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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