Conditions for a Distribution to be Modelled by a Normal Distribution

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The normal distribution is the most useful and widely used statistical distribution, but it can’t be used everywhere. When considering whether or not a distribution can be modelled by the normal distribution we should keep in mind the main features of the normal distribution:

It is a bell shaped curve symmetrical about the mean. It has no skew.

It is very unlikely, though possible, that values occur which are less or more than three standard deviations from the mean.

There are no upper or lower cut off points. In theory a random variable which has a normal distribution may take any values fromto

Of course, if we take a random sample of some variable we would not expect the random sample to be exactly randomly distributed, nor exactly bell shaped and maybe there are some quite extreme values well below or above three standard deviations from the mean, but if the conditions are ‘approximately’ met or a histogram of values is ‘not too far’ from that we would expect from a normal distribution then we can often take the normal distribution to be suitable.

The histogram on the left indicates a normal distribution is plausible. The histogram on the right indicates positive skew so a normal distribution is unlikely to be unsuitable.

We may be able to reject a normal distribution as suitable on theoretical grounds if the set of possible values is limited in some way. Heights and weights may not take values less than zero so in theory these can not be modelled by a normal distribution, though in fact they often are.

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