Normal Probability Distributions and the Standard Normal Distribution

Normal Probability Distributions

Introduction to Normal Distributions

The normal distribution is a fundamental continuous probability distribution in statistics, widely used to model real-world phenomena such as heights, test scores, and lifespans. Its graph, known as the normal curve, is bell-shaped and symmetric.

• Definition: A normal distribution is a continuous probability distribution for a random variable x.

• Applications: Used in nature, industry, and business (e.g., smartphone lifetimes, housing costs).

Properties of a Normal Distribution

• Mean, Median, Mode: All are equal.

• Shape: Bell-shaped and symmetric about the mean (μ).

• Total Area: The area under the curve is 1.

• Asymptotic: The curve approaches but never touches the x-axis.

• Inflection Points: Occur at x = μ - σ and x = μ + σ, where the curve changes from upward to downward.

Probability Density Function (pdf): For continuous distributions, the pdf must satisfy:

• Total area under the curve equals 1.

• The function is never negative.

Parameters: The mean (μ) determines the center, and the standard deviation (σ) determines the spread.

Normal Probability Density Function

The general formula for the normal probability density function is:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$

Additional info: This formula is not required for basic calculations but is fundamental in advanced statistics.

Comparing Normal Curves

Different normal curves can have different means and standard deviations:

• Curves with the same mean are centered at the same point.

• Curves with the same standard deviation have the same spread.

• Greater mean shifts the curve right; greater standard deviation makes the curve wider and flatter.

Example: If curve A is centered at x = 15 and curve B at x = 12, curve A has a greater mean. If curve B is more spread out, it has a greater standard deviation.

Empirical Rule (68-95-99.7 Rule)

The empirical rule describes the proportion of values within certain standard deviations from the mean:

• About 68% of values lie within 1 standard deviation (μ ± σ).

• About 95% within 2 standard deviations (μ ± 2σ).

• About 99.7% within 3 standard deviations (μ ± 3σ).

Example: For test scores with μ = 450 and σ = 27:

• 68% between 423 and 477

• 95% between 396 and 504

• 99.7% between 369 and 531

The Standard Normal Distribution

Definition and Properties

The standard normal distribution is a special case of the normal distribution with mean 0 and standard deviation 1. The horizontal axis is labeled with z-scores, which measure the number of standard deviations a value is from the mean.

• Mean: 0

• Standard deviation: 1

• Total area under the curve: 1

Transforming x to z: Any normal distribution can be converted to the standard normal distribution using:

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

Interpretation: The area under the standard normal curve to the left of a z-score represents the probability that z is less than that value.

Using the Standard Normal Table

The Standard Normal Table (or z-table) lists cumulative areas under the curve to the left of a given z-score.

• Cumulative area is close to 0 for z-scores near -3.49.

• Cumulative area increases as z increases.

• Cumulative area for z = 0 is 0.5000.

• Cumulative area is close to 1 for z-scores near 3.49.

Sample Table: Cumulative Areas for Selected z-scores

z-score	Cumulative Area (to left)
1.15	0.8749
-0.24	0.4052
0	0.5000
1.23	0.8907
-0.75	0.2266
-0.90	0.1611
1.06	0.8554
1.25	0.8944
-1.5	0.0668

Finding Areas Under the Standard Normal Curve

To find probabilities or areas under the curve:

• Area to the left of z: Use the z-table directly.

• Area to the right of z: Subtract the area to the left from 1.

• Area between two z-scores: Subtract the smaller cumulative area from the larger.

Example:

• Area to left of z = 1.23: 0.8907

• Area to right of z = 1.23: 1 - 0.8907 = 0.1093

• Area between z = -0.75 and z = 1.23: 0.8907 - 0.2266 = 0.6641

Technology Applications

Statistical software and calculators (e.g., TI-84 Plus, Excel, Minitab) can compute areas under the normal curve:

• TI-84: normalcdf(lower, upper, μ, σ)

• Excel: =NORM.S.DIST(z, TRUE) for left area; =1-NORM.S.DIST(z, TRUE) for right area

Interpreting Probabilities

For a continuous normal distribution, the area under the curve to the left of a z-score gives the probability that z is less than that value:

$P(z < a) = \text{Area to the left of } a$

Example: If the area to the left of z = -0.90 is 0.1611, then $P(z < -0.90) = 0.1611$.

Unusual and Very Unusual Values

• Values more than 2 standard deviations from the mean (|z| > 2) are considered unusual.

• Values more than 3 standard deviations from the mean (|z| > 3) are considered very unusual.

Summary Table: Types of Areas Under the Standard Normal Curve

Type of Area	How to Find	Example
Left of z	Use z-table	z = 1.15, area = 0.8749
Right of z	1 - area to left	z = 1.06, area = 1 - 0.8554 = 0.1446
Between two z-scores	Subtract smaller area from larger	z = -1.5 to 1.25, area = 0.8944 - 0.0668 = 0.8276

Additional info: Technology may yield slightly different results due to rounding or calculation methods.