Section 7.1

Properties of the Normal Distribution

Understand the characteristics of the bell curve, the role of area as probability, and the continuous uniform distribution.

1

Uniform Probability Distribution

Before diving into the normal curve, simpler continuous distributions help illustrate key concepts. In a Uniform Distribution, values are equally likely across a range .

Key Concept: Area = Probability

For continuous variables, $P(x = c) = 0$. Instead, we calculate probability over an interval. The probability is the Area under the curve (PDF) for that interval.

  • Total Area under the graph must equal 1.
  • Height of the graph must be .
  • For uniform: Height = (making it a rectangle).

Uniform Distribution Explorer

Parameters

Find Probability

Loading chart...
Calculation:
0.3333
2

Graphing the Normal Curve

The Normal Distribution (or Gaussian) is the most important distribution in statistics. It is defined entirely by two parameters: the mean () and the standard deviation ().

Normal Curve Explorer

0

Shifts the curve left/right

1

Controls spread/flatness

Symmetric?Yes

Peak at:x = 0

Inflection Pts:-1, 1

Loading chart...

Effect of

The Mean determines the center. Increasing shifts the entire curve to the right.

Effect of

The Std Dev determines the spread. Increasing flattens the curve (peaks lower).

3

Properties of the Normal Curve

Critical Characteristics

  • Symmetric: The left side is a mirror image of the right side around .
  • Single Peak: The highest point is at . Here, Mean = Median = Mode.
  • Inflection Points: The curve creates "shoulders" (concavity changes) exactly at and .
  • Asymptotic: The tails approach, but never touch, the horizontal axis.

The Empirical Rule

The Empirical Rule showing 68%, 95%, and 99.7% distributions on a normal curve
4

Role of Area in PDF

Area = Probability

Because a continuous variable has infinite possible values, the probability of getting exactly one specific number (e.g., exactly 120.000000 lbs) is zero.

Therefore, we always calculate the probability that the variable falls between two values. This corresponds to the area under the density curve between those points.

Adaptive Engine

LogicLens Practice Suite

Log in to Access Adaptive Practice

Our AI engine generates unique practice problems based on your progress.