The normal distribution, z-scores and percentiles

Learning objectives

Describe a continuous random variable, for which a probability is an area under a density curve, and describe a normal distribution as symmetric, single-peaked and bell-shaped with center μ and spread σ

Standardize a value with z = (x − μ) ÷ σ, state that the standard normal distribution has mean 0 and standard deviation 1, and compare values from different normal distributions using z-scores or percentiles

Apply the empirical rule to estimate the percentage of a normal distribution that lies within one, two or three standard deviations of the mean

Calculate the probability that a normally distributed variable lies in a given interval using a standard normal table or technology

Determine the value, or the boundaries of an interval, corresponding to a given area or percentile of a normal distribution, and determine an unknown mean or standard deviation from a stated percentile

Judge from a dotplot, a histogram or a boxplot whether a normal model is reasonable for a data set, naming the features, such as strong skew or outliers, that would rule it out