Normal Curve and Z Scores

Probability

• Definition: Probability refers to the likelihood that an outcome will occur.

• Symbolization: It is symbolized as "P".

• Use: It is used to predict any random event, as opposed to a fixed event.

Sample Space

• Definition: The sample space is the number of total possible outcomes in a given scenario.

• Function of interest: Denoted as f(x), which expresses how often an outcome of interest occurs.

Steps to Calculate Probability

1. Find the sample space.

2. Find f(x).

3. Calculate Probability:

   • The formula for probability is given by:
     $P(x) = \frac{f(x)}{\text{sample space}}$

Example 1: Rock-Paper-Scissors

• Analysis:

  • f(x) = 1 (only one winning outcome)

  • Sample space = 3 (rock, paper, scissors)

• Calculation:
  $P(x) = \frac{1}{3} = 0.33 = 33\%$

Example 2: Drawing 2's from a Deck of Cards

• Sample Space:

  • Total cards = 52

• Function of interest:

  • f(x) = 4 (four 2's in a deck)

• Calculation:
  $P(x) = \frac{4}{52} = 0.076 = 0.08 = 8\%$

Characteristics of Probability

1. Probability values vary between 0 and 1.

   • Represents fraction, decimal, percent, or proportion, and values must vary within the range of 0 to 1.

2. Normal Distribution (ND)

   • Behavioral data that researchers measure often tend to approximate a normal distribution.

   • It is characterized by having scores that are closer to the mean.

Characteristics of Normal Distribution

1. Mathematically defined: Normal distribution is a mathematically defined concept.

2. Theoretical Nature: Normal distribution serves as a theoretical model.

3. Behavioral Approximation: Often, behavioral data approximates a normal distribution.

4. Central Tendency Measures: The mean, median, and mode are all located at the 50th percentile.

5. Symmetry: Normal distribution is symmetrical, meaning values are evenly distributed around the mean.

6. Mean Values: The mean can equal any real number: ( -\infty < M < +\infty ).

7. Standard Deviation Values: Standard deviation can equal any positive value; data can vary (SD > 0).

8. Area Under Curve: The total area under the normal distribution curve equals 1, and it can never be negative. This area is used to determine probabilities for normally distributed data.

9. Asymptotic Tails: The tails of a normal distribution are asymptotic, meaning they approach the x-axis but never actually touch it.

Research Focus

• Use of the term "normal" in research often relates to the statistical norm, helping researchers understand typical behavior within a population or dataset.