This example of a normal distribution can help your understanding of basic statistics.
The normally distributed statistical process is one of the most basic continuous probability distributions.
A normal distribution is a probability distribution that looks like a bell-shaped curve. It is also called a Gaussian distribution or bell curve. The shape of the normal distribution depends on the mean and standard deviation of the data, in particular, its location and width.
The most common representation of the normal distribution is the so-called "bell curve" in which the mean, median, and mode are at the midpoint of the scale and most of the data are near this midpoint with fewer data points as one moves away from it towards either tail.
Most real world stochastic phenomena (whose mean and variance can be computed) exhibit a behavior that can be characterized by a normally distributed variable.
This graph of the probability density function (PDF) for normally distributed stochastic process with different values of standard deviation (square root of variance) is given below. As the variance or standard deviation increases, the height of the characteristic bell shaped curve (the probability of the variable to cluster closely around the mean value) decreases. A normal distribution also exhibits a normally distributed histogram, provided sample size is very large. This type of distribution has one mode, or peak, at its center and symmetrical tails that extend out in either direction.
This is the formula for the normal distribution. The variables and constant of the equation are:
A hunter organization wants to know the distribution of weights of elk in their hunter preserve. The distribution of weights of elk in the hunter preserve can be modeled with a normal distribution.
The random sample of 8 elk weights is small. This will probably result in a large standard deviation. However, if this is the only sample available, it will give some idea of the actual weights of the elk population
The mean or average is the sum of all the data points divided by the total number of data points. Calculating the mean gives:Variance is a measure of the dispersion of a set of data from its mean Calculating the variance:
One can then find the standard deviation from the variance by simply taking the square root of the variance:
The density function for the normal distribution can then be constructed:
Some other fundamental statistical probability distributions are:
Normally distributed statistical processes display one of the fundamental statistical probability distributions i.e. the normal distribution. The distribution can be completely described with just two parameters: the mean, and the variance or standard deviation. The graph of probability density function of normal distribution is a characteristic bell curve, which is symmetric about the mean. The value of knowing the parameters of a distributions normal curve is in calculating probabilities, which are used to make predictions about future events. So that means if a normal population has a calculated standard deviation from a sample then we would expect the following:
Other important probability distributions include uniform distribution, lognormal distribution, t distribution and gamma distribution.