The Normal Distribution

Summary :

The normal distribution is the most widely used continuous distribution, with a symmetric bell-shaped curve. This chapter introduces the normal distribution and its parameters, the standard normal distribution and z-scores, and how converting any normal value to a z-score lets you find probabilities as areas under the curve.

The Normal Distribution

The normal distribution is a continuous distribution whose graph is a symmetric, bell-shaped curve. It is the most important distribution in statistics and appears across psychology, business, economics, and the sciences. A normal distribution is defined by two parameters: its mean, which locates the centre of the curve, and its standard deviation, which sets how wide or narrow the bell is. The curve is symmetric about the mean, so the mean, median, and mode coincide.

The Standard Normal Distribution

The standard normal distribution is a special normal distribution with a mean of zero and a standard deviation of one. Any normal value can be converted to a standard normal value, called a z-score, by subtracting the mean and dividing by the standard deviation. A z-score measures how many standard deviations a value lies from the mean, and its sign shows whether the value is above or below the mean. This conversion lets a single reference table serve every normal distribution.

Finding Probabilities

Because probability for a continuous distribution is area under the curve, probabilities for a normal variable are found by converting values to z-scores and reading the corresponding area. The area to the left of a z-score gives the probability of being below that value, and areas between two z-scores give the probability of falling in that range. Calculators and tables provide these areas, so any question about a normal variable becomes a question about area under the standard normal curve.