Normal Distribution

In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution that is often used as a first approximation to describe real-valued random variables that tend to cluster around a single mean value. The graph of the associated probability density function is "bell"-shaped, and is known as the Gaussian function or bell curve


where parameter μ is the mean (location of the peak) and σ 2 is the variance (the measure of the width of the distribution). The distribution with μ = 0 and σ 2 = 1 is called the standard normal.

The normal distribution is considered the most prominent probability distribution in statistics. There are several reasons for this. First, the normal distribution is very tractable analytically, that is, a large number of results involving this distribution can be derived in explicit form. Second, the normal distribution arises as the outcome of the central limit theorem, which states that under mild conditions the sum of a large number of random variables is distributed approximately normally. Finally, the "bell" shape of the normal distribution makes it a convenient choice for modelling a large variety of random variables encountered in practice. Get latest updates on the related subject in Statistic homework help and assignment help at

For this reason, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural sciences, and social sciences as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption. Note that a normally-distributed variable has a symmetric distribution about its mean.

Quantities that grow exponentially, such as prices, incomes or populations, are often skewed to the right, and hence may be better described by other distributions, such as the log-normal distribution or Pareto distribution. In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off extremely rapidly. As a result, statistical inference using a normal distribution is not robust to the presence of outliers (data that is unexpectedly far from the mean, due to exceptional circumstances, observational error, etc.). When outliers are expected, data may be better described using a heavy-tailed distribution such as the Student's t-distribution. These latest developments are all covered in the Statistic homework help and assignment help at


From a technical perspective, alternative characterizations are possible, for example:

The normal distribution is the only absolutely continuous distribution all of whose cumulants beyond the first two (i.e. other than the mean and variance) are zero.

For a given mean and variance, the corresponding normal distribution is the continuous distribution with the maximum entropy.


Suppose you must establish regulations concerning the maximum number of people who can occupy a lift.

looking for homework help?...let us know your assignment problem.You know that the total weight of 8 people chosen at random follows a normal distribution with a mean of 550kg and a standard deviation of 150kg.

What’s the probability that the total weight of 8 people exceeds 600kg?

First sketch a diagram.

The mean is 550kg and we are interested in the area that is greater than 600kg.

z = ( x - μ ) / s

Here x = 600kg,
μ , the mean = 550kg
σ , the standard deviation = 150kg

z = ( 600 - 550 ) / 150

z = 50 / 150

z = 0.33

Look in the table down the left hand column for z = 0.3, and across under 0.03.

The number in the table is the tail area for z=0.33 which is 0.3707 .

This is the probability that the weight will exceed 600kg.

Our answer is

"The probability that the total weight of 8 people exceeds 600kg is 0.37 correct to 2 figures."


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