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The t-Distribution and its Relation to the Normal Distribution
Understanding Normal Distribution Tables and Probabilities
The Normal or Gaussian Distribution
When there is only one measurement it is usual to use a standard deviation as an indication of how a measurement differs from the mean, for exa
Can we determine how far a specific person is from the centre (or mean)? We can measure this deviation which will tell us how far a person differs from the “average” but it can be in any direction. So instead of having a distance that is either positive or negative, we have a single number, rather like the distance of a point from the centre of a circle.
The normal (or Gaussian) distribution is well known to most users of statistics : if you are unfamiliar with it, you are advised to brush up on this before reading this article. In its simplest terms the chi squared distribution can be described as the square of the normal distribution.
Normal or Gaussian Distribution
Where the chi squared comes into play is if more than one type of measurement is recorded. For example we may measure the weights and heights of a number of men. We might want to ask whether a person differs from the average. There will be some relationship between a person’s height and weight, so these measurements are not independent, but the relationship will not be perfect. There will be some short fat people and some tall thin people. In Figure 1 we illustrate a typical relationship.
However life is not so straightforward and it is important to understand what is meant by this:
Consider the proportion of a population that is more than 2 standard deviations greater than its mean.For example, if a population of mice has a mean length of 6.8 cm and a standard deviation of 0.6 cm [link widoczny dla zalogowanych], this will correspond to the proportion of mice whose length is more than 8 cm.Using normal distribution tables, or functions in Excel, we can estimate that around 2.3% of the mice should exceed this length, providing the lengths of the mice are normally distributed.Therefore around 4.6% (this number equals twice 2.3%) will ideally have a length more than 2 standard deviations either side of the mean, implying 4.6% of the mice will have a length that is either less than 5.6 cm (= 6.8 – 2 x 0.6) or more than 8 cm (= 6.8 + 2 x chi squared statistic tells us how many mice should have a length greater than the square of this distance from the mean.Since the square of 2 is 4, the portion of samples whose chi-squared value is greater than 4 (with 1 degree of freedom as discussed below) is this seems complicated so we will express this in another way:
The portion of samples in a population more than 2 standard deviations greater than the mean is expected to be 2.3% if they are normally distributed.Hence the portion of samples more than 2 standard deviations either side of the mean is expected to be double that the portion of samples with a chi squared value greater than 4 (the square of 2) is expected to be should we want to use chi squared, since in it simplest form it is related directly to the normal distribution? The reason is that a squared number has no direction. However when only one variable is measured, e.g. the length of a mouse, there is no difference in the answers we get if we use the chi-squared or normal distribution [link widoczny dla zalogowanych], they are identical.
More Than One Measurement : Weight and Height
Many basic statistical methods appear like magic to be unrelated [link widoczny dla zalogowanych] yet a little lateral thinking demonstrates that most are based on the same fundamental underlying principles.