Normal Probability Distributions

Discrete vs. Continuous Probability Distributions

Probability distributions can be classified as either discrete or continuous based on the type of random variable they describe.

• Discrete Probability Distribution: Describes probabilities for discrete random variables (variables that take on countable values, such as the number of heads in coin tosses).

• Continuous Probability Distribution: Describes probabilities for continuous random variables (variables that can take on any value within an interval, such as height or weight).

For continuous distributions, probabilities are represented by areas under a curve rather than by individual probabilities for specific values.

Density Curves

A density curve is the graph of a continuous probability distribution. It must satisfy two main properties:

• The total area under the curve must equal 1.

• Every point on the curve must have a vertical height that is 0 or greater (the curve cannot fall below the x-axis).

Standard Normal Distribution

Definition and Properties

The standard normal distribution is a special case of the normal probability distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. It is symmetric about the mean and describes many natural phenomena.

• The total area under the standard normal curve is 1.

• Probabilities correspond to areas under the curve.

• Values on the horizontal axis are called z-scores, which measure the number of standard deviations a value is from the mean.

Area as Probability

For continuous distributions, the probability that a random variable falls within a certain interval is equal to the area under the density curve over that interval. For the standard normal distribution, this is often written as P(a < z < b).

Finding Areas for the Normal Distribution

Using the Standard Normal Table (Table A-2)

The standard normal table (Table A-2) provides cumulative areas (probabilities) to the left of a given z-score for the standard normal distribution (μ = 0, σ = 1).

• One page is for negative z-scores, the other for positive z-scores.

• The z-score is found along the margins; the area (probability) is in the body of the table.

• The area represents P(Z < z), the probability that a standard normal variable is less than z.

Example: Bone Mineral Density Test

Suppose a bone mineral density test result is measured as a z-score. What is the probability that a randomly selected adult has a result less than 1.27?

• Look up z = 1.27 in Table A-2: Area = 0.8980.

• Interpretation: There is an 89.80% chance that a randomly selected adult has a bone density z-score less than 1.27.

Finding Probabilities for Other Intervals

• P(z > a): Subtract the area to the left of a from 1: P(z > a) = 1 - P(z < a).

• P(a < z < b): Subtract the area to the left of a from the area to the left of b: P(a < z < b) = P(z < b) - P(z < a).

Example: What is the probability of a result between -1 and 1.27?

• P(z < 1.27) = 0.8980

• P(z < -1) = 0.1587

• P(-1 < z < 1.27) = 0.8980 - 0.1587 = 0.7393

Finding Areas Using Technology

Modern technology can be used to find areas under the normal curve:

• Excel:

  • To find the area to the left of x: =NORM.DIST(x, mean, std dev, TRUE)

  • To find the value with a given area to the left: =NORM.INV(probability, mean, std dev)

• Statcrunch: Use the Normal Calculator under STAT → CALCULATORS → NORMAL.

The Empirical (68-95-99.7) Rule

Definition and Application

The Empirical Rule states that for a normal distribution:

• About 68% of the data falls within 1 standard deviation of the mean.

• About 95% falls within 2 standard deviations.

• About 99.7% falls within 3 standard deviations.

Percentiles and Critical Values

Percentiles

The percentile of a number is the percentage of data below that number. For the standard normal distribution, percentiles correspond to areas under the curve to the left of a given z-score.

• For example, the 80th percentile is the value below which 80% of the data falls.

Critical Values

A critical value for the standard normal distribution is a z-score that separates unlikely values from likely values. The notation zα denotes the z-score with an area of α to its right.

• To find zα, look for the value with area 1-α to its left in the standard normal table.

• For example, z0.04 is the z-score with 0.04 area to its right (or 0.96 to its left).

Key Formulas

• Standard Normal Probability:

• Finding z-score: