According to Hayter (2013), the normal distribution is "the most important of all continuous probability distributions and is used extensively as the basis for many statistical inference methods." This is because the normal distribution is a useful tool for describing random phenomena, including the sampling distribution of the sample means as stated in the central limit theorem.
Figure: Normal Distribution
The normal distribution (also known as the bell curve or Gaussian distribution) was discovered during the 18th century by Abraham de Moivre (Tabak, 2005). de Moivre noted many of the curve's characteristics including its symmetry about the mean μ and its "bell-shape" created by two inflection points that are 1 standard deviation (σ) to the right and left of the mean where the curve's slope is the steepest (Tabak, 2005).
The probability density function of a normal distribution is the following:
for -∞ < x < ∞ (Hayter, 2013).
To view a derivation of this formula, click here .
There does not exist a simple closed-form solution to the cumulative distribution function of a normal distribution. So, in order to calculate the probability of a normal random variable being less than some value (the area under the curve and to the left of the value), you have to use tables of the standard normal distribution. The standard normal distribution is the normal distribution with mean μ =0 and variance σ2 =1 (Hayter, 2013). Any normal random variable X~N(μ, σ2) can be transformed into a random variable with a standard normal distribution using the following formula:
Using the applet below created by GeoGebra Materials Team (2014), explore the normal distribution and its probability values.
Looking at the probability values, you will see that there's about a 68% probability a normal random variable takes a value within 1 standard deviation of the mean, 95% probability it takes a value within 2 standard deviations of the mean, and 99.7% probability it takes a value within 3 standard deviations of the mean (Hayter, 2013).
To learn more about the normal distribution, click here to link to a module on Khan Academy.
