History Of Normal Curve And Normal Distribution 142z5e

HISTORY OF NORMAL CURVE AND NORMAL DISTRIBUTION

DEVELOPMENT Some authors attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his "The Doctrine of Chances" the study of the coefficients in the binomial expansion of (a + b)n. De Moivre proved that the middle term in this expansion has the approximate magnitude of

, and that

"If m or ½n be a Quantity infinitely great, then the

Logarithm of the Ratio, which a Term distant from the middle by the Interval ℓ, has to the middle Term, is

Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.

." Although this theorem can be interpreted as the first obscure expression for the normal probability law,Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function. In 1809 Gauss published his monograph "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M′, M′′, … to denote the measurements of some unknown quantity V, and sought the "most probable"

estimator: the one that maximizes the probability φ(M−V) · φ(M′−V) · φ(M′′−V) · … of obtaining the observed experimental results. In his notation φΔ is the probability law of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:

where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear weighted least squares (NWLS) method.

Although Gauss was the first to suggest the normal distribution

law, Laplace made

significant

contributions. It was Laplace who first posed the problem

of

aggregating

several

observations

in

1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of

the integral ∫ e−t ²dt = √π in

normalization

constant

1782,

providing

for

the

the

normal

distribution.[51] Finally, it was Laplace who in 1810 proved

Marquis de Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.

and presented to the Academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution. It is of interest to note that in 1809 an American mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe. In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: "The number

of

particles

whose

velocity,

resolved

in

a

certain

direction,

lies

between x and x + dx is

NAMING Since its introduction, the normal distribution has been known by many different names: the law of error, the law of facility of errors, Laplace's second law, Gaussian law, etc. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual".[56] However, by the end of the 19th century some authors[nb 5] had started using the name normal distribution, where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Peirce (one of

those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under

certain

circumstances."[57] Around

the

turn

of

the

20th

century Pearson popularized the term normal as a designation for this distribution.[58] Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'. —Pearson (1920) Also, it was Pearson who first wrote the distribution in of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:

The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around 1950s, appearing in the popular textbooks by P.G. Hoel (1947) "Introduction to mathematical statistics" and A.M. Mood (1950) "Introduction to the theory of statistics".[59] When the name is used, the "Gaussian distribution" was named after Carl Friedrich Gauss, who introduced the distribution in 1809 as a way of rationalizing the method of least squares as outlined above. Among English speakers, both

"normal distribution" and "Gaussian distribution" are in common use, with different preferred by different communities.

FROM FREQUENCY POLYGON TO GAUSSIAN (NORMAL CURVE) In probability, we saw that the binomial distribution could be used to solve problems such as "If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?" The probability of exactly x heads out of N flips is computed

using

the

formula:

where x is the number of heads (60), N is the number of flips (100), and π is the probability of a head (0.5). Therefore, to solve this problem, you compute the probability of 60 heads, then the probability of 61 heads, 62 heads, etc., and add up all these probabilities. Imagine how long it must have taken to compute binomial probabilities before the advent of calculators and computers. Abraham de Moivre, an 18th century statistician and consultant to gamblers, was often called upon to make these lengthy computations. de Moivre noted that when the number of events (coin flips) increased, the shape of the binomial distribution approached a very smooth curve. Binomial distributions for 2, 4, and 12 flips are shown in Figure 1.

Figure 1. Examples of binomial Figure 1. Examples of binomial distributions. The heights of the blue bars represent the probabilities. de Moivre reasoned that if he could find a mathematical expression for this curve, he would be able to solve problems such as finding the probability of 60 or more heads out of 100 coin flips much more easily. This is exactly what he did, and the curve he discovered is now called the "normal curve."

Figure 2. The normal approximation to the binomial distribution for 12 coin flips. The smooth curve is the normal distribution. Note how well it approximates the binomial probabilities represented by the heights of the blue lines.

The importance of the normal curve stems primarily from the fact that the distributions of many natural phenomena are at least approximately normally distributed. One of the first applications of the normal distribution was to the analysis of errors of measurement made in astronomical observations, errors that occurred because of imperfect instruments and imperfect observers. Galileo in the 17th century noted that these errors were symmetric and that small errors

occurred more frequently than large errors. This led to several hypothesized distributions of errors, but it was not until the early 19th century that it was discovered that these errors followed a normal distribution. Independently, the mathematicians Adrian in 1808 and Gauss in 1809 developed the formula for the normal distribution and showed that errors were fit well by this distribution. This same distribution had been discovered by Laplace in 1778 when he derived the extremely important central limit theorem, the topic of a later section of this chapter. Laplace showed that even if a distribution is not normally distributed, the means of repeated samples from the distribution would be very nearly normally distributed, and that the larger the sample size, the closer the distribution of means would be to a normal distribution. Most statistical procedures for testing differences between means assume normal distributions. Because the distribution of means is very close to normal, these tests work well even if the original distribution is only roughly normal. Quételet was the first to apply the normal distribution to human characteristics. He noted that characteristics such as height, weight, and strength were normally distributed.

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