Statistics Study Notes

Introduction

  • This chapter discusses statistics and its importance, covering various foundational topics essential for understanding statistical concepts.
    • Sections include:
    • A. What are Statistics?
    • B. Importance of Statistics
    • C. Descriptive Statistics
    • D. Inferential Statistics
    • E. Variables
    • F. Percentiles
    • G. Measurement
    • H. Levels of Measurement
    • I. Distributions
    • J. Summation Notation
    • K. Linear Transformations
    • L. Logarithms
    • M. Exercises

Variables by Heidi Ziemer

  • Prerequisites: None
  • Learning Objectives:
    1. Define and distinguish between independent and dependent variables.
    2. Define and distinguish between discrete and continuous variables.
    3. Define and distinguish between qualitative and quantitative variables.

Independent and Dependent Variables

  • Definition of Variables:
    • Properties or characteristics that can vary in value; unlike constants such as π.
  • Independent Variables: Manipulated by the researcher to observe effects.
  • Dependent Variables: The outcomes measured to see the effects of changes in the independent variable.

Examples:

  1. Blueberries and Aging:

    • Independent Variable: Type of dietary supplement (none, blueberry, strawberry, spinach).
    • Dependent Variables: Memory test and motor skills test.

  2. Beta-Carotene and Cancer:

    • Independent Variable: Supplements (beta-carotene vs. placebo).
    • Dependent Variable: Occurrence of cancer rates.

  3. Brightness of Brake Lights:

    • Independent Variable: Brightness of brake lights.
    • Dependent Variable: Time to react and hit brakes.

Levels of an Independent Variable

  • Treatment comparisons can have two levels (experimental and control) or more depending on the number of experimental conditions; e.g., comparing five types of diets yields 5 levels.

Qualitative and Quantitative Variables

  • Qualitative Variables: Attribute-based (e.g., hair color, gender); no numerical ordering.
  • Quantitative Variables: Numeric-based (e.g., height, weight).
    • Example in diets: type of supplement is qualitative; memory test scores are quantitative.

Discrete and Continuous Variables

  • Discrete Variables: Countable values; e.g., number of children (cannot have fractions).
  • Continuous Variables: Measured on a continuum; e.g., time to respond to a question (can have fractions).

Percentiles by David Lane

  • Prerequisites: None
  • Learning Objectives:
    1. Define percentiles.
    2. Use formulas for computing percentiles.

Definition of Percentiles

  • A score that ranks another score relative to a distribution;
  • Example: A score at the 65th percentile means 65% of scores are lower.

Different Definitions of Percentiles

  1. Definition 1: Lowest score greater than 65% of scores.
  2. Definition 2: Smallest score that is greater than or equal to 65% of scores.
  3. Weighted Average Definition: Combines results from Definitions 1 and 2 while handling rounding issues elegantly.

Example Calculations:

  1. Calculating the 25th Percentile:
    • Formula to find rank (R) of Pth percentile: R = \frac{P}{100} \times (N + 1)
    • If R is an integer, the percentile is the corresponding ranked score. If not, interpolate between scores.
    • Interpolation Formula Example: For 2nd score at rank 2 and 3rd at rank 3, interpolate using fractional rank.

Distributions by David M. Lane and Heidi Ziemer

  • Prerequisites: Chapter 1: Variables
  • Learning Objectives:
    1. Define “distribution.”
    2. Interpret frequency distributions.
    3. Distinguish between frequency and probability distributions.
    4. Construct a grouped frequency distribution for continuous variables.
    5. Identify skewness of distributions.
    6. Identify bimodal, leptokurtic, and platykurtic distributions.

Distributions of Discrete Variables

  • Example: Counting M&M colors leads to a frequency table depicting their distributions (counts of each color).

Continuous Variables

  • Use grouped frequency distributions for continuous data, e.g., response times in milliseconds.
  • Group scores within intervals to summarize data effectively and construct histograms.

Probability Densities

  • Continuous distributions represented visually via probability density graphs; normally distributed (bell-shaped) curves.

Key Points:

  • The area under the curve = 1.
  • The chance of obtaining an exact measure is zero.
  • Probabilities are represented by areas under the curve between two points.

Shapes of Distributions

  1. Positive Skew: Longer tail on the right, mean > median.
  2. Negative Skew: Longer tail on the left, mean < median.
  3. Bimodal Distribution: Two peaks.
  4. Leptokurtic: Taller peak, fatter tails.
  5. Platykurtic: Flatter peak, thinner tails.

Summation Notation by David M. Lane

  • Learning Objectives:
    1. Use summation notation for sums of all numbers.
    2. Use notation for subsets of numbers and sum of squares.

Summation Basics

  • Greek letter \Sigma signifies summation; example for grapes:
  • \Sigma{i=1}^{n} Xi = X1 + X2 + \ldots + X_n.
  • Notation for summing squares exemplified and requires careful interpretation (difference between total sum squaring and squaring totals).

Exercises

  • Practice Problems for reinforcement, covering topics from chapters:
    1. Inferential vs. descriptive statistics application.
    2. Identifying independent/dependent variables.
    3. Interpreting percentiles and computational exercises.