Statistics: Density Curves, Normal Distributions, and Standardization

Density Curve

A graphical representation of the distribution of a continuous random variable, with properties such as being above the X-axis and having an area of 1, but not necessarily bell-shaped.

Mean, Median, and Mode of a Density Curve

In a density curve, the mean, median, and mode can differ in their positions; the mean is the balance point, the median is the midpoint, and the mode is the highest point.

Uniform Distribution

A type of probability distribution in which all outcomes are equally likely; represented by a flat, horizontal line.

Random Distribution

A distribution where outcomes occur in a random manner without a predictable pattern.

Bell Curve

A graphical representation of a normal distribution, characterized by its bell-shaped curve that is symmetric around the mean.

Normal Distribution

A probability distribution that is symmetric about the mean, where most observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions.

Empirical

Based on observation or experience rather than theory or pure logic.

Empirical Rule

A statistical rule stating that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Inflection Points

Points on a curve where the curvature changes direction, indicating a change in the rate of increase or decrease.

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1, often used for standardization.

Standardizing

The process of converting a normal distribution value to a z-score using the formula z = (x - μ) / σ, and vice versa for un-standardization.

Z-Score

A statistical measurement that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations from the mean.

Table of Standard Normal Probabilities

A table that provides the probabilities associated with z-scores in a standard normal distribution, used to find proportions and z-scores.

Normal Curve Notation: N(μ, σ)

A notation indicating a normal distribution with mean (μ) and standard deviation (σ), where μ represents the average and σ represents the spread of the distribution.

Histogram and Normal Curve Connection

A histogram of normally distributed data approximates the shape of a normal curve, demonstrating that the normal curve is a good model for many types of data due to its properties.

Standard Normal Curve Relationship

Any normal curve can be transformed into the standard normal curve through standardization, which allows for easier comparison and calculation of probabilities.

Simple Standardization

To find a z-score from a normal distribution and a value, subtract the mean from the value and divide by the standard deviation: z = (x - μ) / σ.

Simple Un-Standardization

To find the corresponding value from a z-score, multiply the z-score by the standard deviation and add the mean: x = (z * σ) + μ.

Using the Table of Standard Normal Probabilities

To find a proportion given a z-score, or to find a z-score given a proportion, refer to the table which lists probabilities associated with z-scores.

Basic Normal Distribution Problems

Solve problems involving normal distributions with integer inputs, such as interpreting scores like N(18, 6) where 18 is the mean and 6 is the standard deviation.