This chapter uses the example of shoe sizes forming a bell curve to introduce the normal distribution, noting it is used, and sometimes abused, across psychology, business, economics, and the sciences. It covers recognizing and applying the normal probability distribution, the standard normal distribution, and converting between the two to compare probabilities.
A bell curve across many fields
The chapter introduces the normal distribution through the example of shoe sizes graphing into a bell-shaped curve, noting that the same pattern appears across psychology, business, economics, and the sciences, and that some instructors even use it to help set grades, while emphasizing that recognizing when it does and does not apply is itself a skill.
The standard normal distribution
Students learn to recognize and apply the standard normal probability distribution appropriately, the reference version of the normal distribution used to standardize any normally distributed value so that probabilities can be looked up or computed consistently regardless of the original mean and standard deviation.
Comparing probabilities across distributions
The chapter covers comparing normal probabilities by converting values to the standard normal distribution, a technique that lets students compare two values from different normal distributions, such as scores on two different tests, on a common scale before drawing any conclusion about which is relatively higher or lower.
This chapter presents the normal distribution as the most important of all continuous distributions, appearing across psychology, business, economics, and the sciences, while cautioning that it cannot be applied to everything. It covers the distribution’s two parameters, the standard normal distribution, and how to use the normal distribution to find probabilities.
The most important, and most abused, distribution
The chapter opens by calling the normal distribution the most important of all distributions, noting its bell-shaped graph appears across psychology, business, economics, and the sciences and that some instructors even use it to help set grade curves, while explicitly cautioning that it is also widely misapplied and cannot be assumed for every real-world quantity.
Two parameters and the standard normal distribution
A normal distribution is fully described by two numerical parameters, its mean μ and its standard deviation σ, and the chapter introduces the standard normal distribution, the special case with mean zero and standard deviation one, as the reference distribution used to compute probabilities for any normal distribution by converting, or standardizing, values onto it.
Using the normal distribution
Once a quantity is established as normally distributed with a given mean and standard deviation, the chapter shows how to use the distribution to find probabilities for specified ranges of values, representing them as shaded areas under the bell curve, building the practical skill of moving between raw values and the probabilities they correspond to.
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.
