Page 186 - Probability Demystified

CHAPTER 9 The Normal Distribution 175

He studied the probability of tossing coins, rolling dice, and other forms of
gambling. In 1716, he wrote a book on gambling entitled The Doctrine of
Chances. In addition to his tutoring, wealthy patrons came to him to find out
what the payoff amount should be for various gambling games. He made
many contributions to mathematics. In probability, he tossed a large number
of coins many times and recorded the number of heads that resulted on each
trial. He found that approximately 68% of the results fell within a predictable
distance (now called the standard deviation) on either side of the mean and
that 95% of the results fell within two predictable distances on either side of
the mean. In addition, he noticed that the shape of the distribution was bell-
shaped, and he derived the equation for the normal curve in 1733, but his
work in this area of mathematics went relatively unnoticed for a long period
of time.
In 1781, a French mathematician, Pierre Simon Laplace (1749–1827) was
studying the gender of infants attempting to prove that the number of males
born was slightly more than the number of females born. (This fact has been
verified today.) Laplace noticed that the distribution of male births was also
bell-shaped and that the outcomes followed a particular pattern. Laplace also
developed a formula for the normal distribution, and it is thought that he was
unaware of de Moivre’s earlier work.
About 30 years later in 1809, a German mathematician, Carl Friedrich
Gauss (1777–1855) deduced that the errors in the measurements of the
planets due to imperfections in the lenses in telescopes and the human eye
were approximately bell-shaped. The theory was called Gauss’ Law of Error.
Gauss developed a complex measure of variation for the data and also an
equation for the normal distribution curve. The curve is sometimes called the
Gaussian distribution in his honor. In addition to mathematics, Gauss also
made many contributions to astronomy.
During the 1800s at least seven different measures of variation were used
to describe distributions. It wasn’t until 1893 that the statistician Karl
Pearson coined the term ‘‘standard deviation.’’
Around 1830, researchers began to notice that the normal distribution
could be used to describe other phenomena. For example, in 1846 Adolphe
Quetelet (1796–1874) began to measure the chest sizes of Scottish soldiers.
He was trying to develop the concept of the ‘‘average man,’’ and found that
the normal distribution curve was applicable to these measurements.
Incidentally, Quetelet also developed the concept of body mass index,
which is still used today.
A German experimental psychologist, Hermann Ebbinghaus (1855–1913)
found that the normal distribution was applicable to measures of intelligence
and memorization in humans.