Chapter 7: The Normal Distribution – Properties and Applications

Normal Distribution

Section 7.1 – Properties of the Normal Distribution

The normal distribution is a fundamental concept in statistics, describing how data values are distributed in many natural and social phenomena. This section introduces the properties of continuous probability distributions, with a focus on the normal distribution and its related concepts.

Continuous Probability Distributions

• Continuous probability distribution: A probability model for random variables that can take any value within a given range. The probability of outcomes is represented by the area under a curve.

• Uniform (Rectangular) Distribution: A type of continuous distribution where all outcomes in a range are equally likely.

• The total area under the curve for all possible values equals 1.

• The curve is described by a probability density function (pdf), denoted as f(x). The pdf is used to compute probabilities for continuous random variables.

Cumulative Distribution Function (cdf)

• The cumulative distribution function (cdf) gives the probability that a random variable X is less than or equal to a certain value.

• Probability is found for intervals, not individual values: P(c < x < d) is the probability that X is between c and d, represented by the area under the curve between these points.

• P(x = c) = 0 for continuous variables, since the area at a single point is zero.

Common Phrases Indicating Inequalities

The following table summarizes common phrases used to describe inequalities in probability statements:

>	≥	<	≤
Is greater than More than Larger than	Is greater than or equal to Is at least Is the minimum	Is less than Is smaller than Has fewer than	Is less than or equal to Is at most Is the maximum

The Normal Distribution

• The normal distribution is the most important continuous probability distribution in statistics.

• Its graph is bell-shaped and symmetrical about the mean.

• The area under the curve equals 1.

• It is defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$).

• If a random variable X is normally distributed with mean $\mu$ and standard deviation $\sigma$, we write: $X \sim N(\mu, \sigma)$

• The cumulative distribution function for the normal distribution is $P(X < x)$, which can be calculated using technology or standard normal tables.

The Standard Normal Distribution and Z-Scores

• The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

• Values are converted to z-scores to standardize them:

$z = \frac{x - \mu}{\sigma}$

• A z-score indicates how many standard deviations a value x is from the mean.

• If x is greater than the mean, z is positive; if x is less than the mean, z is negative; if x equals the mean, z is zero.

The Empirical Rule

The Empirical Rule (or 68-95-99.7 Rule) describes the percentage of data within certain standard deviations of the mean in a bell-shaped (normal) distribution:

• About 68% of data falls within 1 standard deviation ($\mu \pm \sigma$)

• About 95% within 2 standard deviations ($\mu \pm 2\sigma$)

• About 99.7% within 3 standard deviations ($\mu \pm 3\sigma$)

Example: If the mean test score is 70 with a standard deviation of 10, about 68% of students scored between 60 and 80.

Section 7.2 – Using the Normal Distribution

This section explains how to use the normal distribution to calculate probabilities and interpret areas under the curve.

Calculating Probabilities with the Normal Distribution

• The area to the left of a value x under the normal curve represents $P(X < x)$.

• The area to the right of x is $P(X > x) = 1 - P(X < x)$.

• For continuous distributions, $P(X < x)$ is the same as $P(X \leq x)$, and $P(X > x)$ is the same as $P(X \geq x)$.

• Calculators or statistical tables are commonly used to find these probabilities.

Example: To find the probability that a randomly selected value is less than x, calculate $P(X < x)$ using the standard normal table or a calculator.

Additional info: In practice, normal probabilities are often found using technology, such as statistical calculators or software, which can compute areas under the curve for any specified interval.