Why infinite possible normal distribution

The graph of this function is called the standard normal curve. In this way, the standard normal curve also describes a valid probability density function. Using the definitions for mean and variance as it relates to continuous probability density functions, we can show that the associated mean for a standard normal distribution is 0, and has a standard deviation of 1.

If upon looking at a distribution it appears to be unimodal, symmetric, and shaped in such a way that visually it appears to be "normally distributed", we can use the empirical rule to develop a test that can be applied to any data value in that distribution to determine whether or not it should be treated as an outlier. Recall, according to the Empirical Rule, in any normal distribution we have the vast majority of our data i. Incredibly important in statistics is the ability to compare how rare or unlikely two data values from two different distributions might be.

For example, which is more unlikely, a lb sumo wrestler or a 7. It may seem like we are comparing apples and oranges here -- and in a sense, we are. However, when the distributions involved are both normal distributions, there is a way to make this comparison in a quantitative way. Of course, we need not limit ourselves to distances measured as integer multiples of the standard deviation.

As such, we can compare the rarity of two values -- even two values coming from two different distributions -- by comparing how many of their respective standard deviations they are from their respective means.

Some of the inferences we will draw about a population given sample data depend upon the underlying distribution being a normal distribution. There are multiple ways to check how "normal" a given data set is, but for now it will suffice to.

It is important to note that in these tables, the probabilities are the area to the LEFT of the z-score. If you need to find the area to the right of a z-score Z greater than some value , you need to subtract the value in the table from one. Because the normal distribution is symmetric, we therefore know that the probability that z is greater than one also equals 0. To calculate the probability that z falls between 1 and -1, we take 1 — 2 0.

This solutions jives with the three sigma rule stated earlier!!! We can convert any and all normal distributions to the standard normal distribution using the equation below.

Example Normal Problem We want to determine the probability that a randomly selected blue crab has a weight greater than 1 kg.

How do we determine this probability? We calculate our z-score to be The standard deviation of a sample of that population may be written as s y , or just s. Aside from their mean and standard deviation, every normal population is identical. Therefore, if you rescale normal populations to allow for these two parameters, every normal population is completely identical. The commonest way to rescale a normal population is to subtract the mean from each observation, and divide by the standard deviation.

This will produce a standard normal population , which has a mean of zero and a standard deviation of one. This is mathematically the simplest of all - and, because is so useful, the standard normal distribution has its own special symbols and terminology. Central limit As we said above, the main reason why the normal distribution is so important in statistics is that many sample statistics, including the mean, tend towards a normal distribution, irrespective of the population distribution.

The way in which a statistic's normal tendency depends upon sample size is described by what is known as central limit theorem. Non-mathematically, there are three factors which determine how large a sample you need in order to assume a statistic is approximately normal. Which statistic you are using - although you may not have much choice in this matter. How normal is the population that is sampled - which, to some extent, depends upon what sort of variable you are dealing with.

How 'approximate', or unrealistic, an answer you and your critics are prepared to accept. Another reason the normal distribution is so popular is because its properties are well known - at least to mathematicians. In particular, there are various formulae which estimate the proportion of a normal population in a defined interval. These are known as probability functions. Related topics : Skewness and kurtosis Chauvenet's criterion for identifying outliers The log normal distribution Gaussian smoothing.