# Statistics

## The Normal Distribution, also known as the Gaussian Distribution or Bell Curve

The normal distribution is a general shape of probability distribution with a particular symmetrical degree of clustering around the mean, which resembles the shape of a bell. The distribution is also often known as the Gaussian distribution; as Karl Gauss was the first mathematician to truly bring it to prominence. The normal distribution is an important distribution because many randomly occurring events follow its distribution of outcomes. In addition, central limit theorem states that the distribution of the sample means of a sufficiently large number of samples will always approximate to the normal distribution. For example, if a coin is tossed 50 times and the number of heads recorded then, if this test is repeated a large number of times, the resulting distribution of the number of heads recorded per 50 tosses would be approximately normally distributed with a mean of 25 heads per 50 tosses.

The normal distribution itself is always symmetric around its mean, and extends to infinity in each direction. The total probability under the curve is always equal to 1. In addition, the fact that the normal distribution has a specific shape means that any normal distribution is fully described by its mean and standard deviation. The mean and standard deviation of a given distribution allow that distribution to be converted into a standard normal distribution, with mean 0 and standard deviation 1, where every data point is already described and known. This also means that all normal distributions have around 68% of their data within 1 standard deviation either side of the mean; 95% of the data within around 2 standard deviations each side the mean and 99.7% of the data within 3 standard deviations either side of the mean.

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This standard shape, and the fact that central limit theorem can be used to create a normal distribution for most variables, means that the normal distribution is used in many areas of business. For example, modern operations management techniques assume that process variations are normally distributed. This is the basis for the six sigma, or six standard deviations, quality control technique: management attempts to ensure that production is flawless for at least six standard deviations each side of the mean. In addition, employee performance is often fitted to a normal distribution as part of peer ranking systems.

It is important to recognise that, whilst many random variables may appear to follow a normal distribution, in fact very few actual variables will follow said distribution exactly. This is largely because the normal distribution extends to infinity in both directions, and measures such as salary and performance will not tend to extend to infinity in either direction. However, given that 99.7% of observations of the normal distribution will be within three standard deviations of the mean, any errors in the distribution are usually accepted as being negligible; hence an approximation to the normal distribution will often be acceptable for problem solving.