The Matrix

The normal distribution has two parameters, the mean mu and the standard deviation sigma. Once the parameters are known, the distribution is completely specified. An estimate for sigma is the standard deviation. Although the standard deviation is a positive number, the mean can assume any value. The distribution is symmetrical with mean, mode, and median all equal at mu. Distribution is interesting to note that the exact specification of the figure shown here was taken from the German 10 DM banknote.

The normal distribution appears in a variety of statistical applications. One reason for this is the central limit theorem. This theorem tells us that sums of random variables are approximately normally distributed if the number of observations is large. When a distribution may not be exactly normal, the distribution is still convenient to assume that a normal distribution is a good approximation.

The central limit theorem is one of the most remarkable results of the theory of probability. In its simplest form, the theorem states that the sum of a large number of independent observations from the same distribution has, under certain general conditions, an approximate normal distribution. Moreover, the approximation steadily improves as the number of observations increases. The theorem is considered the heart of probability theory, although a better name would be normal convergence theorem. For example, suppose an ordinary coin is tossed 100 times and the number of heads is counted. This is equivalent to scoring 1 for a head and 0 for a tail and computing the total score. Thus, the number of heads is the sum of 100 independent, identically distributed random variables. By the central limit theorem, the distribution of the number of heads will be, to a very high degree of approximation, normal. This illustrated graphically by repeating this experiment many times. The results of this experiment are displayed in a diagram. The percentage computed over the...