W10 IT Normal Distribution

Normal Distributions

This pdf is the most popular distribution for continuous random variables

Normal Distributions

1. Moivre

2. Laplace

• First described de  1.___ in 1733

• Elaborated in 1812 by 2. ___

• Describes some natural phenomena

• More importantly, describes sampling characteristics of totals and means

• The normal distribution (also called the Gaussian distribution) is a continuous probability distribution that describes how data values are distributed around the mean.

• It’s one of the most important distributions in statistics because many natural and social phenomena (like test scores, heights, or measurement errors) tend to follow it.

Normal Probability Density Function

• Recall: continuous random variables are described with probability density function (pdfs) curves

•  Normal pdfs are recognized by their typical bell-shape

• Figure: Age distribution of a pediatric population with overlying Normal pdf

Area Under the Curve 

• pdfs should be viewed almost like a histogram

• Top Figure: The darker bars of the histogram

• Bottom Figure: shaded area under the curve (AUC)

μ - expected value (mean “mu”) - μ controls location

σ - standard deviation (sigma) - σ controls spread

Normal pdfs have two parameters

Mean and Standard Deviation of Normal Density

The graph is a bell-shaped and symmetrical curve. The mean, median, and mode are all equal and located at the center.

Standard Deviation

• Points of inflections one σ below and above μ

• Practice sketching Normal curves

• Feel inflection points (where slopes change)

• Label horizontal axis with σ landmarks

• The mean and standard deviation from

the pdf (denoted μ and σ) are parameters

• The mean and standard deviation from

a sample (“xbar” and s) are statistics

• Statistics and parameters are related,

  but are not the same thing!

Two types of means and standard deviations

• 68% of the AUC within ±1σ of μ

• 95% of the AUC within ±2σ of μ

• 99.7% of the AUC within ±3σ of μ

Rule for Normal Distributions

Because the Normal

curve is symmetrical

and the total AUC is

exactly 1...

Symmetry in the Tails

1. Bell-shaped and symmetrical

2. Mean = Median = Mode

3. Total area = 1 (or 100%)

4. Approaches but never touches the

   x-axis

5. Spread depends on standard

   deviation (σ)

6. Empirical Rule (68–95–99.7 Rule)

Properties of a Normal Curve

1, The left and right sides of the curve are

mirror images.

All three measures of central tendency

are equal.

The entire area under the curve

represents all possible outcomes.

The curve extends infinitely in both

directions.

Larger σ = wider curve; smaller σ =

narrower curve.

About 68% of data fall within 1σ of the

mean, 95% within 2σ, and 99.7% within

3σ.

The area under the curve represents probability.

• Since the total area = 1 (or 100%), each portion of the curve

corresponds to a specific probability range.

•  To find probabilities, we convert raw scores (x) to z-scores using:

• Then, use a Z-table (or calculator) to find the area/probability.

• Where: x = a given value of a particular variable.