Chapter 7: The Normal Probability Distribution – Properties and Applications

Properties of the Normal Distribution

Introduction

The normal distribution is a fundamental concept in statistics, describing how data values are distributed for many natural phenomena. This chapter introduces the normal curve, its properties, and the role of area in the normal density function.

Graphing a Normal Curve

A normal curve is a graphical model used to represent continuous random variables that are normally distributed. The curve is bell-shaped and symmetric, and is defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$).

• Model: In mathematics, a model can be an equation, table, or graph that describes reality. The normal curve is a model for continuous random variables.

• Normal Distribution: A variable is normally distributed if its relative frequency histogram approximates the shape of a normal curve.

• Inflection Points: The curve has inflection points at $x = \mu - \sigma$ and $x = \mu + \sigma$, where the curvature changes direction.

• Effect of Parameters: Changing $\mu$ shifts the curve horizontally, while changing $\sigma$ alters the spread and height of the curve.

Example:

If the mean increases from 0 to 3, the curve shifts right by 3 units. If the standard deviation increases from 1 to 2, the curve becomes flatter and more spread out.

Properties of the Normal Curve

The normal density curve has several key properties that make it useful for statistical analysis.

• Symmetry: The curve is symmetric about its mean ($\mu$).

• Single Peak: The mean, median, and mode are all equal, and the highest point occurs at $x = \mu$.

• Inflection Points: Located at $x = \mu - \sigma$ and $x = \mu + \sigma$.

• Total Area: The area under the curve is 1, representing the total probability.

• Equal Areas: The area to the left and right of the mean are both 0.5.

• Asymptotic: As $x$ increases or decreases without bound, the curve approaches but never touches the horizontal axis.

Empirical Rule:

• Approximately 68% of the area is between $x = \mu - \sigma$ and $x = \mu + \sigma$.

• Approximately 95% of the area is between $x = \mu - 2\sigma$ and $x = \mu + 2\sigma$.

• Approximately 99.7% of the area is between $x = \mu - 3\sigma$ and $x = \mu + 3\sigma$.

Role of Area in the Normal Density Function

The area under the normal curve for a given interval represents either the proportion of the population with a characteristic or the probability that a randomly selected individual has that characteristic.

• Probability Density Function (pdf): The normal pdf is given by: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2}$

• Interpretation: The area under the curve between two values of $x$ gives the probability or proportion of observations within that interval.

Example:

If the area under the curve to the right of $x = 200$ is 0.2903, then 29.03% of the population has values greater than 200, and the probability that a randomly selected individual exceeds 200 is 0.2903.

Applications of the Normal Distribution

Standardizing a Normal Random Variable

To compare values from different normal distributions, we standardize them using the z-score:

• Z-score Formula: $z = \frac{x - \mu}{\sigma}$

• The standard normal distribution has $\mu = 0$ and $\sigma = 1$.

• Any normal random variable $X$ can be transformed to a standard normal variable $Z$.

Example:

If IQ scores are normally distributed with $\mu = 100$ and $\sigma = 15$, an individual with an IQ of 120 has a z-score of $z = \frac{120-100}{15} = 1.33$.

Finding and Interpreting Areas Under the Normal Curve

Areas under the normal curve can be found using tables or technology, and represent probabilities or proportions.

• Complement Rule: The area to the right of a z-score is $1 -$ (area to the left).

• Percentiles: The kth percentile divides the lower k% from the upper (100-k)% of the data.

Example:

If the area to the left of $z = -1.17$ is 0.1210, then 12.10% of the population is below this value (12th percentile).

Finding the Value of a Normal Random Variable

Given a percentile, you can find the corresponding value of the random variable using the normal model.

• Procedure:

  1. Find the area (percentile) in the standard normal table.

  2. Find the corresponding z-score.

  3. Convert z-score to original value: $x = \mu + z\sigma$

Example:

If the 20th percentile for heights of three-year-old females (mean = 38.72, SD = 3.17) is 36.1 inches, then 20% of the population is below this height.

Special Notation: $z_\alpha$

$z_\alpha$ is the z-score such that the area to the right of $z_\alpha$ is $\alpha$.

• For example, if the area to the right is 0.10, the area to the left is 0.90. The closest z-score is $z_{0.10} = 1.28$.

Probability for Continuous Random Variables

For any continuous random variable, the probability of a specific value is zero. Probabilities are only meaningful for intervals.

• $P(X = a) = 0$

• $P(X < b) = P(a < X < b) = P(a < X < b)$

Summary Table: Empirical Rule for Normal Distribution

Interval	Approximate Area
$\mu - \sigma$ to $\mu + \sigma$	68%
$\mu - 2\sigma$ to $\mu + 2\sigma$	95%
$\mu - 3\sigma$ to $\mu + 3\sigma$	99.7%

Additional info: These notes expand on the textbook slides and images, providing full academic context, definitions, and examples for each concept.