Study Notes on The Normal Distribution

Section 5.5: The Normal Distribution

Continuous Probability Distributions

  • A continuous random variable can assume any numerical value across one or more intervals.

    • Examples:

    • Car mileage.

    • Temperature.

  • A continuous probability distribution is used to assign probabilities to intervals of values.

Density Curves

  • A density curve is a mathematical model representing a distribution.

  • Key Properties:

    • The total area under the density curve is defined to equal 1 (or 100%).

    • The area under the curve for any range of values represents the proportion of observations within that range.

  • Visualization Example:

    • A histogram of a sample may include a smoothed density curve that depicts the theoretical population distribution.

Characteristics of Density Curves

  • The function f(x) denotes the height of the density curve at a specific value of x.

  • Two Essential Requirements of Density Curves:

    1. f(x) must be greater than or equal to 0 (f(x) ≥ 0).

    2. The total area under the curve must equal 1.

  • A density curve illustrates the overall pattern of a distribution, with the area under the curve representing the proportion of all observations that are located within a specified range.

  • Graphical Representation:

    • An example might show the area = 1.

Variability in Density Curves

  • Density curves can take on virtually any shape:

    • Some shapes are well-known mathematically; others are less common.

Mean and Median of Density Curves

  • Median:

    • The median of a density curve is defined as the equal-areas point, which divides the area under the curve into two equal halves.

  • Mean:

    • The mean is referred to as the balance point; it is the point at which the curve would balance if it were made of solid material.

  • For symmetric density curves, the mean and median are identical.

  • For skewed curves, the mean is influenced by the direction of the long tail:

    • The mean is pulled in the direction of the skewness.

The Normal Probability Distribution

  • Natural Constants:

    • e = 2.71828… (base of the natural logarithm).

    • π (pi) ≈ 3.14159…

  • Definition:

    • Normal (or Gaussian) distributions are a family of symmetrical, bell-shaped density curves defined specifically by:

    • Mean (m).

    • Standard deviation (s).

  • A normal random variable can take any real value spanning from -infinity to +infinity.

Families of Density Curves

  • Variations in means and standard deviations:

    • Examples with different means (m = 10, 15, and 20) while keeping standard deviations constant (s = 3).

    • Other examples maintain the same mean (m = 15) but have varying standard deviations (s = 2, 4, and 6).

Position and Shape of the Normal Curve

  • Different Means with Equal Standard Deviations:

    • If mean μ₁ is greater than mean μ₂, the curve with mean μ₁ is shifted to the right compared to the curve with mean μ₂.

  • Same Mean with Different Standard Deviations:

    • When standard deviation σ₁ is greater than σ₂, the curve corresponding to σ₁ is flatter and more spread out than the curve for σ₂.

Normal Distribution Characteristics

  • Notation:

    • Represented as X hicksim N(m, s) where m is the mean and s is the standard deviation.

  • Probability Analysis:

    • In a continuous distribution, since there are infinite possible values, the probability of any single specific value occurring is essentially zero:

    • P(X = x) = 0

    • Consequently, we only determine probabilities over ranges of values:

    • P(X < x)

Normal Probabilities Calculation

  • Specific Probability Expression:

    • To find the probability that a random variable x falls between two values a and b:

    • P(a ext{ ≤ } x ext{ ≤ } b)

Empirical Rule (68-95-99.7 Rule)

  • For a normal distribution:

    • Approximately 68.26% of observations fall within one standard deviation from the mean (µ ± 1σ).

    • About 95.44% fall within two standard deviations (µ ± 2σ).

    • Nearly 99.73% fall within three standard deviations (µ ± 3σ).

  • Graph Representation:

    • Illustrates the percentage of observed values within specified intervals around the mean.

Standardization and Z-Scores

  • A z-score quantifies the number of standard deviations a data value x is from the mean m.

  • Standardizing Steps:

    1. When x is above the mean, the z-score is positive.

    2. When x is below the mean, the z-score is negative.

    3. If x is one standard deviation above the mean, z = 1; if two standard deviations above the mean, z = 2.

Finding Normal Curve Areas Using Z-Scores

  • Z-score formula:
    Z = \frac{x - \mu}{\sigma}

  • Example Parameters:

    • Considering intervals with respect to the mean \mu and standard deviations:

    • \mu - 30

    • \mu + 20

    • \mu + \sigma

    • \mu + 2\sigma

  • Standard Normal Curve:

    • A normal curve can be standardized, whereby:

    • The mean is 0 and the standard deviation is 1, allowing for universal application of z-scores.

  • Graphical Reference:

    • Standard normal curve illustration with z-scores marked from -3 to +3 relative to the mean.