Bell Curve

Bell Curve
The normal distribution was first introduced by Abraham de Moivre in an article in the year 1733, which was reprinted in the second edition of his The Doctrine of Chances, 1738 in the context of approximating certain binomial distributions for large n. His result was extended by Laplace in his book Analytical Theory of Probabilities (1812), and is now called the theorem of de Moivre-Laplace.
Laplace used the normal distribution in the analysis of errors of experiments. The important method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have used the method since 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors. The fact the distribution is sometimes called Gaussian is an example of Stigler's Law.
The name "bell curve" goes back to Esprit Jouffret who first used the term "bell surface" in 1872 for a bivariate normal with independent components. The name "normal distribution" was coined independently by Charles Sanders Peirce, Francis Galton and Wilhelm Lexis around 1875. Despite this terminology, other probability distributions may be more appropriate in some contexts. (Wikipedia)

Characterization
There are various ways to characterize a probability distribution. The most visual is the probability density function (PDF). Equivalent ways are the cumulative distribution function, the moments, the cumulate, the characteristic function, the moment-generating function, the cumulant-generating function, and Maxwell's theorem.
To indicate that a real-valued random variable X is normally distributed with mean μ and variance σ2 ≥ 0.
While it is certainly useful for certain limit theorems (e.g. asymptotic normality of estimators) and for the theory of Gaussian processes to consider the probability distribution concentrated at μ as a distribution with mean μ and variance σ2 = 0. This degenerate case is often excluded from the considerations because no density with respect to the Lebesgue measure exists.
The normal distribution may also be parameterized using a precision parameter τ, defined as the reciprocal of σ2. This parameterization has an advantage in numerical applications where σ2 is very close to zero and is more convenient to work with in analysis as τ is a natural parameter of the normal distribution.
The continuous probability density function of the normal distribution is the Gaussian function where σ > 0 is the standard deviation, the real parameter μ is the expected value, and is the density function of the "standard" normal distribution: i.e., the normal distribution with μ = 0 and σ = 1. The integral of over the real line is equal to one as shown in the Gaussian integral article.
As a Gaussian function with the denominator in the exponent equal to 2, the standard normal density function is an eigenfunction of the Fourier transform.
The probability density function has notable properties including:
symmetry about its mean μ the mode and median both equal the mean μ
the inflection points of the curve occur one standard deviation away from the mean, i.e. at μ − σ and μ + σ. (Wikipedia)

Education
Education can be seen as a product or a process and considered in a broad sense or a technical sense. According to a philosophy of education grading on a bell curve (or simply known as curving) is a method of assigning grades designed to yield a desired distribution of grades among the students in a class. Strictly speaking, grading "on a bell curve" refers to the assigning of grades according to the frequency distribution (also called the Gaussian distribution), whose graphical representation is referred to as the normal curve or the bell curve. Because bell curve grading assigns grades to students based on their relative performance in comparison to classmates' performance, the term "bell curve grading" came, by extension, to be more loosely applied to any method of assigning grades that makes use of comparison between students' performances, though this type of grading does not necessarily actually make use of any frequency distribution such as the bell-shaped normal distribution.
In true use of bell curve grading, students' scores are scaled according to the frequency distribution represented by the normal curve. The instructor can decide what grade occupies the center of the distribution. This is the grade an average score will earn, and will be the most common. Traditionally, in the ABCDF system this is the 'C' grade. The instructor can also decide what portion of the frequency distribution each grade occupies and whether or not high and low grades are symmetrically assigned an area under the curve; for example, if the top 15% of students earn an 'A,' do the bottom 15% fail or might only the bottom 5% fail? In a system of pure curve grading, the number of students who will receive each grade is already determined at the beginning of a course.Other forms of "curved" grading vary, but one of the most common is to add to all students' absolute scores: the difference between the top student's score and the maximum possible score. For example, if the top score on an exam is 55 out of 60, all students' absolute scores (meaning they have not been adjusted relative to other students' scores in any way) will be increased by 5 before being compared to a pre-determined set of grading benchmarks (for example the common A>90%>B>80% etc. system). In addition to this method, instructors could make the maximum possible score the top student's score. This method prevents unusually hard assignments (usually exams) from unfairly reducing students' grades but relies on the assumption that the top student's performance is a good measure of an assignment's difficulty.

Benefits and Shortcomings
Viewed practically, curved grading is beneficial because it automatically factors in the difficulty a group of test-takers had with a test. If the majority of students have high (or low) scores then the middling grade will be adjusted there and higher or lower grades awarded based on this performance. In addition, the curve ameliorates the problem of deciding grades that fall very near a grade margin. Clustering of marks establish where the margin should be placed. However, grading in this way is essentially normative; scores are referenced to the performance of group member. There must always be at least one student who has a lower score than all others, even if that score is quite high when evaluated against specific performance criteria or standards. Conversely, if all students perform poorly relative to a larger population, even the highest graded students may be failing to meet standards. Thus, curved grading makes it difficult to compare groups of students to one another.
In probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that clusters around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve.
The normal distribution can be used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches. Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data is used.
For theoretical reasons (such as the central limit theorem), any variable that is the sum of a large number of independent factors is likely to be normally distributed. For this reason, the normal distribution is used throughout statistics, natural science, and social science as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption.

References
http://en.wikipedia.org/