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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