 177.6 208 2.08 0.73236 4.88245 178.1 209 2.09 0.73716 4.91442 178.6 210 2.1 0.74193 4.94624 179.1 211 2.11 0.74668 4.97791 179.6 212 2.12 0.75141 5.00944 180.1 213 2.13 0.75612 5.04081 180.6 214 2.14 0.76080 5.07203 181.1 215 2.15 0.76546 5.10311 181.6 216 2.16 0.77010 5.13405 182.1 217 2.17 0.77472 5.16484 182.6 218 2.18 0.77932 5.19549 183.1 219 2.19 0.78390 5.22601 183.6 220 2.2 0.78845 5.25638 184.1 221 2.21 0.79299 5.28661 184.5 222 2.22 0.79750 5.31671 185.0 223 2.23 0.80200 5.34667 185.5 224 2.24 0.80647 5.37650 186.0 225 2.25 0.81093 5.40620 186.4 226 2.26 0.81536 5.43576 186.9 227 2.27 0.81977 5.46519 187.4 228 2.28 0.82417 5.49450 187.9 229 2.29 0.82855 5.52367 188.3 230 2.3 0.83290 5.55272 188.8 231 2.31 0.83724 5.58165 189.3 232 2.32 0.84156 5.61044 189.7 233 2.33 0.84586 5.63912 190.2 234 2.34 0.85015 5.66767 190.6 235 2.35 0.85441 5.69610 191.1 236 2.36 0.85866 5.72441 191.5 237 2.37 0.86288 5.75259 192.0 238 2.38 0.86710 5.78066 192.4 239 2.39 0.87129 5.80862 192.9 240 2.4 0.87546 5.83645 193.3 241 2.41 0.87962 5.86417 193.8 242 2.42 0.88376 5.89178 194.2 243 2.43 0.88789 5.91927 194.7 244 2.44 0.89199 5.94665 195.1 245 2.45 0.89608 5.97392 195.5 246 2.46 0.90016 6.00107 196.0 247 2.47 0.90421 6.02812 196.4 248 2.48 0.90825 6.05505 196.8 249 2.49 0.91228 6.08188 197.3 250 2.5 0.91629 6.10860 197.7

Portions of articles copyright John Scoville

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