Psychological Statistics Review for Exam 1

Lecture 2: Introduction to Statistics

Key Terms to Know

• Population vs. Sample

• Parameter vs. Statistic

• Descriptive vs. Inferential Statistics

• Theory vs. Hypothesis

• Four Scales of Measurement:

  • Nominal

  • Ordinal

  • Interval

  • Ratio

• Characteristics of Each Scale:

  • Comparisons between each type of scale

Scales of Measurement

Lecture 3: Research Methods

Key Concepts

• Correlational vs. Experimental Research: Methodologies to measure relationships between variables

  • Correlation: Measure two variables without manipulation; does not imply causation.

  • Experimental Research: Manipulation of one variable (independent variable) to measure the effect on a dependent variable.

Key Terms: Introduction to Statistics

• Correlational methods of research

• Experimental methods of research

• Independent variable

• Dependent variable

• Control condition

• Experimental condition

• Notation meanings: Σ, X, N, n

• Order of operations (PEMDAS)

• Definitions of population, sample, data, dataset, parameter, statistic, descriptive statistics, inferential statistics, sampling error, theory, hypothesis

• Scales: nominal, ordinal, interval, ratio

Lecture 3 & 4: Frequency Distributions

Overview

• Frequency Distribution: Organizes data by grouping into tables/graphs; a descriptive statistic

  • Frequency Distribution Table: Shows frequency of scores and structured overall data

  • Categories in column X, frequency count in column f

Calculations

• Includes number of scores, sum of scores, proportions, and percentages.

• Percentile/Percentile Rank: Indicates the percentage of scores in the distribution that are equal to or below a given value.

• Graphical Representation: Frequency distribution tables can be represented as histograms.

Frequency Distribution Graph: Histogram

• Example Graph:

  • X Values: (e.g., quiz scores)

  • Frequency Counts: (Display as a bar graph with x-axis for quiz scores and y-axis for frequency)

Lectures 4 & 5: Central Tendency

Definition

• Central Tendency: A measures of central location that identifies a single value representative of the group.

Key Measures

1. Mean:

   • Population Mean (μ): ext{μ} = rac{ ext{Σ}X}{N}

   • Sample Mean (M): ext{M} = rac{ ext{Σ}X}{n}

   • Concept: "Balance point"; represents dividing total equally.

   • Characteristics: The mean can change with any score adjustment, including adding/removing a score or applying a constant.

2. Median:

   • Definition: Midpoint when scores are arranged; divides distribution into two equal groups.

3. Mode:

   • Definition: Most frequent score; must exist within the distribution.

   • Distributions: Bimodal and multimodal distributions allow for the existence of multiple modes.

Lecture 6: Graphs

Objectives

• Goal of Graphs: Present data clearly without distortion, make large datasets understandable, highlight underlying messages of data to aid interpretation.

• Impact of Poor Graphs: Can obscure true understanding of data.

Types of Graphs

• Pie Charts

• Histograms (for frequency distributions)

• Bar Graphs

• Line Graphs

• Scatter Plots

Lecture 7: Variability

Definition

• Variability: Describes the spread of scores in relation to each other or to the mean.

Measures of Variability

1. Range:

   • Calculated as: ext{Range} = X{max} - X{min}

   • Notes: Effective but imprecise, susceptible to extreme values.

2. Interquartile Range (IQR):

   • Advantages: Excludes extremes; calculated as: ext{IQR} = Q3 - Q1

   • Denotes the middle 50% of data.

3. Standard Deviation:

   • Average distance from the mean; measures dispersion.

   • Calculation of Deviation Scores: For population: (X - μ) and for sample: (X - M) .

   • To calculate average deviation: square deviation scores to eliminate sign reversal and compute variance.

4. Variance:

   • Defined as the mean of squared deviation scores.

   • Population Variance Formula: ext{Population Variance} = rac{ ext{SS}}{N}

   • Here, SS refers to "Sum of Squared Deviations".

Lecture 8: Variance & Standard Deviation

Definitional vs. Computational Formulas

1. Population:

   • Definitional: ext{SS} = ext{∑}(X - μ)^2

   • Computational: ext{SS} = ext{∑}X^2 - rac{( ext{∑}X)^2}{N}

2. Sample:

   • Definitional: ext{SS} = ext{∑}(X - M)^2

   • Computational: ext{SS} = ext{∑}X^2 - rac{( ext{∑}X)^2}{n}

Lecture 9: Variance & Standard Deviation (continued)

Standards for Use

• SS: Acts as the numerator in variance formulas.

• Estimation Adjustments: Variance accounts for sampling error via adjustment to correct underestimation of population variability.

  • Sample variance calculation: ext{Sample Variance} = rac{ ext{SS}}{n-1}

• Relationship: Standard deviation is the square root of variance.

How Scores Influence Variability

• Changing individual scores affects standard deviation distinctively:

  • Adjusting one score changes variability.

  • Adding a constant to each score does not affect variability, but multiplying does.

Lecture 10-12: Z-Scores

Concept

• Z-Score: Metric that describes the exact location of a score relative to the mean regarding standard deviations.

  • Components:

    • Sign (+/-): Indicating position above/below the mean.

    • Number: Distance from the mean expressed in standard deviations.

Formulas

1. Score to Z-Score Transformation:
   z = rac{X - μ}{σ}

2. Z-Score to Original Score Transformation: X = μ + zσ

   • For samples, adapt by using Sample Mean (M) and Sample Standard Deviation (s).

Z-Score Distribution

• Properties: Z-Score distribution maintains the shape of the original data, with mean = 0 and standard deviation = 1.

• Purpose: Allows for comparison across different distributions.

Lecture 12: Z-Scores (continued)

Transformation Steps

1. Transform original scores to z-scores (X to z).

2. Convert z-scores back to new X-values (z to X).

• Note: The relative position for both transformations remains unchanged.

Questions?

• Reminder: Don’t forget your calculator and a pen!

• Lab: Review worksheet for additional practice and applications.