Properties of Normal Distributions (College Board AP® Statistics)

Continuous random variables

What is a continuous random variable?

• A continuous random variable (CRV) is a random variable whose value lies within one or more intervals on a number line

  • The number of values that a CRV could be, within a given interval, are infinite

    • e.g. the weight of a squirrel is a continuous random variable, a squirrel selected at random could take on any value within the given interval

• In some cases it can be easy to confuse continuous and discrete variables

  • e.g. foot length is a continuous variable, as it can be measured to any specified degree of accuracy

  • but shoe size is a discrete variable as it can be counted (4, 5, 6, etc.)

What is a continuous probability distribution?

• A continuous probability distribution describes the probabilities of the occurrences of each possible outcome associated with a continuous random variable

• As there are an infinite number of possible values for a continuous random variable within a given interval

  • the probability that it assumes one specific value is 0

    • i.e. for a value in the given interval,

  • Instead we talk about the probability of being within a certain range

• A continuous probability distribution can be represented by a continuous graph, , called a probability density function 

  • the values for lie along the horizontal axis

  • and the probability density is given on the vertical axis

• The area under the graph between the points and is equal to

  • The total area under the graph equals 1

• As for any value , it does not matter if we use strict or weak inequalities

  • for any value of

Properties of normal distributions

What is a normal distribution?

• A normal distribution is a continuous probability distribution

• If a continuous random variable follows a normal distribution, then its shape will be:

  • symmetrical

  • mound-shaped (bell-shaped)

• If follows a normal distribution then

  • is the mean

  • is the standard deviation

What are the important properties of a normal distribution?

• The normal distribution is symmetrical about

  • Mean = Median = Mode =

• The empirical rules state that the data in a normal distribution is distributed such that (approximately):

  • 68% of the data lies within one standard deviation of the mean

  • 95% of the data lies within two standard deviations of the mean

  • 99.7% lies within three standard deviations of the mean

What can be modeled using a normal distribution?

• A lot of real-life continuous variables can be modeled by a normal distribution if:

  • the population is large enough

  • and the variable is roughly symmetrical with one mode

• For a normal distribution, the continuous random variable can take any real value

  • However, values far from the mean (more than 4 standard deviations away) have a probability density of practically zero

  • This allows us to model variables that are not defined for all real values, e.g. height and weight

What cannot be modeled using a normal distribution?

• Variables which have more than one mode or no mode

  • e.g. the number given by a random number generator

• Variables which are not symmetrical

  • e.g. how long a human lives for

• Variables in which acceptable ranges of values (within 3 standard deviations of the mean) become negative when they are meant to be positive

  • e.g. length modeled by a normal distribution with mean 10 cm and standard deviation 8 cm

  • 3 standard deviations below the mean would be and a length cannot be negative