Normal distribution parameters

The family of normal distributions consists of symmetric bell-shaped distributions that are defined by two parameters, µ and σ. The mean and standard deviation of a normal distribution are equal to µ and σ, respectively.

Normal distributions as models for data

A normal distribution is sometimes used as a population to model the variability in a data set. (The data are assumed to be a random sample from this population.) On the basis of a single set of data, there is rarely enough information about the shape of the underlying distribution to be sure that a normal distribution is the 'correct' population, but it is often a close enough approximation.

Teabag weight

The data histogram below shows the weights of 50 teabags that were filled by one machine in an hour. It is important to the company to both ensure that the average weight of the teabags is over the nominal 5.5 grams per bag and to ensure that the variability of the weights is as small as possible — customer satisfaction will be affected if there are badly under-filled teabags.

There are only 50 observations, so it is impossible to be sure of the shape of the underlying population distribution. However the histogram is not badly skew, so a normal distribution is a reasonable model.

Adjust the normal parameters with the sliders to match the shapes of the histogram and normal curve as closely as possible.

Many data sets cannot be modelled by a normal distribution. A normal distribution would not be an appropriate model for ...

Data with skew distributions can often be transformed into a fairly symmetrical form. A normal distribution may be a reasonable model for the transformed data.

Do not assume that all data sets that you meet can be modelled adequately by normal distributions.

Normal distributions describe many summary statistics

A more important reason for the importance of the normal distribution in statistics is that...

Many summary statistics have normal distributions (at least approximately).

We demonstrated earlier that the mean of a random sample has a distribution that is close to normal when the sample size is moderate or large, irrespective of the shape of the distribution of the individual values.

In a similar way, the distributions of the following summary statistics are approximately normal when sample size is moderate or large...

Since most statistical methods require an understanding of the variability of such summary statistics, it is important that you become familiar with the properties of normal distributions.