Descriptive Statistics Overview

Descriptive Statistics

• Procedures that summarize, organize, and clarify data sets, commonly presented in tables, graphs, or as summary statistics.
• Two primary purposes: to highlight observable patterns and to maintain conciseness.

Measures of Central Tendency

• Statistical measures aimed at identifying a central or representative score within a distribution.
• Types include:
  • Mean: Average computed by summing all scores and dividing by the number of scores.
  • Median: The middle value when data is ordered numerically.
  • Mode: The most frequently occurring value in a data set.
• Normal Distribution: A theoretical distribution where the mean, median, and mode are equal and symmetrically aligned.

Z-Scores

• Standardized scores indicate how far an individual value deviates from the mean, expressed in units of standard deviation.
• Positive Z-scores are above the mean; negative Z-scores are below.
• Z-scores help describe the position of data points in a distribution without altering the distribution's shape.

The Empirical Rule

• About 68% of data values fall within 1 standard deviation of the mean.
• About 95% fall within 2 standard deviations.
• About 99.7% fall within 3 standard deviations; known as the "68-95-99.7" rule.

Outliers

• Extreme values that are significantly different from other observations in a dataset.
• Outliers can substantially affect statistical measures, particularly the mean, making it less representative of the data set.

Measures of Variability

• Indicates the spread or dispersion of scores in a dataset.
• Key measures include:
  • Range: Difference between the highest and lowest scores.
  • Sample Variance: Average of the squared deviations from the sample mean.
  • Standard Deviation: Average distance scores deviate from the mean, derived from the square root of the sample variance.
• In a normal distribution, most scores fall within one standard deviation of the mean, and nearly all within three.

Types of Variables

• Continuous Variable: Can take any value along a continuum, allowing measurement in fractions (e.g., race times).
• Discrete Variable: Limited to whole units or categories with no intermediate values (e.g., number of students).

Scales of Measurement

• Framework defined by S.S. Stevens in the 1940s to describe how numbers can be used in different contexts.
• The four scales are:
  • Nominal: Assigns numbers to categorize or label subjects without indicating quantitative value.
  • Ordinal: Provides a rank order with meaningful relationships, but differences between ranks may not be uniform.
  • Interval: Involves ordered values with equal spacing but no true zero.
  • Ratio: Similar to interval but includes a meaningful zero point, allowing for interpretation of ratios.