Normal Curve

A Brief History of the Normal Curve

The discovery of the normal curve, also known as the “bell-shape” curve or the Gaussian curve, can be dated to the 17th century, when Galileo Galilei, an Italian physicist and astronomer, noted that the measurement errors in astronomical observations were very systematic and that small errors were more likely to occur than large errors. In 1778, Pierre-Simon Laplace, while working on his famous central limit theorem, noted that the sampling distribution of the sample mean approximated a normal distribution and that the larger the sample size, the closer the distribution would be to a normal distribution, no matter what the population distribution might be. Also in the 18th century, a French statistician, Abraham de Moivre, who was often asked to do statistical consulting for gamblers, found that when the number[Page 691] of events (e.g., coin flips) increased, the shape of the binomial distribution would approximate a symmetrical and smooth curve. However, the mathematical formula for this curve was not discovered until the 19th century, by Adrian Marie Legendre in 1808 and Carl Friedrich Gauss in 1809. The German 10 deutsche mark bill (see Figure 1) had Gauss's picture on it, along with the well-known bell-shaped normal curve and its formula.

Important Properties of a Normal Curve

This family of curves has the following characteristics, which are important to know. For those who are unfamiliar with the calculus calculation, all functions involving integration can be safely skipped. Graphical illustrations are used to facilitate an understanding of the concepts presented, and knowledge of basic algebra is assumed.

1.
The mode, the mean, and the median are all at the same point on the abscissa, the horizontal axis of the curve (see Figure 2). That is to say, mode = mean = median for a normal distribution.

2.

The curve is symmetrical about the point on the abscissas that denotes the mean, the mode, or the median, with equal numbers of observations above and below the point (see Figure 2).



Figure 2 Typical Normal PDF Curve

3.

The skewness and the kurtosis for a normal distribution are both 0. However, statisticians who omit the subtraction of 3 from the kurtosis formula will report the kurtosis value to be 3 for a normal distribution.

4.

The area jointly determined by two points (a, b) on the abscissa and the normal probability density curve indicates the probability that an observation falls within the intervals of [a, b], [a, b), (a, b], or (a, b). In other words,



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