Normal Distribution

The normal probability density is the familiar bell-shaped curve; areas under the curve represent the likelihood that repeated measurements of CERTAIN TYPES of processes will take on values in a particular range.
The failure to recognize that not every process is well-described by a normal density is one of the most common errors in studying natural and social phenomena.
The probability Prob(-infinity < Y < u) that an event Y takes on values less than u is given by the area under the curve to the left of u. This function Prob(-infinity < Y < u) is called the normal probability distribution.
Related to this is the probability, Prob(v < Y < u), that repeated measurements of a process Y will take on values between v and u. This is given by the area under the curve between v and u, and can be viewed as the difference
Prob(v < Y < u) = Prob(-∞ < Y < u) - Prob(-∞ < Y < v)
Below are graphs of a normal density, with shaded areas indicating the normal distribution. Click here for a graph of the normal distribution function.
Prob(-∞ < Y < u) Prob(v < Y < u)
The particular shape of the curve is determined by two parameters:
  the mean, μ, or average value, and
  the standard deviation, σ, a measure of how widely the measurements spread around the mean.
With these parameters, and recalling the area under a curve is given by an integtal, we can write the formula for the probability distribution Prob(-∞ < Y < u).
How, other than by seeing how closely the histogram matches the normal curve with the mean and standard deviation of the data set, can we test if the data are normally distributed? An answer is provided by kurtosis.