Terman Study Thoughts

Emil Kirkegaard posted mean and SD for participants in the Terman study in a comment to http://infoproc.blogspot.com/2014/07/success-ability-and-all-that.html
“The average Termite was 152, sd=15.”

This seemed high if IQ is a normal distribution and Terman Study participants can be just viewed as a truncated sample of it over 135 so I decided to simulate that and see how the numbers compared. This was apparently a mistake. Changed cutoff to 140.

cutoff <- 140

#make this reproducible
set.seed(2)
iq <- rnorm(1e6, mean=100, sd=15)
summary(iq)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    23.8    89.9   100.0   100.0   110.0   174.0

studyIQ <- iq[iq>cutoff]
summary(studyIQ)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     140     141     143     144     146     174

mean(studyIQ)

## [1] 144.5

sd(studyIQ)

## [1] 4.22

plot(density(studyIQ), xlab="IQ", main="Simulated IQ Density for Terman Study")
abline(v=mean(studyIQ))

Let’s try a lognormal distribution. See http://en.wikipedia.org/wiki/Log-normal_distribution for details of how this is parameterized. I’m not sure I have the right distribution…

m <- 100
v <- 15^2
ml <- log(m^2 / (v + m^2))
sl <- sqrt(log(1+v/m^2))
#make this reproducible
set.seed(2)
iql <- m * rlnorm(1e6, meanlog=ml, sdlog=sl)
mean(iql)

## [1] 98.9

sd(iql)

## [1] 14.83

summary(iql)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    45.8    88.5    97.8    98.9   108.0   204.0

studyIQL <- iql[iql>cutoff]
summary(studyIQL)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     140     142     145     147     150     204

mean(studyIQL)

## [1] 147.1

sd(studyIQL)

## [1] 6.816

plot(density(studyIQL), xlab="IQ", main="Simulated Lognormal IQ Density for Terman Study")
abline(v=mean(studyIQL))